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Matching statistic: St000076
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St000076: Alternating sign matrices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> 0
[[1,0],[0,1]]
=> 0
[[0,1],[1,0]]
=> 1
[[1,0,0],[0,1,0],[0,0,1]]
=> 0
[[0,1,0],[1,0,0],[0,0,1]]
=> 1
[[1,0,0],[0,0,1],[0,1,0]]
=> 1
[[0,1,0],[1,-1,1],[0,1,0]]
=> 2
[[0,0,1],[1,0,0],[0,1,0]]
=> 3
[[0,1,0],[0,0,1],[1,0,0]]
=> 3
[[0,0,1],[0,1,0],[1,0,0]]
=> 4
[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> 0
[[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> 1
[[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> 1
[[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
=> 2
[[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> 3
[[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]]
=> 3
[[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> 4
[[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> 1
[[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> 2
[[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> 2
[[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> 3
[[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]]
=> 4
[[0,1,0,0],[0,0,1,0],[1,0,-1,1],[0,0,1,0]]
=> 4
[[0,0,1,0],[0,1,0,0],[1,0,-1,1],[0,0,1,0]]
=> 5
[[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> 3
[[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
=> 4
[[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
=> 5
[[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> 6
[[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
=> 5
[[0,0,1,0],[0,1,-1,1],[1,0,0,0],[0,0,1,0]]
=> 6
[[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
=> 7
[[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]]
=> 3
[[0,1,0,0],[1,-1,1,0],[0,0,0,1],[0,1,0,0]]
=> 4
[[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
=> 5
[[0,1,0,0],[0,0,1,0],[1,-1,0,1],[0,1,0,0]]
=> 5
[[0,0,1,0],[0,1,0,0],[1,-1,0,1],[0,1,0,0]]
=> 6
[[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> 4
[[0,1,0,0],[1,-1,0,1],[0,0,1,0],[0,1,0,0]]
=> 5
[[0,0,1,0],[1,0,-1,1],[0,0,1,0],[0,1,0,0]]
=> 6
[[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
=> 7
[[0,1,0,0],[0,0,0,1],[1,-1,1,0],[0,1,0,0]]
=> 6
[[0,0,1,0],[0,1,-1,1],[1,-1,1,0],[0,1,0,0]]
=> 7
[[0,0,0,1],[0,1,0,0],[1,-1,1,0],[0,1,0,0]]
=> 8
[[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> 8
[[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
=> 9
[[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> 6
[[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]]
=> 7
[[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]]
=> 7
[[0,0,1,0],[0,1,-1,1],[0,0,1,0],[1,0,0,0]]
=> 8
[[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
=> 9
Description
The rank of the alternating sign matrix in the alternating sign matrix poset.
This rank is the sum of the entries of the monotone triangle minus $\binom{n+2}{3}$, which is the smallest sum of the entries in the set of all monotone triangles with bottom row $1\dots n$.
Alternatively, $rank(A)=\frac{1}{2} \sum_{i,j=1}^n (i-j)^2 a_{ij}$, see [3, thm.5.1].
Matching statistic: St000380
Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000380: Integer partitions ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 16%
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000380: Integer partitions ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 16%
Values
[[1]]
=> [[1]]
=> ([],1)
=> [1]
=> 2 = 0 + 2
[[1,0],[0,1]]
=> [[1,1],[2]]
=> ([],1)
=> [1]
=> 2 = 0 + 2
[[0,1],[1,0]]
=> [[1,2],[2]]
=> ([(0,1)],2)
=> [2]
=> 3 = 1 + 2
[[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ([],1)
=> [1]
=> 2 = 0 + 2
[[0,1,0],[1,0,0],[0,0,1]]
=> [[1,1,2],[2,2],[3]]
=> ([(0,1)],2)
=> [2]
=> 3 = 1 + 2
[[1,0,0],[0,0,1],[0,1,0]]
=> [[1,1,1],[2,3],[3]]
=> ([(0,1)],2)
=> [2]
=> 3 = 1 + 2
[[0,1,0],[1,-1,1],[0,1,0]]
=> [[1,1,2],[2,3],[3]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 4 = 2 + 2
[[0,0,1],[1,0,0],[0,1,0]]
=> [[1,1,3],[2,3],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 5 = 3 + 2
[[0,1,0],[0,0,1],[1,0,0]]
=> [[1,2,2],[2,3],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 5 = 3 + 2
[[0,0,1],[0,1,0],[1,0,0]]
=> [[1,2,3],[2,3],[3]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> [5,3]
=> 6 = 4 + 2
[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,2],[3,3],[4]]
=> ([],1)
=> [1]
=> 2 = 0 + 2
[[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,2],[2,2,2],[3,3],[4]]
=> ([(0,1)],2)
=> [2]
=> 3 = 1 + 2
[[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,3],[3,3],[4]]
=> ([(0,1)],2)
=> [2]
=> 3 = 1 + 2
[[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
=> [[1,1,1,2],[2,2,3],[3,3],[4]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 4 = 2 + 2
[[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> [[1,1,1,3],[2,2,3],[3,3],[4]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 5 = 3 + 2
[[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]]
=> [[1,1,2,2],[2,2,3],[3,3],[4]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 5 = 3 + 2
[[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> [[1,1,2,3],[2,2,3],[3,3],[4]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> [5,3]
=> 6 = 4 + 2
[[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [[1,1,1,1],[2,2,2],[3,4],[4]]
=> ([(0,1)],2)
=> [2]
=> 3 = 1 + 2
[[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> [[1,1,1,2],[2,2,2],[3,4],[4]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> [3,1]
=> 4 = 2 + 2
[[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> [[1,1,1,1],[2,2,3],[3,4],[4]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 4 = 2 + 2
[[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> [[1,1,1,2],[2,2,3],[3,4],[4]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 5 = 3 + 2
[[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]]
=> [[1,1,1,3],[2,2,3],[3,4],[4]]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> [5,1]
=> 6 = 4 + 2
[[0,1,0,0],[0,0,1,0],[1,0,-1,1],[0,0,1,0]]
=> [[1,1,2,2],[2,2,3],[3,4],[4]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 6 = 4 + 2
[[0,0,1,0],[0,1,0,0],[1,0,-1,1],[0,0,1,0]]
=> [[1,1,2,3],[2,2,3],[3,4],[4]]
=> ([(0,3),(0,7),(1,8),(2,8),(3,9),(4,5),(5,1),(6,2),(7,4),(7,9),(9,6)],10)
=> [6,4]
=> 7 = 5 + 2
[[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> [[1,1,1,1],[2,2,4],[3,4],[4]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 5 = 3 + 2
[[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
=> [[1,1,1,2],[2,2,4],[3,4],[4]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [5,2]
=> 6 = 4 + 2
[[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
=> [[1,1,1,3],[2,2,4],[3,4],[4]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> [6,3]
=> 7 = 5 + 2
[[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [[1,1,1,4],[2,2,4],[3,4],[4]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> [7,1]
=> 8 = 6 + 2
[[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
=> [[1,1,2,2],[2,2,4],[3,4],[4]]
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> [6,3,1]
=> 7 = 5 + 2
[[0,0,1,0],[0,1,-1,1],[1,0,0,0],[0,0,1,0]]
=> [[1,1,2,3],[2,2,4],[3,4],[4]]
=> ([(0,6),(1,9),(1,10),(2,8),(3,7),(4,3),(4,12),(5,2),(5,12),(6,4),(6,5),(7,9),(7,11),(8,10),(8,11),(9,13),(10,13),(11,13),(12,1),(12,7),(12,8)],14)
=> [7,4,3]
=> 8 = 6 + 2
[[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
=> [[1,1,2,4],[2,2,4],[3,4],[4]]
=> ([(0,7),(0,8),(1,12),(2,11),(3,10),(4,10),(4,11),(5,3),(6,1),(6,13),(7,9),(8,5),(9,2),(9,4),(10,14),(11,6),(11,14),(13,12),(14,13)],15)
=> [8,6,1]
=> ? = 7 + 2
[[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]]
=> [[1,1,1,1],[2,3,3],[3,4],[4]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 5 = 3 + 2
[[0,1,0,0],[1,-1,1,0],[0,0,0,1],[0,1,0,0]]
=> [[1,1,1,2],[2,3,3],[3,4],[4]]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> [5,1]
=> 6 = 4 + 2
[[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
=> [[1,1,1,3],[2,3,3],[3,4],[4]]
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> [6,1]
=> 7 = 5 + 2
[[0,1,0,0],[0,0,1,0],[1,-1,0,1],[0,1,0,0]]
=> [[1,1,2,2],[2,3,3],[3,4],[4]]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> [6,1]
=> 7 = 5 + 2
[[0,0,1,0],[0,1,0,0],[1,-1,0,1],[0,1,0,0]]
=> [[1,1,2,3],[2,3,3],[3,4],[4]]
=> ([(0,3),(0,8),(1,10),(2,9),(3,11),(4,2),(5,4),(6,7),(7,1),(7,9),(8,5),(8,11),(9,10),(11,6)],12)
=> [7,5]
=> 8 = 6 + 2
[[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> [[1,1,1,1],[2,3,4],[3,4],[4]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> [5,3]
=> 6 = 4 + 2
[[0,1,0,0],[1,-1,0,1],[0,0,1,0],[0,1,0,0]]
=> [[1,1,1,2],[2,3,4],[3,4],[4]]
=> ([(0,5),(0,6),(1,8),(2,9),(3,8),(3,9),(4,1),(5,4),(6,7),(7,2),(7,3),(8,10),(9,10)],11)
=> [6,4,1]
=> 7 = 5 + 2
[[0,0,1,0],[1,0,-1,1],[0,0,1,0],[0,1,0,0]]
=> [[1,1,1,3],[2,3,4],[3,4],[4]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> [7,3]
=> 8 = 6 + 2
[[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
=> [[1,1,1,4],[2,3,4],[3,4],[4]]
=> ([(0,5),(0,10),(1,16),(2,15),(3,14),(4,13),(5,12),(6,2),(6,13),(7,4),(7,14),(8,1),(9,6),(10,11),(10,12),(11,3),(11,7),(12,9),(13,15),(14,8),(15,16)],17)
=> [8,6,3]
=> 9 = 7 + 2
[[0,1,0,0],[0,0,0,1],[1,-1,1,0],[0,1,0,0]]
=> [[1,1,2,2],[2,3,4],[3,4],[4]]
=> ([(0,6),(0,7),(1,9),(2,12),(3,9),(3,12),(4,10),(5,1),(6,5),(7,8),(8,2),(8,3),(9,11),(11,10),(12,4),(12,11)],13)
=> [7,5,1]
=> 8 = 6 + 2
[[0,0,1,0],[0,1,-1,1],[1,-1,1,0],[0,1,0,0]]
=> [[1,1,2,3],[2,3,4],[3,4],[4]]
=> ([(0,9),(0,11),(1,18),(2,17),(3,19),(4,13),(4,19),(5,12),(5,13),(6,16),(7,14),(8,5),(8,18),(9,10),(10,3),(10,4),(11,1),(11,8),(12,17),(13,15),(15,16),(16,14),(17,7),(18,2),(18,12),(19,6),(19,15)],20)
=> [8,6,4,2]
=> 9 = 7 + 2
[[0,0,0,1],[0,1,0,0],[1,-1,1,0],[0,1,0,0]]
=> [[1,1,2,4],[2,3,4],[3,4],[4]]
=> ([(0,12),(0,13),(1,16),(2,15),(3,23),(4,19),(5,17),(5,20),(6,4),(7,5),(7,15),(7,16),(8,10),(9,7),(10,2),(11,1),(11,23),(12,8),(12,22),(13,14),(13,22),(14,3),(14,11),(15,17),(15,21),(16,20),(16,21),(17,24),(19,18),(20,19),(20,24),(21,24),(22,9),(23,6),(24,18)],25)
=> [9,7,5,4]
=> ? = 8 + 2
[[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> [[1,1,3,3],[2,3,4],[3,4],[4]]
=> ([(0,8),(2,11),(2,12),(3,10),(4,9),(5,4),(5,14),(6,3),(6,14),(7,1),(8,5),(8,6),(9,11),(9,13),(10,12),(10,13),(11,15),(12,15),(13,15),(14,2),(14,9),(14,10),(15,7)],16)
=> [9,4,3]
=> ? = 8 + 2
[[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
=> [[1,1,3,4],[2,3,4],[3,4],[4]]
=> ([(0,13),(0,15),(1,17),(2,16),(2,17),(3,19),(4,16),(4,18),(5,22),(6,21),(7,20),(8,23),(8,29),(9,4),(9,28),(10,3),(10,28),(11,6),(12,7),(12,24),(13,14),(14,1),(14,2),(15,9),(15,10),(16,25),(17,12),(17,25),(18,26),(18,29),(19,23),(19,26),(20,27),(22,27),(23,30),(24,20),(24,22),(25,24),(26,30),(27,21),(28,8),(28,18),(28,19),(29,5),(29,30),(30,11)],31)
=> [10,8,6,4,3]
=> ? = 9 + 2
[[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> [[1,2,2,2],[2,3,3],[3,4],[4]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> [7,1]
=> 8 = 6 + 2
[[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]]
=> [[1,2,2,3],[2,3,3],[3,4],[4]]
=> ([(0,5),(0,10),(1,16),(2,15),(3,14),(4,13),(5,12),(6,2),(6,13),(7,4),(7,14),(8,1),(9,6),(10,11),(10,12),(11,3),(11,7),(12,9),(13,15),(14,8),(15,16)],17)
=> [8,6,3]
=> 9 = 7 + 2
[[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]]
=> [[1,2,2,2],[2,3,4],[3,4],[4]]
=> ([(0,7),(0,8),(1,12),(2,11),(3,10),(4,10),(4,11),(5,3),(6,1),(6,13),(7,9),(8,5),(9,2),(9,4),(10,14),(11,6),(11,14),(13,12),(14,13)],15)
=> [8,6,1]
=> ? = 7 + 2
[[0,0,1,0],[0,1,-1,1],[0,0,1,0],[1,0,0,0]]
=> [[1,2,2,3],[2,3,4],[3,4],[4]]
=> ([(0,10),(0,12),(1,16),(2,17),(3,21),(4,22),(5,14),(6,13),(6,14),(7,13),(7,15),(8,7),(8,21),(9,2),(9,18),(10,11),(11,5),(11,6),(12,3),(12,8),(13,19),(14,9),(14,19),(15,22),(16,20),(17,20),(18,16),(18,17),(19,18),(21,4),(21,15),(22,1)],23)
=> [9,7,5,2]
=> ? = 8 + 2
[[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
=> [[1,2,2,4],[2,3,4],[3,4],[4]]
=> ([(0,14),(0,15),(1,19),(2,18),(3,29),(4,30),(5,22),(6,23),(7,24),(7,25),(8,9),(9,7),(9,18),(9,19),(10,5),(11,6),(12,2),(12,29),(13,1),(13,30),(14,16),(14,28),(15,17),(15,28),(16,3),(16,12),(17,4),(17,13),(18,24),(18,27),(19,25),(19,27),(20,26),(21,26),(22,20),(23,21),(24,22),(24,31),(25,23),(25,31),(27,31),(28,8),(29,10),(30,11),(31,20),(31,21)],32)
=> [10,8,6,4,4]
=> ? = 9 + 2
[[0,0,1,0],[0,0,0,1],[0,1,0,0],[1,0,0,0]]
=> [[1,2,3,3],[2,3,4],[3,4],[4]]
=> ([(0,13),(0,15),(1,17),(2,16),(2,17),(3,19),(4,16),(4,18),(5,22),(6,21),(7,20),(8,23),(8,29),(9,4),(9,28),(10,3),(10,28),(11,6),(12,7),(12,24),(13,14),(14,1),(14,2),(15,9),(15,10),(16,25),(17,12),(17,25),(18,26),(18,29),(19,23),(19,26),(20,27),(22,27),(23,30),(24,20),(24,22),(25,24),(26,30),(27,21),(28,8),(28,18),(28,19),(29,5),(29,30),(30,11)],31)
=> [10,8,6,4,3]
=> ? = 9 + 2
[[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> [[1,2,3,4],[2,3,4],[3,4],[4]]
=> ([(0,5),(0,16),(0,17),(1,23),(2,19),(3,11),(3,21),(4,10),(4,20),(5,12),(5,13),(6,50),(7,51),(8,24),(8,58),(9,25),(9,59),(10,14),(10,52),(11,15),(11,53),(12,26),(12,60),(13,27),(13,60),(14,54),(15,55),(16,4),(16,29),(17,3),(17,29),(18,48),(18,49),(19,32),(19,33),(20,46),(20,52),(21,47),(21,53),(22,34),(22,35),(23,18),(23,54),(23,55),(24,40),(24,42),(25,41),(25,43),(26,46),(26,56),(27,47),(27,57),(28,63),(29,1),(30,62),(31,61),(32,61),(33,61),(34,6),(34,62),(35,7),(35,62),(36,58),(37,59),(38,32),(39,33),(40,44),(41,45),(42,38),(43,39),(44,31),(45,31),(46,36),(47,37),(48,40),(48,63),(49,41),(49,63),(50,38),(51,39),(52,8),(52,36),(53,9),(53,37),(54,28),(54,48),(55,28),(55,49),(56,30),(56,34),(57,30),(57,35),(58,42),(58,50),(59,43),(59,51),(60,22),(60,56),(60,57),(62,2),(63,44),(63,45)],64)
=> [11,9,9,7,7,5,5,5,3,3]
=> ? = 10 + 2
[[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,2,2],[3,3,3],[4,4],[5]]
=> ([],1)
=> [1]
=> 2 = 0 + 2
[[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,2],[2,2,2,2],[3,3,3],[4,4],[5]]
=> ([(0,1)],2)
=> [2]
=> 3 = 1 + 2
[[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,2,3],[3,3,3],[4,4],[5]]
=> ([(0,1)],2)
=> [2]
=> 3 = 1 + 2
[[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,2],[2,2,2,3],[3,3,3],[4,4],[5]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 4 = 2 + 2
[[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,3],[2,2,2,3],[3,3,3],[4,4],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 5 = 3 + 2
[[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,2,2],[2,2,2,3],[3,3,3],[4,4],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 5 = 3 + 2
[[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,2,3],[2,2,2,3],[3,3,3],[4,4],[5]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> [5,3]
=> 6 = 4 + 2
[[0,1,0,0,0],[0,0,1,0,0],[1,0,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,2,2],[2,2,2,3],[3,3,4],[4,4],[5]]
=> ?
=> ?
=> ? = 4 + 2
[[0,0,1,0,0],[0,1,-1,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,2,3],[2,2,2,4],[3,3,4],[4,4],[5]]
=> ?
=> ?
=> ? = 6 + 2
[[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,2,4],[2,2,2,4],[3,3,4],[4,4],[5]]
=> ([(0,7),(0,8),(1,12),(2,11),(3,10),(4,10),(4,11),(5,3),(6,1),(6,13),(7,9),(8,5),(9,2),(9,4),(10,14),(11,6),(11,14),(13,12),(14,13)],15)
=> [8,6,1]
=> ? = 7 + 2
[[0,1,0,0,0],[1,-1,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,2],[2,2,3,3],[3,3,4],[4,4],[5]]
=> ?
=> ?
=> ? = 4 + 2
[[0,1,0,0,0],[0,0,1,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,2,2],[2,2,3,3],[3,3,4],[4,4],[5]]
=> ?
=> ?
=> ? = 5 + 2
[[0,0,1,0,0],[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,2,3],[2,2,3,3],[3,3,4],[4,4],[5]]
=> ?
=> ?
=> ? = 6 + 2
[[0,0,1,0,0],[1,0,-1,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,3],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ?
=> ?
=> ? = 6 + 2
[[0,1,0,0,0],[0,0,0,1,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,2,2],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ?
=> ?
=> ? = 6 + 2
[[0,0,0,1,0],[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,2,4],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ?
=> ?
=> ? = 8 + 2
[[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,3,3],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ([(0,8),(2,11),(2,12),(3,10),(4,9),(5,4),(5,14),(6,3),(6,14),(7,1),(8,5),(8,6),(9,11),(9,13),(10,12),(10,13),(11,15),(12,15),(13,15),(14,2),(14,9),(14,10),(15,7)],16)
=> [9,4,3]
=> ? = 8 + 2
[[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,3,4],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ([(0,13),(0,15),(1,17),(2,16),(2,17),(3,19),(4,16),(4,18),(5,22),(6,21),(7,20),(8,23),(8,29),(9,4),(9,28),(10,3),(10,28),(11,6),(12,7),(12,24),(13,14),(14,1),(14,2),(15,9),(15,10),(16,25),(17,12),(17,25),(18,26),(18,29),(19,23),(19,26),(20,27),(22,27),(23,30),(24,20),(24,22),(25,24),(26,30),(27,21),(28,8),(28,18),(28,19),(29,5),(29,30),(30,11)],31)
=> [10,8,6,4,3]
=> ? = 9 + 2
[[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1]]
=> [[1,1,2,2,2],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ([(0,7),(0,8),(1,12),(2,11),(3,10),(4,10),(4,11),(5,3),(6,1),(6,13),(7,9),(8,5),(9,2),(9,4),(10,14),(11,6),(11,14),(13,12),(14,13)],15)
=> [8,6,1]
=> ? = 7 + 2
[[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1]]
=> [[1,1,2,2,3],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ?
=> ?
=> ? = 8 + 2
[[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1]]
=> [[1,1,2,2,4],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ([(0,14),(0,15),(1,19),(2,18),(3,29),(4,30),(5,22),(6,23),(7,24),(7,25),(8,9),(9,7),(9,18),(9,19),(10,5),(11,6),(12,2),(12,29),(13,1),(13,30),(14,16),(14,28),(15,17),(15,28),(16,3),(16,12),(17,4),(17,13),(18,24),(18,27),(19,25),(19,27),(20,26),(21,26),(22,20),(23,21),(24,22),(24,31),(25,23),(25,31),(27,31),(28,8),(29,10),(30,11),(31,20),(31,21)],32)
=> [10,8,6,4,4]
=> ? = 9 + 2
[[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1]]
=> [[1,1,2,3,3],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ([(0,13),(0,15),(1,17),(2,16),(2,17),(3,19),(4,16),(4,18),(5,22),(6,21),(7,20),(8,23),(8,29),(9,4),(9,28),(10,3),(10,28),(11,6),(12,7),(12,24),(13,14),(14,1),(14,2),(15,9),(15,10),(16,25),(17,12),(17,25),(18,26),(18,29),(19,23),(19,26),(20,27),(22,27),(23,30),(24,20),(24,22),(25,24),(26,30),(27,21),(28,8),(28,18),(28,19),(29,5),(29,30),(30,11)],31)
=> [10,8,6,4,3]
=> ? = 9 + 2
[[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1]]
=> [[1,1,2,3,4],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ([(0,5),(0,16),(0,17),(1,23),(2,19),(3,11),(3,21),(4,10),(4,20),(5,12),(5,13),(6,50),(7,51),(8,24),(8,58),(9,25),(9,59),(10,14),(10,52),(11,15),(11,53),(12,26),(12,60),(13,27),(13,60),(14,54),(15,55),(16,4),(16,29),(17,3),(17,29),(18,48),(18,49),(19,32),(19,33),(20,46),(20,52),(21,47),(21,53),(22,34),(22,35),(23,18),(23,54),(23,55),(24,40),(24,42),(25,41),(25,43),(26,46),(26,56),(27,47),(27,57),(28,63),(29,1),(30,62),(31,61),(32,61),(33,61),(34,6),(34,62),(35,7),(35,62),(36,58),(37,59),(38,32),(39,33),(40,44),(41,45),(42,38),(43,39),(44,31),(45,31),(46,36),(47,37),(48,40),(48,63),(49,41),(49,63),(50,38),(51,39),(52,8),(52,36),(53,9),(53,37),(54,28),(54,48),(55,28),(55,49),(56,30),(56,34),(57,30),(57,35),(58,42),(58,50),(59,43),(59,51),(60,22),(60,56),(60,57),(62,2),(63,44),(63,45)],64)
=> [11,9,9,7,7,5,5,5,3,3]
=> ? = 10 + 2
[[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,2,3],[2,2,2,3],[3,3,3],[4,5],[5]]
=> ([(0,3),(0,6),(0,7),(1,8),(1,12),(2,8),(2,11),(3,9),(3,10),(4,2),(4,13),(5,1),(5,14),(6,4),(6,9),(7,5),(7,10),(8,15),(9,13),(10,14),(11,15),(12,15),(13,11),(14,12)],16)
=> ?
=> ? = 5 + 2
[[0,1,0,0,0],[0,0,1,0,0],[1,0,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[1,1,1,2,2],[2,2,2,3],[3,3,4],[4,5],[5]]
=> ?
=> ?
=> ? = 5 + 2
[[0,0,1,0,0],[0,1,0,0,0],[1,0,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[1,1,1,2,3],[2,2,2,3],[3,3,4],[4,5],[5]]
=> ([(0,3),(0,8),(1,10),(2,10),(3,9),(4,6),(5,4),(5,11),(6,2),(7,1),(8,5),(8,9),(9,11),(11,7)],12)
=> ?
=> ? = 6 + 2
[[0,0,1,0,0],[1,0,-1,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[1,1,1,1,3],[2,2,2,4],[3,3,4],[4,5],[5]]
=> ([(0,4),(0,6),(1,10),(2,9),(3,7),(4,8),(5,3),(5,11),(6,1),(6,8),(7,9),(8,5),(8,10),(10,11),(11,2),(11,7)],12)
=> ?
=> ? = 6 + 2
[[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[1,1,1,1,4],[2,2,2,4],[3,3,4],[4,5],[5]]
=> ([(0,4),(0,7),(2,10),(3,9),(4,8),(5,3),(5,11),(6,1),(7,5),(7,8),(8,11),(9,10),(10,6),(11,2),(11,9)],12)
=> ?
=> ? = 7 + 2
[[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[1,1,1,2,2],[2,2,2,4],[3,3,4],[4,5],[5]]
=> ?
=> ?
=> ? = 6 + 2
[[0,0,1,0,0],[0,1,-1,1,0],[1,0,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[1,1,1,2,3],[2,2,2,4],[3,3,4],[4,5],[5]]
=> ?
=> ?
=> ? = 7 + 2
[[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[1,1,1,2,4],[2,2,2,4],[3,3,4],[4,5],[5]]
=> ?
=> ?
=> ? = 8 + 2
[[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [[1,1,1,1,1],[2,2,3,3],[3,3,4],[4,5],[5]]
=> ?
=> ?
=> ? = 4 + 2
[[0,1,0,0,0],[1,-1,1,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [[1,1,1,1,2],[2,2,3,3],[3,3,4],[4,5],[5]]
=> ?
=> ?
=> ? = 5 + 2
[[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [[1,1,1,1,3],[2,2,3,3],[3,3,4],[4,5],[5]]
=> ?
=> ?
=> ? = 6 + 2
[[0,1,0,0,0],[0,0,1,0,0],[1,-1,0,1,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [[1,1,1,2,2],[2,2,3,3],[3,3,4],[4,5],[5]]
=> ?
=> ?
=> ? = 6 + 2
[[0,0,1,0,0],[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [[1,1,1,2,3],[2,2,3,3],[3,3,4],[4,5],[5]]
=> ?
=> ?
=> ? = 7 + 2
[[0,1,0,0,0],[1,-1,0,1,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [[1,1,1,1,2],[2,2,3,4],[3,3,4],[4,5],[5]]
=> ([(0,4),(0,7),(1,9),(2,10),(3,9),(3,10),(4,11),(5,8),(6,1),(7,5),(7,11),(8,2),(8,3),(9,12),(10,12),(11,6)],13)
=> ?
=> ? = 6 + 2
[[0,0,1,0,0],[1,0,-1,1,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [[1,1,1,1,3],[2,2,3,4],[3,3,4],[4,5],[5]]
=> ?
=> ?
=> ? = 7 + 2
[[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [[1,1,1,1,4],[2,2,3,4],[3,3,4],[4,5],[5]]
=> ?
=> ?
=> ? = 8 + 2
[[0,1,0,0,0],[0,0,0,1,0],[1,-1,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [[1,1,1,2,2],[2,2,3,4],[3,3,4],[4,5],[5]]
=> ?
=> ?
=> ? = 7 + 2
[[0,0,1,0,0],[0,1,-1,1,0],[1,-1,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [[1,1,1,2,3],[2,2,3,4],[3,3,4],[4,5],[5]]
=> ([(0,10),(0,11),(1,13),(1,14),(2,21),(3,20),(4,14),(4,20),(5,17),(6,16),(7,18),(8,12),(9,1),(9,22),(10,5),(10,15),(11,8),(11,15),(12,3),(12,4),(13,21),(14,19),(15,9),(15,17),(17,22),(18,16),(19,18),(20,7),(20,19),(21,6),(22,2),(22,13)],23)
=> ?
=> ? = 8 + 2
[[0,0,0,1,0],[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [[1,1,1,2,4],[2,2,3,4],[3,3,4],[4,5],[5]]
=> ?
=> ?
=> ? = 9 + 2
[[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [[1,1,1,3,3],[2,2,3,4],[3,3,4],[4,5],[5]]
=> ?
=> ?
=> ? = 9 + 2
[[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [[1,1,1,3,4],[2,2,3,4],[3,3,4],[4,5],[5]]
=> ?
=> ?
=> ? = 10 + 2
[[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[1,0,0,-1,1],[0,0,0,1,0]]
=> [[1,1,2,2,2],[2,2,3,3],[3,3,4],[4,5],[5]]
=> ?
=> ?
=> ? = 7 + 2
[[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[1,0,0,-1,1],[0,0,0,1,0]]
=> [[1,1,2,2,3],[2,2,3,3],[3,3,4],[4,5],[5]]
=> ?
=> ?
=> ? = 8 + 2
[[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[1,0,0,-1,1],[0,0,0,1,0]]
=> [[1,1,2,2,2],[2,2,3,4],[3,3,4],[4,5],[5]]
=> ?
=> ?
=> ? = 8 + 2
[[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[1,0,0,-1,1],[0,0,0,1,0]]
=> [[1,1,2,2,3],[2,2,3,4],[3,3,4],[4,5],[5]]
=> ?
=> ?
=> ? = 9 + 2
Description
Half of the maximal perimeter of a rectangle fitting into the diagram of an integer partition.
Put differently, this is the smallest number $n$ such that the partition fits into the triangular partition $(n-1,n-2,\dots,1)$.
Matching statistic: St001392
Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St001392: Integer partitions ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 13%
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St001392: Integer partitions ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 13%
Values
[[1]]
=> [[1]]
=> ([],1)
=> [1]
=> 0
[[1,0],[0,1]]
=> [[1,1],[2]]
=> ([],1)
=> [1]
=> 0
[[0,1],[1,0]]
=> [[1,2],[2]]
=> ([(0,1)],2)
=> [2]
=> 1
[[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ([],1)
=> [1]
=> 0
[[0,1,0],[1,0,0],[0,0,1]]
=> [[1,1,2],[2,2],[3]]
=> ([(0,1)],2)
=> [2]
=> 1
[[1,0,0],[0,0,1],[0,1,0]]
=> [[1,1,1],[2,3],[3]]
=> ([(0,1)],2)
=> [2]
=> 1
[[0,1,0],[1,-1,1],[0,1,0]]
=> [[1,1,2],[2,3],[3]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 2
[[0,0,1],[1,0,0],[0,1,0]]
=> [[1,1,3],[2,3],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 3
[[0,1,0],[0,0,1],[1,0,0]]
=> [[1,2,2],[2,3],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 3
[[0,0,1],[0,1,0],[1,0,0]]
=> [[1,2,3],[2,3],[3]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> [5,3]
=> 4
[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,2],[3,3],[4]]
=> ([],1)
=> [1]
=> 0
[[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,2],[2,2,2],[3,3],[4]]
=> ([(0,1)],2)
=> [2]
=> 1
[[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,3],[3,3],[4]]
=> ([(0,1)],2)
=> [2]
=> 1
[[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
=> [[1,1,1,2],[2,2,3],[3,3],[4]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 2
[[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> [[1,1,1,3],[2,2,3],[3,3],[4]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 3
[[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]]
=> [[1,1,2,2],[2,2,3],[3,3],[4]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 3
[[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> [[1,1,2,3],[2,2,3],[3,3],[4]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> [5,3]
=> 4
[[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [[1,1,1,1],[2,2,2],[3,4],[4]]
=> ([(0,1)],2)
=> [2]
=> 1
[[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> [[1,1,1,2],[2,2,2],[3,4],[4]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> [3,1]
=> 2
[[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> [[1,1,1,1],[2,2,3],[3,4],[4]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 2
[[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> [[1,1,1,2],[2,2,3],[3,4],[4]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 3
[[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]]
=> [[1,1,1,3],[2,2,3],[3,4],[4]]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> [5,1]
=> 4
[[0,1,0,0],[0,0,1,0],[1,0,-1,1],[0,0,1,0]]
=> [[1,1,2,2],[2,2,3],[3,4],[4]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 4
[[0,0,1,0],[0,1,0,0],[1,0,-1,1],[0,0,1,0]]
=> [[1,1,2,3],[2,2,3],[3,4],[4]]
=> ([(0,3),(0,7),(1,8),(2,8),(3,9),(4,5),(5,1),(6,2),(7,4),(7,9),(9,6)],10)
=> [6,4]
=> 5
[[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> [[1,1,1,1],[2,2,4],[3,4],[4]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 3
[[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
=> [[1,1,1,2],[2,2,4],[3,4],[4]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [5,2]
=> 4
[[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
=> [[1,1,1,3],[2,2,4],[3,4],[4]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> [6,3]
=> 5
[[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [[1,1,1,4],[2,2,4],[3,4],[4]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> [7,1]
=> 6
[[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
=> [[1,1,2,2],[2,2,4],[3,4],[4]]
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> [6,3,1]
=> 5
[[0,0,1,0],[0,1,-1,1],[1,0,0,0],[0,0,1,0]]
=> [[1,1,2,3],[2,2,4],[3,4],[4]]
=> ([(0,6),(1,9),(1,10),(2,8),(3,7),(4,3),(4,12),(5,2),(5,12),(6,4),(6,5),(7,9),(7,11),(8,10),(8,11),(9,13),(10,13),(11,13),(12,1),(12,7),(12,8)],14)
=> [7,4,3]
=> 6
[[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
=> [[1,1,2,4],[2,2,4],[3,4],[4]]
=> ([(0,7),(0,8),(1,12),(2,11),(3,10),(4,10),(4,11),(5,3),(6,1),(6,13),(7,9),(8,5),(9,2),(9,4),(10,14),(11,6),(11,14),(13,12),(14,13)],15)
=> [8,6,1]
=> 7
[[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]]
=> [[1,1,1,1],[2,3,3],[3,4],[4]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 3
[[0,1,0,0],[1,-1,1,0],[0,0,0,1],[0,1,0,0]]
=> [[1,1,1,2],[2,3,3],[3,4],[4]]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> [5,1]
=> 4
[[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
=> [[1,1,1,3],[2,3,3],[3,4],[4]]
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> [6,1]
=> 5
[[0,1,0,0],[0,0,1,0],[1,-1,0,1],[0,1,0,0]]
=> [[1,1,2,2],[2,3,3],[3,4],[4]]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> [6,1]
=> 5
[[0,0,1,0],[0,1,0,0],[1,-1,0,1],[0,1,0,0]]
=> [[1,1,2,3],[2,3,3],[3,4],[4]]
=> ([(0,3),(0,8),(1,10),(2,9),(3,11),(4,2),(5,4),(6,7),(7,1),(7,9),(8,5),(8,11),(9,10),(11,6)],12)
=> [7,5]
=> 6
[[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> [[1,1,1,1],[2,3,4],[3,4],[4]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> [5,3]
=> 4
[[0,1,0,0],[1,-1,0,1],[0,0,1,0],[0,1,0,0]]
=> [[1,1,1,2],[2,3,4],[3,4],[4]]
=> ([(0,5),(0,6),(1,8),(2,9),(3,8),(3,9),(4,1),(5,4),(6,7),(7,2),(7,3),(8,10),(9,10)],11)
=> [6,4,1]
=> 5
[[0,0,1,0],[1,0,-1,1],[0,0,1,0],[0,1,0,0]]
=> [[1,1,1,3],[2,3,4],[3,4],[4]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> [7,3]
=> 6
[[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
=> [[1,1,1,4],[2,3,4],[3,4],[4]]
=> ([(0,5),(0,10),(1,16),(2,15),(3,14),(4,13),(5,12),(6,2),(6,13),(7,4),(7,14),(8,1),(9,6),(10,11),(10,12),(11,3),(11,7),(12,9),(13,15),(14,8),(15,16)],17)
=> [8,6,3]
=> 7
[[0,1,0,0],[0,0,0,1],[1,-1,1,0],[0,1,0,0]]
=> [[1,1,2,2],[2,3,4],[3,4],[4]]
=> ([(0,6),(0,7),(1,9),(2,12),(3,9),(3,12),(4,10),(5,1),(6,5),(7,8),(8,2),(8,3),(9,11),(11,10),(12,4),(12,11)],13)
=> [7,5,1]
=> 6
[[0,0,1,0],[0,1,-1,1],[1,-1,1,0],[0,1,0,0]]
=> [[1,1,2,3],[2,3,4],[3,4],[4]]
=> ([(0,9),(0,11),(1,18),(2,17),(3,19),(4,13),(4,19),(5,12),(5,13),(6,16),(7,14),(8,5),(8,18),(9,10),(10,3),(10,4),(11,1),(11,8),(12,17),(13,15),(15,16),(16,14),(17,7),(18,2),(18,12),(19,6),(19,15)],20)
=> [8,6,4,2]
=> ? = 7
[[0,0,0,1],[0,1,0,0],[1,-1,1,0],[0,1,0,0]]
=> [[1,1,2,4],[2,3,4],[3,4],[4]]
=> ([(0,12),(0,13),(1,16),(2,15),(3,23),(4,19),(5,17),(5,20),(6,4),(7,5),(7,15),(7,16),(8,10),(9,7),(10,2),(11,1),(11,23),(12,8),(12,22),(13,14),(13,22),(14,3),(14,11),(15,17),(15,21),(16,20),(16,21),(17,24),(19,18),(20,19),(20,24),(21,24),(22,9),(23,6),(24,18)],25)
=> [9,7,5,4]
=> ? = 8
[[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> [[1,1,3,3],[2,3,4],[3,4],[4]]
=> ([(0,8),(2,11),(2,12),(3,10),(4,9),(5,4),(5,14),(6,3),(6,14),(7,1),(8,5),(8,6),(9,11),(9,13),(10,12),(10,13),(11,15),(12,15),(13,15),(14,2),(14,9),(14,10),(15,7)],16)
=> [9,4,3]
=> 8
[[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
=> [[1,1,3,4],[2,3,4],[3,4],[4]]
=> ([(0,13),(0,15),(1,17),(2,16),(2,17),(3,19),(4,16),(4,18),(5,22),(6,21),(7,20),(8,23),(8,29),(9,4),(9,28),(10,3),(10,28),(11,6),(12,7),(12,24),(13,14),(14,1),(14,2),(15,9),(15,10),(16,25),(17,12),(17,25),(18,26),(18,29),(19,23),(19,26),(20,27),(22,27),(23,30),(24,20),(24,22),(25,24),(26,30),(27,21),(28,8),(28,18),(28,19),(29,5),(29,30),(30,11)],31)
=> [10,8,6,4,3]
=> ? = 9
[[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> [[1,2,2,2],[2,3,3],[3,4],[4]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> [7,1]
=> 6
[[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]]
=> [[1,2,2,3],[2,3,3],[3,4],[4]]
=> ([(0,5),(0,10),(1,16),(2,15),(3,14),(4,13),(5,12),(6,2),(6,13),(7,4),(7,14),(8,1),(9,6),(10,11),(10,12),(11,3),(11,7),(12,9),(13,15),(14,8),(15,16)],17)
=> [8,6,3]
=> 7
[[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]]
=> [[1,2,2,2],[2,3,4],[3,4],[4]]
=> ([(0,7),(0,8),(1,12),(2,11),(3,10),(4,10),(4,11),(5,3),(6,1),(6,13),(7,9),(8,5),(9,2),(9,4),(10,14),(11,6),(11,14),(13,12),(14,13)],15)
=> [8,6,1]
=> 7
[[0,0,1,0],[0,1,-1,1],[0,0,1,0],[1,0,0,0]]
=> [[1,2,2,3],[2,3,4],[3,4],[4]]
=> ([(0,10),(0,12),(1,16),(2,17),(3,21),(4,22),(5,14),(6,13),(6,14),(7,13),(7,15),(8,7),(8,21),(9,2),(9,18),(10,11),(11,5),(11,6),(12,3),(12,8),(13,19),(14,9),(14,19),(15,22),(16,20),(17,20),(18,16),(18,17),(19,18),(21,4),(21,15),(22,1)],23)
=> [9,7,5,2]
=> ? = 8
[[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
=> [[1,2,2,4],[2,3,4],[3,4],[4]]
=> ([(0,14),(0,15),(1,19),(2,18),(3,29),(4,30),(5,22),(6,23),(7,24),(7,25),(8,9),(9,7),(9,18),(9,19),(10,5),(11,6),(12,2),(12,29),(13,1),(13,30),(14,16),(14,28),(15,17),(15,28),(16,3),(16,12),(17,4),(17,13),(18,24),(18,27),(19,25),(19,27),(20,26),(21,26),(22,20),(23,21),(24,22),(24,31),(25,23),(25,31),(27,31),(28,8),(29,10),(30,11),(31,20),(31,21)],32)
=> [10,8,6,4,4]
=> ? = 9
[[0,0,1,0],[0,0,0,1],[0,1,0,0],[1,0,0,0]]
=> [[1,2,3,3],[2,3,4],[3,4],[4]]
=> ([(0,13),(0,15),(1,17),(2,16),(2,17),(3,19),(4,16),(4,18),(5,22),(6,21),(7,20),(8,23),(8,29),(9,4),(9,28),(10,3),(10,28),(11,6),(12,7),(12,24),(13,14),(14,1),(14,2),(15,9),(15,10),(16,25),(17,12),(17,25),(18,26),(18,29),(19,23),(19,26),(20,27),(22,27),(23,30),(24,20),(24,22),(25,24),(26,30),(27,21),(28,8),(28,18),(28,19),(29,5),(29,30),(30,11)],31)
=> [10,8,6,4,3]
=> ? = 9
[[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> [[1,2,3,4],[2,3,4],[3,4],[4]]
=> ([(0,5),(0,16),(0,17),(1,23),(2,19),(3,11),(3,21),(4,10),(4,20),(5,12),(5,13),(6,50),(7,51),(8,24),(8,58),(9,25),(9,59),(10,14),(10,52),(11,15),(11,53),(12,26),(12,60),(13,27),(13,60),(14,54),(15,55),(16,4),(16,29),(17,3),(17,29),(18,48),(18,49),(19,32),(19,33),(20,46),(20,52),(21,47),(21,53),(22,34),(22,35),(23,18),(23,54),(23,55),(24,40),(24,42),(25,41),(25,43),(26,46),(26,56),(27,47),(27,57),(28,63),(29,1),(30,62),(31,61),(32,61),(33,61),(34,6),(34,62),(35,7),(35,62),(36,58),(37,59),(38,32),(39,33),(40,44),(41,45),(42,38),(43,39),(44,31),(45,31),(46,36),(47,37),(48,40),(48,63),(49,41),(49,63),(50,38),(51,39),(52,8),(52,36),(53,9),(53,37),(54,28),(54,48),(55,28),(55,49),(56,30),(56,34),(57,30),(57,35),(58,42),(58,50),(59,43),(59,51),(60,22),(60,56),(60,57),(62,2),(63,44),(63,45)],64)
=> [11,9,9,7,7,5,5,5,3,3]
=> ? = 10
[[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,2,2],[3,3,3],[4,4],[5]]
=> ([],1)
=> [1]
=> 0
[[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,2],[2,2,2,2],[3,3,3],[4,4],[5]]
=> ([(0,1)],2)
=> [2]
=> 1
[[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,2,3],[3,3,3],[4,4],[5]]
=> ([(0,1)],2)
=> [2]
=> 1
[[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,2],[2,2,2,3],[3,3,3],[4,4],[5]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 2
[[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,3],[2,2,2,3],[3,3,3],[4,4],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 3
[[0,1,0,0,0],[0,0,1,0,0],[1,0,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,2,2],[2,2,2,3],[3,3,4],[4,4],[5]]
=> ?
=> ?
=> ? = 4
[[0,0,1,0,0],[0,1,-1,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,2,3],[2,2,2,4],[3,3,4],[4,4],[5]]
=> ?
=> ?
=> ? = 6
[[0,1,0,0,0],[1,-1,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,2],[2,2,3,3],[3,3,4],[4,4],[5]]
=> ?
=> ?
=> ? = 4
[[0,1,0,0,0],[0,0,1,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,2,2],[2,2,3,3],[3,3,4],[4,4],[5]]
=> ?
=> ?
=> ? = 5
[[0,0,1,0,0],[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,2,3],[2,2,3,3],[3,3,4],[4,4],[5]]
=> ?
=> ?
=> ? = 6
[[0,0,1,0,0],[1,0,-1,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,3],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ?
=> ?
=> ? = 6
[[0,1,0,0,0],[0,0,0,1,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,2,2],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ?
=> ?
=> ? = 6
[[0,0,1,0,0],[0,1,-1,1,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,2,3],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ([(0,9),(0,11),(1,18),(2,17),(3,19),(4,13),(4,19),(5,12),(5,13),(6,16),(7,14),(8,5),(8,18),(9,10),(10,3),(10,4),(11,1),(11,8),(12,17),(13,15),(15,16),(16,14),(17,7),(18,2),(18,12),(19,6),(19,15)],20)
=> [8,6,4,2]
=> ? = 7
[[0,0,0,1,0],[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,2,4],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ?
=> ?
=> ? = 8
[[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,3,4],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ([(0,13),(0,15),(1,17),(2,16),(2,17),(3,19),(4,16),(4,18),(5,22),(6,21),(7,20),(8,23),(8,29),(9,4),(9,28),(10,3),(10,28),(11,6),(12,7),(12,24),(13,14),(14,1),(14,2),(15,9),(15,10),(16,25),(17,12),(17,25),(18,26),(18,29),(19,23),(19,26),(20,27),(22,27),(23,30),(24,20),(24,22),(25,24),(26,30),(27,21),(28,8),(28,18),(28,19),(29,5),(29,30),(30,11)],31)
=> [10,8,6,4,3]
=> ? = 9
[[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1]]
=> [[1,1,2,2,3],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ?
=> ?
=> ? = 8
[[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1]]
=> [[1,1,2,2,4],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ([(0,14),(0,15),(1,19),(2,18),(3,29),(4,30),(5,22),(6,23),(7,24),(7,25),(8,9),(9,7),(9,18),(9,19),(10,5),(11,6),(12,2),(12,29),(13,1),(13,30),(14,16),(14,28),(15,17),(15,28),(16,3),(16,12),(17,4),(17,13),(18,24),(18,27),(19,25),(19,27),(20,26),(21,26),(22,20),(23,21),(24,22),(24,31),(25,23),(25,31),(27,31),(28,8),(29,10),(30,11),(31,20),(31,21)],32)
=> [10,8,6,4,4]
=> ? = 9
[[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1]]
=> [[1,1,2,3,3],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ([(0,13),(0,15),(1,17),(2,16),(2,17),(3,19),(4,16),(4,18),(5,22),(6,21),(7,20),(8,23),(8,29),(9,4),(9,28),(10,3),(10,28),(11,6),(12,7),(12,24),(13,14),(14,1),(14,2),(15,9),(15,10),(16,25),(17,12),(17,25),(18,26),(18,29),(19,23),(19,26),(20,27),(22,27),(23,30),(24,20),(24,22),(25,24),(26,30),(27,21),(28,8),(28,18),(28,19),(29,5),(29,30),(30,11)],31)
=> [10,8,6,4,3]
=> ? = 9
[[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1]]
=> [[1,1,2,3,4],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ([(0,5),(0,16),(0,17),(1,23),(2,19),(3,11),(3,21),(4,10),(4,20),(5,12),(5,13),(6,50),(7,51),(8,24),(8,58),(9,25),(9,59),(10,14),(10,52),(11,15),(11,53),(12,26),(12,60),(13,27),(13,60),(14,54),(15,55),(16,4),(16,29),(17,3),(17,29),(18,48),(18,49),(19,32),(19,33),(20,46),(20,52),(21,47),(21,53),(22,34),(22,35),(23,18),(23,54),(23,55),(24,40),(24,42),(25,41),(25,43),(26,46),(26,56),(27,47),(27,57),(28,63),(29,1),(30,62),(31,61),(32,61),(33,61),(34,6),(34,62),(35,7),(35,62),(36,58),(37,59),(38,32),(39,33),(40,44),(41,45),(42,38),(43,39),(44,31),(45,31),(46,36),(47,37),(48,40),(48,63),(49,41),(49,63),(50,38),(51,39),(52,8),(52,36),(53,9),(53,37),(54,28),(54,48),(55,28),(55,49),(56,30),(56,34),(57,30),(57,35),(58,42),(58,50),(59,43),(59,51),(60,22),(60,56),(60,57),(62,2),(63,44),(63,45)],64)
=> [11,9,9,7,7,5,5,5,3,3]
=> ? = 10
[[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,2,3],[2,2,2,3],[3,3,3],[4,5],[5]]
=> ([(0,3),(0,6),(0,7),(1,8),(1,12),(2,8),(2,11),(3,9),(3,10),(4,2),(4,13),(5,1),(5,14),(6,4),(6,9),(7,5),(7,10),(8,15),(9,13),(10,14),(11,15),(12,15),(13,11),(14,12)],16)
=> ?
=> ? = 5
[[0,1,0,0,0],[0,0,1,0,0],[1,0,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[1,1,1,2,2],[2,2,2,3],[3,3,4],[4,5],[5]]
=> ?
=> ?
=> ? = 5
[[0,0,1,0,0],[0,1,0,0,0],[1,0,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[1,1,1,2,3],[2,2,2,3],[3,3,4],[4,5],[5]]
=> ([(0,3),(0,8),(1,10),(2,10),(3,9),(4,6),(5,4),(5,11),(6,2),(7,1),(8,5),(8,9),(9,11),(11,7)],12)
=> ?
=> ? = 6
[[0,0,1,0,0],[1,0,-1,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[1,1,1,1,3],[2,2,2,4],[3,3,4],[4,5],[5]]
=> ([(0,4),(0,6),(1,10),(2,9),(3,7),(4,8),(5,3),(5,11),(6,1),(6,8),(7,9),(8,5),(8,10),(10,11),(11,2),(11,7)],12)
=> ?
=> ? = 6
[[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[1,1,1,1,4],[2,2,2,4],[3,3,4],[4,5],[5]]
=> ([(0,4),(0,7),(2,10),(3,9),(4,8),(5,3),(5,11),(6,1),(7,5),(7,8),(8,11),(9,10),(10,6),(11,2),(11,9)],12)
=> ?
=> ? = 7
[[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[1,1,1,2,2],[2,2,2,4],[3,3,4],[4,5],[5]]
=> ?
=> ?
=> ? = 6
[[0,0,1,0,0],[0,1,-1,1,0],[1,0,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[1,1,1,2,3],[2,2,2,4],[3,3,4],[4,5],[5]]
=> ?
=> ?
=> ? = 7
[[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[1,1,1,2,4],[2,2,2,4],[3,3,4],[4,5],[5]]
=> ?
=> ?
=> ? = 8
[[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [[1,1,1,1,1],[2,2,3,3],[3,3,4],[4,5],[5]]
=> ?
=> ?
=> ? = 4
[[0,1,0,0,0],[1,-1,1,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [[1,1,1,1,2],[2,2,3,3],[3,3,4],[4,5],[5]]
=> ?
=> ?
=> ? = 5
[[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [[1,1,1,1,3],[2,2,3,3],[3,3,4],[4,5],[5]]
=> ?
=> ?
=> ? = 6
[[0,1,0,0,0],[0,0,1,0,0],[1,-1,0,1,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [[1,1,1,2,2],[2,2,3,3],[3,3,4],[4,5],[5]]
=> ?
=> ?
=> ? = 6
[[0,0,1,0,0],[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [[1,1,1,2,3],[2,2,3,3],[3,3,4],[4,5],[5]]
=> ?
=> ?
=> ? = 7
[[0,1,0,0,0],[1,-1,0,1,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [[1,1,1,1,2],[2,2,3,4],[3,3,4],[4,5],[5]]
=> ([(0,4),(0,7),(1,9),(2,10),(3,9),(3,10),(4,11),(5,8),(6,1),(7,5),(7,11),(8,2),(8,3),(9,12),(10,12),(11,6)],13)
=> ?
=> ? = 6
[[0,0,1,0,0],[1,0,-1,1,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [[1,1,1,1,3],[2,2,3,4],[3,3,4],[4,5],[5]]
=> ?
=> ?
=> ? = 7
[[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [[1,1,1,1,4],[2,2,3,4],[3,3,4],[4,5],[5]]
=> ?
=> ?
=> ? = 8
[[0,1,0,0,0],[0,0,0,1,0],[1,-1,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [[1,1,1,2,2],[2,2,3,4],[3,3,4],[4,5],[5]]
=> ?
=> ?
=> ? = 7
[[0,0,1,0,0],[0,1,-1,1,0],[1,-1,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [[1,1,1,2,3],[2,2,3,4],[3,3,4],[4,5],[5]]
=> ([(0,10),(0,11),(1,13),(1,14),(2,21),(3,20),(4,14),(4,20),(5,17),(6,16),(7,18),(8,12),(9,1),(9,22),(10,5),(10,15),(11,8),(11,15),(12,3),(12,4),(13,21),(14,19),(15,9),(15,17),(17,22),(18,16),(19,18),(20,7),(20,19),(21,6),(22,2),(22,13)],23)
=> ?
=> ? = 8
[[0,0,0,1,0],[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [[1,1,1,2,4],[2,2,3,4],[3,3,4],[4,5],[5]]
=> ?
=> ?
=> ? = 9
[[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [[1,1,1,3,3],[2,2,3,4],[3,3,4],[4,5],[5]]
=> ?
=> ?
=> ? = 9
[[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [[1,1,1,3,4],[2,2,3,4],[3,3,4],[4,5],[5]]
=> ?
=> ?
=> ? = 10
[[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[1,0,0,-1,1],[0,0,0,1,0]]
=> [[1,1,2,2,2],[2,2,3,3],[3,3,4],[4,5],[5]]
=> ?
=> ?
=> ? = 7
[[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[1,0,0,-1,1],[0,0,0,1,0]]
=> [[1,1,2,2,3],[2,2,3,3],[3,3,4],[4,5],[5]]
=> ?
=> ?
=> ? = 8
[[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[1,0,0,-1,1],[0,0,0,1,0]]
=> [[1,1,2,2,2],[2,2,3,4],[3,3,4],[4,5],[5]]
=> ?
=> ?
=> ? = 8
[[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[1,0,0,-1,1],[0,0,0,1,0]]
=> [[1,1,2,2,3],[2,2,3,4],[3,3,4],[4,5],[5]]
=> ?
=> ?
=> ? = 9
[[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[1,0,0,-1,1],[0,0,0,1,0]]
=> [[1,1,2,2,4],[2,2,3,4],[3,3,4],[4,5],[5]]
=> ?
=> ?
=> ? = 10
[[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[1,0,0,-1,1],[0,0,0,1,0]]
=> [[1,1,2,3,3],[2,2,3,4],[3,3,4],[4,5],[5]]
=> ?
=> ?
=> ? = 10
[[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,-1,1],[0,0,0,1,0]]
=> [[1,1,2,3,4],[2,2,3,4],[3,3,4],[4,5],[5]]
=> ([(0,16),(0,17),(0,28),(1,22),(2,19),(3,11),(3,21),(4,54),(5,55),(6,26),(6,57),(7,25),(7,67),(8,23),(8,64),(9,24),(9,65),(10,14),(10,56),(11,15),(11,58),(12,3),(12,37),(13,27),(13,66),(14,59),(15,60),(16,13),(16,53),(17,6),(17,61),(18,51),(18,52),(19,33),(19,34),(20,35),(20,36),(21,50),(21,58),(22,18),(22,59),(22,60),(23,41),(23,43),(24,42),(24,44),(25,50),(25,63),(26,29),(26,46),(27,29),(27,45),(28,12),(28,53),(28,61),(29,70),(30,71),(31,69),(32,68),(33,68),(34,68),(35,4),(35,69),(36,5),(36,69),(37,1),(38,33),(39,34),(40,64),(41,48),(42,49),(43,38),(44,39),(45,62),(45,70),(46,56),(46,70),(47,65),(48,32),(49,32),(50,40),(51,42),(51,71),(52,41),(52,71),(53,7),(53,66),(54,38),(55,39),(56,9),(56,47),(57,10),(57,46),(58,8),(58,40),(59,30),(59,51),(60,30),(60,52),(61,37),(61,57),(62,31),(62,36),(63,31),(63,35),(64,43),(64,54),(65,44),(65,55),(66,45),(66,67),(67,20),(67,62),(67,63),(69,2),(70,47),(71,48),(71,49)],72)
=> ?
=> ? = 11
[[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [[1,1,1,1,3],[2,2,2,3],[3,3,5],[4,5],[5]]
=> ([(0,5),(0,6),(1,3),(1,13),(2,4),(2,12),(3,10),(4,8),(5,11),(6,2),(6,11),(8,9),(9,7),(10,7),(11,1),(11,12),(12,8),(12,13),(13,9),(13,10)],14)
=> ?
=> ? = 6
Description
The largest nonnegative integer which is not a part and is smaller than the largest part of the partition.
Matching statistic: St000147
Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 13%
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 13%
Values
[[1]]
=> [[1]]
=> ([],1)
=> [1]
=> 1 = 0 + 1
[[1,0],[0,1]]
=> [[1,1],[2]]
=> ([],1)
=> [1]
=> 1 = 0 + 1
[[0,1],[1,0]]
=> [[1,2],[2]]
=> ([(0,1)],2)
=> [2]
=> 2 = 1 + 1
[[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ([],1)
=> [1]
=> 1 = 0 + 1
[[0,1,0],[1,0,0],[0,0,1]]
=> [[1,1,2],[2,2],[3]]
=> ([(0,1)],2)
=> [2]
=> 2 = 1 + 1
[[1,0,0],[0,0,1],[0,1,0]]
=> [[1,1,1],[2,3],[3]]
=> ([(0,1)],2)
=> [2]
=> 2 = 1 + 1
[[0,1,0],[1,-1,1],[0,1,0]]
=> [[1,1,2],[2,3],[3]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 3 = 2 + 1
[[0,0,1],[1,0,0],[0,1,0]]
=> [[1,1,3],[2,3],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4 = 3 + 1
[[0,1,0],[0,0,1],[1,0,0]]
=> [[1,2,2],[2,3],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4 = 3 + 1
[[0,0,1],[0,1,0],[1,0,0]]
=> [[1,2,3],[2,3],[3]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> [5,3]
=> 5 = 4 + 1
[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,2],[3,3],[4]]
=> ([],1)
=> [1]
=> 1 = 0 + 1
[[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,2],[2,2,2],[3,3],[4]]
=> ([(0,1)],2)
=> [2]
=> 2 = 1 + 1
[[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,3],[3,3],[4]]
=> ([(0,1)],2)
=> [2]
=> 2 = 1 + 1
[[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
=> [[1,1,1,2],[2,2,3],[3,3],[4]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 3 = 2 + 1
[[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> [[1,1,1,3],[2,2,3],[3,3],[4]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4 = 3 + 1
[[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]]
=> [[1,1,2,2],[2,2,3],[3,3],[4]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4 = 3 + 1
[[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> [[1,1,2,3],[2,2,3],[3,3],[4]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> [5,3]
=> 5 = 4 + 1
[[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [[1,1,1,1],[2,2,2],[3,4],[4]]
=> ([(0,1)],2)
=> [2]
=> 2 = 1 + 1
[[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> [[1,1,1,2],[2,2,2],[3,4],[4]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> [3,1]
=> 3 = 2 + 1
[[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> [[1,1,1,1],[2,2,3],[3,4],[4]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 3 = 2 + 1
[[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> [[1,1,1,2],[2,2,3],[3,4],[4]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4 = 3 + 1
[[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]]
=> [[1,1,1,3],[2,2,3],[3,4],[4]]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> [5,1]
=> 5 = 4 + 1
[[0,1,0,0],[0,0,1,0],[1,0,-1,1],[0,0,1,0]]
=> [[1,1,2,2],[2,2,3],[3,4],[4]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 5 = 4 + 1
[[0,0,1,0],[0,1,0,0],[1,0,-1,1],[0,0,1,0]]
=> [[1,1,2,3],[2,2,3],[3,4],[4]]
=> ([(0,3),(0,7),(1,8),(2,8),(3,9),(4,5),(5,1),(6,2),(7,4),(7,9),(9,6)],10)
=> [6,4]
=> 6 = 5 + 1
[[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> [[1,1,1,1],[2,2,4],[3,4],[4]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4 = 3 + 1
[[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
=> [[1,1,1,2],[2,2,4],[3,4],[4]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [5,2]
=> 5 = 4 + 1
[[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
=> [[1,1,1,3],[2,2,4],[3,4],[4]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> [6,3]
=> 6 = 5 + 1
[[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [[1,1,1,4],[2,2,4],[3,4],[4]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> [7,1]
=> 7 = 6 + 1
[[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
=> [[1,1,2,2],[2,2,4],[3,4],[4]]
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> [6,3,1]
=> 6 = 5 + 1
[[0,0,1,0],[0,1,-1,1],[1,0,0,0],[0,0,1,0]]
=> [[1,1,2,3],[2,2,4],[3,4],[4]]
=> ([(0,6),(1,9),(1,10),(2,8),(3,7),(4,3),(4,12),(5,2),(5,12),(6,4),(6,5),(7,9),(7,11),(8,10),(8,11),(9,13),(10,13),(11,13),(12,1),(12,7),(12,8)],14)
=> [7,4,3]
=> 7 = 6 + 1
[[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
=> [[1,1,2,4],[2,2,4],[3,4],[4]]
=> ([(0,7),(0,8),(1,12),(2,11),(3,10),(4,10),(4,11),(5,3),(6,1),(6,13),(7,9),(8,5),(9,2),(9,4),(10,14),(11,6),(11,14),(13,12),(14,13)],15)
=> [8,6,1]
=> 8 = 7 + 1
[[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]]
=> [[1,1,1,1],[2,3,3],[3,4],[4]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4 = 3 + 1
[[0,1,0,0],[1,-1,1,0],[0,0,0,1],[0,1,0,0]]
=> [[1,1,1,2],[2,3,3],[3,4],[4]]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> [5,1]
=> 5 = 4 + 1
[[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
=> [[1,1,1,3],[2,3,3],[3,4],[4]]
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> [6,1]
=> 6 = 5 + 1
[[0,1,0,0],[0,0,1,0],[1,-1,0,1],[0,1,0,0]]
=> [[1,1,2,2],[2,3,3],[3,4],[4]]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> [6,1]
=> 6 = 5 + 1
[[0,0,1,0],[0,1,0,0],[1,-1,0,1],[0,1,0,0]]
=> [[1,1,2,3],[2,3,3],[3,4],[4]]
=> ([(0,3),(0,8),(1,10),(2,9),(3,11),(4,2),(5,4),(6,7),(7,1),(7,9),(8,5),(8,11),(9,10),(11,6)],12)
=> [7,5]
=> 7 = 6 + 1
[[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> [[1,1,1,1],[2,3,4],[3,4],[4]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> [5,3]
=> 5 = 4 + 1
[[0,1,0,0],[1,-1,0,1],[0,0,1,0],[0,1,0,0]]
=> [[1,1,1,2],[2,3,4],[3,4],[4]]
=> ([(0,5),(0,6),(1,8),(2,9),(3,8),(3,9),(4,1),(5,4),(6,7),(7,2),(7,3),(8,10),(9,10)],11)
=> [6,4,1]
=> 6 = 5 + 1
[[0,0,1,0],[1,0,-1,1],[0,0,1,0],[0,1,0,0]]
=> [[1,1,1,3],[2,3,4],[3,4],[4]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> [7,3]
=> 7 = 6 + 1
[[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
=> [[1,1,1,4],[2,3,4],[3,4],[4]]
=> ([(0,5),(0,10),(1,16),(2,15),(3,14),(4,13),(5,12),(6,2),(6,13),(7,4),(7,14),(8,1),(9,6),(10,11),(10,12),(11,3),(11,7),(12,9),(13,15),(14,8),(15,16)],17)
=> [8,6,3]
=> 8 = 7 + 1
[[0,1,0,0],[0,0,0,1],[1,-1,1,0],[0,1,0,0]]
=> [[1,1,2,2],[2,3,4],[3,4],[4]]
=> ([(0,6),(0,7),(1,9),(2,12),(3,9),(3,12),(4,10),(5,1),(6,5),(7,8),(8,2),(8,3),(9,11),(11,10),(12,4),(12,11)],13)
=> [7,5,1]
=> 7 = 6 + 1
[[0,0,1,0],[0,1,-1,1],[1,-1,1,0],[0,1,0,0]]
=> [[1,1,2,3],[2,3,4],[3,4],[4]]
=> ([(0,9),(0,11),(1,18),(2,17),(3,19),(4,13),(4,19),(5,12),(5,13),(6,16),(7,14),(8,5),(8,18),(9,10),(10,3),(10,4),(11,1),(11,8),(12,17),(13,15),(15,16),(16,14),(17,7),(18,2),(18,12),(19,6),(19,15)],20)
=> [8,6,4,2]
=> ? = 7 + 1
[[0,0,0,1],[0,1,0,0],[1,-1,1,0],[0,1,0,0]]
=> [[1,1,2,4],[2,3,4],[3,4],[4]]
=> ([(0,12),(0,13),(1,16),(2,15),(3,23),(4,19),(5,17),(5,20),(6,4),(7,5),(7,15),(7,16),(8,10),(9,7),(10,2),(11,1),(11,23),(12,8),(12,22),(13,14),(13,22),(14,3),(14,11),(15,17),(15,21),(16,20),(16,21),(17,24),(19,18),(20,19),(20,24),(21,24),(22,9),(23,6),(24,18)],25)
=> [9,7,5,4]
=> ? = 8 + 1
[[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> [[1,1,3,3],[2,3,4],[3,4],[4]]
=> ([(0,8),(2,11),(2,12),(3,10),(4,9),(5,4),(5,14),(6,3),(6,14),(7,1),(8,5),(8,6),(9,11),(9,13),(10,12),(10,13),(11,15),(12,15),(13,15),(14,2),(14,9),(14,10),(15,7)],16)
=> [9,4,3]
=> 9 = 8 + 1
[[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
=> [[1,1,3,4],[2,3,4],[3,4],[4]]
=> ([(0,13),(0,15),(1,17),(2,16),(2,17),(3,19),(4,16),(4,18),(5,22),(6,21),(7,20),(8,23),(8,29),(9,4),(9,28),(10,3),(10,28),(11,6),(12,7),(12,24),(13,14),(14,1),(14,2),(15,9),(15,10),(16,25),(17,12),(17,25),(18,26),(18,29),(19,23),(19,26),(20,27),(22,27),(23,30),(24,20),(24,22),(25,24),(26,30),(27,21),(28,8),(28,18),(28,19),(29,5),(29,30),(30,11)],31)
=> [10,8,6,4,3]
=> ? = 9 + 1
[[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> [[1,2,2,2],[2,3,3],[3,4],[4]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> [7,1]
=> 7 = 6 + 1
[[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]]
=> [[1,2,2,3],[2,3,3],[3,4],[4]]
=> ([(0,5),(0,10),(1,16),(2,15),(3,14),(4,13),(5,12),(6,2),(6,13),(7,4),(7,14),(8,1),(9,6),(10,11),(10,12),(11,3),(11,7),(12,9),(13,15),(14,8),(15,16)],17)
=> [8,6,3]
=> 8 = 7 + 1
[[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]]
=> [[1,2,2,2],[2,3,4],[3,4],[4]]
=> ([(0,7),(0,8),(1,12),(2,11),(3,10),(4,10),(4,11),(5,3),(6,1),(6,13),(7,9),(8,5),(9,2),(9,4),(10,14),(11,6),(11,14),(13,12),(14,13)],15)
=> [8,6,1]
=> 8 = 7 + 1
[[0,0,1,0],[0,1,-1,1],[0,0,1,0],[1,0,0,0]]
=> [[1,2,2,3],[2,3,4],[3,4],[4]]
=> ([(0,10),(0,12),(1,16),(2,17),(3,21),(4,22),(5,14),(6,13),(6,14),(7,13),(7,15),(8,7),(8,21),(9,2),(9,18),(10,11),(11,5),(11,6),(12,3),(12,8),(13,19),(14,9),(14,19),(15,22),(16,20),(17,20),(18,16),(18,17),(19,18),(21,4),(21,15),(22,1)],23)
=> [9,7,5,2]
=> ? = 8 + 1
[[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
=> [[1,2,2,4],[2,3,4],[3,4],[4]]
=> ([(0,14),(0,15),(1,19),(2,18),(3,29),(4,30),(5,22),(6,23),(7,24),(7,25),(8,9),(9,7),(9,18),(9,19),(10,5),(11,6),(12,2),(12,29),(13,1),(13,30),(14,16),(14,28),(15,17),(15,28),(16,3),(16,12),(17,4),(17,13),(18,24),(18,27),(19,25),(19,27),(20,26),(21,26),(22,20),(23,21),(24,22),(24,31),(25,23),(25,31),(27,31),(28,8),(29,10),(30,11),(31,20),(31,21)],32)
=> [10,8,6,4,4]
=> ? = 9 + 1
[[0,0,1,0],[0,0,0,1],[0,1,0,0],[1,0,0,0]]
=> [[1,2,3,3],[2,3,4],[3,4],[4]]
=> ([(0,13),(0,15),(1,17),(2,16),(2,17),(3,19),(4,16),(4,18),(5,22),(6,21),(7,20),(8,23),(8,29),(9,4),(9,28),(10,3),(10,28),(11,6),(12,7),(12,24),(13,14),(14,1),(14,2),(15,9),(15,10),(16,25),(17,12),(17,25),(18,26),(18,29),(19,23),(19,26),(20,27),(22,27),(23,30),(24,20),(24,22),(25,24),(26,30),(27,21),(28,8),(28,18),(28,19),(29,5),(29,30),(30,11)],31)
=> [10,8,6,4,3]
=> ? = 9 + 1
[[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> [[1,2,3,4],[2,3,4],[3,4],[4]]
=> ([(0,5),(0,16),(0,17),(1,23),(2,19),(3,11),(3,21),(4,10),(4,20),(5,12),(5,13),(6,50),(7,51),(8,24),(8,58),(9,25),(9,59),(10,14),(10,52),(11,15),(11,53),(12,26),(12,60),(13,27),(13,60),(14,54),(15,55),(16,4),(16,29),(17,3),(17,29),(18,48),(18,49),(19,32),(19,33),(20,46),(20,52),(21,47),(21,53),(22,34),(22,35),(23,18),(23,54),(23,55),(24,40),(24,42),(25,41),(25,43),(26,46),(26,56),(27,47),(27,57),(28,63),(29,1),(30,62),(31,61),(32,61),(33,61),(34,6),(34,62),(35,7),(35,62),(36,58),(37,59),(38,32),(39,33),(40,44),(41,45),(42,38),(43,39),(44,31),(45,31),(46,36),(47,37),(48,40),(48,63),(49,41),(49,63),(50,38),(51,39),(52,8),(52,36),(53,9),(53,37),(54,28),(54,48),(55,28),(55,49),(56,30),(56,34),(57,30),(57,35),(58,42),(58,50),(59,43),(59,51),(60,22),(60,56),(60,57),(62,2),(63,44),(63,45)],64)
=> [11,9,9,7,7,5,5,5,3,3]
=> ? = 10 + 1
[[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,2,2],[3,3,3],[4,4],[5]]
=> ([],1)
=> [1]
=> 1 = 0 + 1
[[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,2],[2,2,2,2],[3,3,3],[4,4],[5]]
=> ([(0,1)],2)
=> [2]
=> 2 = 1 + 1
[[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,2,3],[3,3,3],[4,4],[5]]
=> ([(0,1)],2)
=> [2]
=> 2 = 1 + 1
[[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,2],[2,2,2,3],[3,3,3],[4,4],[5]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 3 = 2 + 1
[[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,3],[2,2,2,3],[3,3,3],[4,4],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4 = 3 + 1
[[0,1,0,0,0],[0,0,1,0,0],[1,0,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,2,2],[2,2,2,3],[3,3,4],[4,4],[5]]
=> ?
=> ?
=> ? = 4 + 1
[[0,0,1,0,0],[0,1,-1,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,2,3],[2,2,2,4],[3,3,4],[4,4],[5]]
=> ?
=> ?
=> ? = 6 + 1
[[0,1,0,0,0],[1,-1,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,2],[2,2,3,3],[3,3,4],[4,4],[5]]
=> ?
=> ?
=> ? = 4 + 1
[[0,1,0,0,0],[0,0,1,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,2,2],[2,2,3,3],[3,3,4],[4,4],[5]]
=> ?
=> ?
=> ? = 5 + 1
[[0,0,1,0,0],[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,2,3],[2,2,3,3],[3,3,4],[4,4],[5]]
=> ?
=> ?
=> ? = 6 + 1
[[0,0,1,0,0],[1,0,-1,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,3],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ?
=> ?
=> ? = 6 + 1
[[0,1,0,0,0],[0,0,0,1,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,2,2],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ?
=> ?
=> ? = 6 + 1
[[0,0,1,0,0],[0,1,-1,1,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,2,3],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ([(0,9),(0,11),(1,18),(2,17),(3,19),(4,13),(4,19),(5,12),(5,13),(6,16),(7,14),(8,5),(8,18),(9,10),(10,3),(10,4),(11,1),(11,8),(12,17),(13,15),(15,16),(16,14),(17,7),(18,2),(18,12),(19,6),(19,15)],20)
=> [8,6,4,2]
=> ? = 7 + 1
[[0,0,0,1,0],[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,2,4],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ?
=> ?
=> ? = 8 + 1
[[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,3,4],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ([(0,13),(0,15),(1,17),(2,16),(2,17),(3,19),(4,16),(4,18),(5,22),(6,21),(7,20),(8,23),(8,29),(9,4),(9,28),(10,3),(10,28),(11,6),(12,7),(12,24),(13,14),(14,1),(14,2),(15,9),(15,10),(16,25),(17,12),(17,25),(18,26),(18,29),(19,23),(19,26),(20,27),(22,27),(23,30),(24,20),(24,22),(25,24),(26,30),(27,21),(28,8),(28,18),(28,19),(29,5),(29,30),(30,11)],31)
=> [10,8,6,4,3]
=> ? = 9 + 1
[[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1]]
=> [[1,1,2,2,3],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ?
=> ?
=> ? = 8 + 1
[[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1]]
=> [[1,1,2,2,4],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ([(0,14),(0,15),(1,19),(2,18),(3,29),(4,30),(5,22),(6,23),(7,24),(7,25),(8,9),(9,7),(9,18),(9,19),(10,5),(11,6),(12,2),(12,29),(13,1),(13,30),(14,16),(14,28),(15,17),(15,28),(16,3),(16,12),(17,4),(17,13),(18,24),(18,27),(19,25),(19,27),(20,26),(21,26),(22,20),(23,21),(24,22),(24,31),(25,23),(25,31),(27,31),(28,8),(29,10),(30,11),(31,20),(31,21)],32)
=> [10,8,6,4,4]
=> ? = 9 + 1
[[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1]]
=> [[1,1,2,3,3],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ([(0,13),(0,15),(1,17),(2,16),(2,17),(3,19),(4,16),(4,18),(5,22),(6,21),(7,20),(8,23),(8,29),(9,4),(9,28),(10,3),(10,28),(11,6),(12,7),(12,24),(13,14),(14,1),(14,2),(15,9),(15,10),(16,25),(17,12),(17,25),(18,26),(18,29),(19,23),(19,26),(20,27),(22,27),(23,30),(24,20),(24,22),(25,24),(26,30),(27,21),(28,8),(28,18),(28,19),(29,5),(29,30),(30,11)],31)
=> [10,8,6,4,3]
=> ? = 9 + 1
[[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1]]
=> [[1,1,2,3,4],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ([(0,5),(0,16),(0,17),(1,23),(2,19),(3,11),(3,21),(4,10),(4,20),(5,12),(5,13),(6,50),(7,51),(8,24),(8,58),(9,25),(9,59),(10,14),(10,52),(11,15),(11,53),(12,26),(12,60),(13,27),(13,60),(14,54),(15,55),(16,4),(16,29),(17,3),(17,29),(18,48),(18,49),(19,32),(19,33),(20,46),(20,52),(21,47),(21,53),(22,34),(22,35),(23,18),(23,54),(23,55),(24,40),(24,42),(25,41),(25,43),(26,46),(26,56),(27,47),(27,57),(28,63),(29,1),(30,62),(31,61),(32,61),(33,61),(34,6),(34,62),(35,7),(35,62),(36,58),(37,59),(38,32),(39,33),(40,44),(41,45),(42,38),(43,39),(44,31),(45,31),(46,36),(47,37),(48,40),(48,63),(49,41),(49,63),(50,38),(51,39),(52,8),(52,36),(53,9),(53,37),(54,28),(54,48),(55,28),(55,49),(56,30),(56,34),(57,30),(57,35),(58,42),(58,50),(59,43),(59,51),(60,22),(60,56),(60,57),(62,2),(63,44),(63,45)],64)
=> [11,9,9,7,7,5,5,5,3,3]
=> ? = 10 + 1
[[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,2,3],[2,2,2,3],[3,3,3],[4,5],[5]]
=> ([(0,3),(0,6),(0,7),(1,8),(1,12),(2,8),(2,11),(3,9),(3,10),(4,2),(4,13),(5,1),(5,14),(6,4),(6,9),(7,5),(7,10),(8,15),(9,13),(10,14),(11,15),(12,15),(13,11),(14,12)],16)
=> ?
=> ? = 5 + 1
[[0,1,0,0,0],[0,0,1,0,0],[1,0,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[1,1,1,2,2],[2,2,2,3],[3,3,4],[4,5],[5]]
=> ?
=> ?
=> ? = 5 + 1
[[0,0,1,0,0],[0,1,0,0,0],[1,0,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[1,1,1,2,3],[2,2,2,3],[3,3,4],[4,5],[5]]
=> ([(0,3),(0,8),(1,10),(2,10),(3,9),(4,6),(5,4),(5,11),(6,2),(7,1),(8,5),(8,9),(9,11),(11,7)],12)
=> ?
=> ? = 6 + 1
[[0,0,1,0,0],[1,0,-1,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[1,1,1,1,3],[2,2,2,4],[3,3,4],[4,5],[5]]
=> ([(0,4),(0,6),(1,10),(2,9),(3,7),(4,8),(5,3),(5,11),(6,1),(6,8),(7,9),(8,5),(8,10),(10,11),(11,2),(11,7)],12)
=> ?
=> ? = 6 + 1
[[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[1,1,1,1,4],[2,2,2,4],[3,3,4],[4,5],[5]]
=> ([(0,4),(0,7),(2,10),(3,9),(4,8),(5,3),(5,11),(6,1),(7,5),(7,8),(8,11),(9,10),(10,6),(11,2),(11,9)],12)
=> ?
=> ? = 7 + 1
[[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[1,1,1,2,2],[2,2,2,4],[3,3,4],[4,5],[5]]
=> ?
=> ?
=> ? = 6 + 1
[[0,0,1,0,0],[0,1,-1,1,0],[1,0,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[1,1,1,2,3],[2,2,2,4],[3,3,4],[4,5],[5]]
=> ?
=> ?
=> ? = 7 + 1
[[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[1,1,1,2,4],[2,2,2,4],[3,3,4],[4,5],[5]]
=> ?
=> ?
=> ? = 8 + 1
[[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [[1,1,1,1,1],[2,2,3,3],[3,3,4],[4,5],[5]]
=> ?
=> ?
=> ? = 4 + 1
[[0,1,0,0,0],[1,-1,1,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [[1,1,1,1,2],[2,2,3,3],[3,3,4],[4,5],[5]]
=> ?
=> ?
=> ? = 5 + 1
[[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [[1,1,1,1,3],[2,2,3,3],[3,3,4],[4,5],[5]]
=> ?
=> ?
=> ? = 6 + 1
[[0,1,0,0,0],[0,0,1,0,0],[1,-1,0,1,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [[1,1,1,2,2],[2,2,3,3],[3,3,4],[4,5],[5]]
=> ?
=> ?
=> ? = 6 + 1
[[0,0,1,0,0],[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [[1,1,1,2,3],[2,2,3,3],[3,3,4],[4,5],[5]]
=> ?
=> ?
=> ? = 7 + 1
[[0,1,0,0,0],[1,-1,0,1,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [[1,1,1,1,2],[2,2,3,4],[3,3,4],[4,5],[5]]
=> ([(0,4),(0,7),(1,9),(2,10),(3,9),(3,10),(4,11),(5,8),(6,1),(7,5),(7,11),(8,2),(8,3),(9,12),(10,12),(11,6)],13)
=> ?
=> ? = 6 + 1
[[0,0,1,0,0],[1,0,-1,1,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [[1,1,1,1,3],[2,2,3,4],[3,3,4],[4,5],[5]]
=> ?
=> ?
=> ? = 7 + 1
[[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [[1,1,1,1,4],[2,2,3,4],[3,3,4],[4,5],[5]]
=> ?
=> ?
=> ? = 8 + 1
[[0,1,0,0,0],[0,0,0,1,0],[1,-1,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [[1,1,1,2,2],[2,2,3,4],[3,3,4],[4,5],[5]]
=> ?
=> ?
=> ? = 7 + 1
[[0,0,1,0,0],[0,1,-1,1,0],[1,-1,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [[1,1,1,2,3],[2,2,3,4],[3,3,4],[4,5],[5]]
=> ([(0,10),(0,11),(1,13),(1,14),(2,21),(3,20),(4,14),(4,20),(5,17),(6,16),(7,18),(8,12),(9,1),(9,22),(10,5),(10,15),(11,8),(11,15),(12,3),(12,4),(13,21),(14,19),(15,9),(15,17),(17,22),(18,16),(19,18),(20,7),(20,19),(21,6),(22,2),(22,13)],23)
=> ?
=> ? = 8 + 1
[[0,0,0,1,0],[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [[1,1,1,2,4],[2,2,3,4],[3,3,4],[4,5],[5]]
=> ?
=> ?
=> ? = 9 + 1
[[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [[1,1,1,3,3],[2,2,3,4],[3,3,4],[4,5],[5]]
=> ?
=> ?
=> ? = 9 + 1
[[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [[1,1,1,3,4],[2,2,3,4],[3,3,4],[4,5],[5]]
=> ?
=> ?
=> ? = 10 + 1
[[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[1,0,0,-1,1],[0,0,0,1,0]]
=> [[1,1,2,2,2],[2,2,3,3],[3,3,4],[4,5],[5]]
=> ?
=> ?
=> ? = 7 + 1
[[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[1,0,0,-1,1],[0,0,0,1,0]]
=> [[1,1,2,2,3],[2,2,3,3],[3,3,4],[4,5],[5]]
=> ?
=> ?
=> ? = 8 + 1
[[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[1,0,0,-1,1],[0,0,0,1,0]]
=> [[1,1,2,2,2],[2,2,3,4],[3,3,4],[4,5],[5]]
=> ?
=> ?
=> ? = 8 + 1
[[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[1,0,0,-1,1],[0,0,0,1,0]]
=> [[1,1,2,2,3],[2,2,3,4],[3,3,4],[4,5],[5]]
=> ?
=> ?
=> ? = 9 + 1
[[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[1,0,0,-1,1],[0,0,0,1,0]]
=> [[1,1,2,2,4],[2,2,3,4],[3,3,4],[4,5],[5]]
=> ?
=> ?
=> ? = 10 + 1
[[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[1,0,0,-1,1],[0,0,0,1,0]]
=> [[1,1,2,3,3],[2,2,3,4],[3,3,4],[4,5],[5]]
=> ?
=> ?
=> ? = 10 + 1
[[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,-1,1],[0,0,0,1,0]]
=> [[1,1,2,3,4],[2,2,3,4],[3,3,4],[4,5],[5]]
=> ([(0,16),(0,17),(0,28),(1,22),(2,19),(3,11),(3,21),(4,54),(5,55),(6,26),(6,57),(7,25),(7,67),(8,23),(8,64),(9,24),(9,65),(10,14),(10,56),(11,15),(11,58),(12,3),(12,37),(13,27),(13,66),(14,59),(15,60),(16,13),(16,53),(17,6),(17,61),(18,51),(18,52),(19,33),(19,34),(20,35),(20,36),(21,50),(21,58),(22,18),(22,59),(22,60),(23,41),(23,43),(24,42),(24,44),(25,50),(25,63),(26,29),(26,46),(27,29),(27,45),(28,12),(28,53),(28,61),(29,70),(30,71),(31,69),(32,68),(33,68),(34,68),(35,4),(35,69),(36,5),(36,69),(37,1),(38,33),(39,34),(40,64),(41,48),(42,49),(43,38),(44,39),(45,62),(45,70),(46,56),(46,70),(47,65),(48,32),(49,32),(50,40),(51,42),(51,71),(52,41),(52,71),(53,7),(53,66),(54,38),(55,39),(56,9),(56,47),(57,10),(57,46),(58,8),(58,40),(59,30),(59,51),(60,30),(60,52),(61,37),(61,57),(62,31),(62,36),(63,31),(63,35),(64,43),(64,54),(65,44),(65,55),(66,45),(66,67),(67,20),(67,62),(67,63),(69,2),(70,47),(71,48),(71,49)],72)
=> ?
=> ? = 11 + 1
[[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [[1,1,1,1,3],[2,2,2,3],[3,3,5],[4,5],[5]]
=> ([(0,5),(0,6),(1,3),(1,13),(2,4),(2,12),(3,10),(4,8),(5,11),(6,2),(6,11),(8,9),(9,7),(10,7),(11,1),(11,12),(12,8),(12,13),(13,9),(13,10)],14)
=> ?
=> ? = 6 + 1
Description
The largest part of an integer partition.
Matching statistic: St000093
Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St000093: Graphs ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 16%
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St000093: Graphs ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 16%
Values
[[1]]
=> [[1]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[1,0],[0,1]]
=> [[1,1],[2]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[0,1],[1,0]]
=> [[1,2],[2]]
=> ([(0,1)],2)
=> ([],2)
=> 2 = 1 + 1
[[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[0,1,0],[1,0,0],[0,0,1]]
=> [[1,1,2],[2,2],[3]]
=> ([(0,1)],2)
=> ([],2)
=> 2 = 1 + 1
[[1,0,0],[0,0,1],[0,1,0]]
=> [[1,1,1],[2,3],[3]]
=> ([(0,1)],2)
=> ([],2)
=> 2 = 1 + 1
[[0,1,0],[1,-1,1],[0,1,0]]
=> [[1,1,2],[2,3],[3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 3 = 2 + 1
[[0,0,1],[1,0,0],[0,1,0]]
=> [[1,1,3],[2,3],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 4 = 3 + 1
[[0,1,0],[0,0,1],[1,0,0]]
=> [[1,2,2],[2,3],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 4 = 3 + 1
[[0,0,1],[0,1,0],[1,0,0]]
=> [[1,2,3],[2,3],[3]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ([(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
=> 5 = 4 + 1
[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,2],[3,3],[4]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,2],[2,2,2],[3,3],[4]]
=> ([(0,1)],2)
=> ([],2)
=> 2 = 1 + 1
[[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,3],[3,3],[4]]
=> ([(0,1)],2)
=> ([],2)
=> 2 = 1 + 1
[[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
=> [[1,1,1,2],[2,2,3],[3,3],[4]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 3 = 2 + 1
[[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> [[1,1,1,3],[2,2,3],[3,3],[4]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 4 = 3 + 1
[[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]]
=> [[1,1,2,2],[2,2,3],[3,3],[4]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 4 = 3 + 1
[[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> [[1,1,2,3],[2,2,3],[3,3],[4]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ([(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
=> 5 = 4 + 1
[[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [[1,1,1,1],[2,2,2],[3,4],[4]]
=> ([(0,1)],2)
=> ([],2)
=> 2 = 1 + 1
[[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> [[1,1,1,2],[2,2,2],[3,4],[4]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 3 = 2 + 1
[[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> [[1,1,1,1],[2,2,3],[3,4],[4]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 3 = 2 + 1
[[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> [[1,1,1,2],[2,2,3],[3,4],[4]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 4 = 3 + 1
[[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]]
=> [[1,1,1,3],[2,2,3],[3,4],[4]]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(4,5)],6)
=> 5 = 4 + 1
[[0,1,0,0],[0,0,1,0],[1,0,-1,1],[0,0,1,0]]
=> [[1,1,2,2],[2,2,3],[3,4],[4]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 5 = 4 + 1
[[0,0,1,0],[0,1,0,0],[1,0,-1,1],[0,0,1,0]]
=> [[1,1,2,3],[2,2,3],[3,4],[4]]
=> ([(0,3),(0,7),(1,8),(2,8),(3,9),(4,5),(5,1),(6,2),(7,4),(7,9),(9,6)],10)
=> ([(2,9),(3,6),(3,7),(3,8),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,9),(7,9),(8,9)],10)
=> 6 = 5 + 1
[[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> [[1,1,1,1],[2,2,4],[3,4],[4]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 4 = 3 + 1
[[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
=> [[1,1,1,2],[2,2,4],[3,4],[4]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> 5 = 4 + 1
[[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
=> [[1,1,1,3],[2,2,4],[3,4],[4]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
=> 6 = 5 + 1
[[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [[1,1,1,4],[2,2,4],[3,4],[4]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ([(6,7)],8)
=> ? = 6 + 1
[[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
=> [[1,1,2,2],[2,2,4],[3,4],[4]]
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> ([(3,6),(3,9),(4,5),(4,9),(5,8),(6,8),(7,8),(7,9),(8,9)],10)
=> 6 = 5 + 1
[[0,0,1,0],[0,1,-1,1],[1,0,0,0],[0,0,1,0]]
=> [[1,1,2,3],[2,2,4],[3,4],[4]]
=> ([(0,6),(1,9),(1,10),(2,8),(3,7),(4,3),(4,12),(5,2),(5,12),(6,4),(6,5),(7,9),(7,11),(8,10),(8,11),(9,13),(10,13),(11,13),(12,1),(12,7),(12,8)],14)
=> ([(3,12),(3,13),(4,5),(4,13),(5,12),(6,9),(6,10),(6,11),(7,8),(7,10),(7,11),(7,12),(8,9),(8,11),(8,13),(9,10),(9,12),(10,13),(11,12),(11,13),(12,13)],14)
=> 7 = 6 + 1
[[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
=> [[1,1,2,4],[2,2,4],[3,4],[4]]
=> ([(0,7),(0,8),(1,12),(2,11),(3,10),(4,10),(4,11),(5,3),(6,1),(6,13),(7,9),(8,5),(9,2),(9,4),(10,14),(11,6),(11,14),(13,12),(14,13)],15)
=> ([(2,11),(3,10),(3,11),(4,12),(4,13),(4,14),(5,12),(5,13),(5,14),(6,8),(6,12),(6,13),(6,14),(7,9),(7,12),(7,13),(7,14),(8,9),(8,10),(8,11),(9,12),(9,13),(9,14),(10,12),(10,13),(10,14),(11,12),(11,13),(11,14)],15)
=> ? = 7 + 1
[[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]]
=> [[1,1,1,1],[2,3,3],[3,4],[4]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 4 = 3 + 1
[[0,1,0,0],[1,-1,1,0],[0,0,0,1],[0,1,0,0]]
=> [[1,1,1,2],[2,3,3],[3,4],[4]]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(4,5)],6)
=> 5 = 4 + 1
[[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
=> [[1,1,1,3],[2,3,3],[3,4],[4]]
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ([(5,6)],7)
=> 6 = 5 + 1
[[0,1,0,0],[0,0,1,0],[1,-1,0,1],[0,1,0,0]]
=> [[1,1,2,2],[2,3,3],[3,4],[4]]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ([(5,6)],7)
=> 6 = 5 + 1
[[0,0,1,0],[0,1,0,0],[1,-1,0,1],[0,1,0,0]]
=> [[1,1,2,3],[2,3,3],[3,4],[4]]
=> ([(0,3),(0,8),(1,10),(2,9),(3,11),(4,2),(5,4),(6,7),(7,1),(7,9),(8,5),(8,11),(9,10),(11,6)],12)
=> ([(2,8),(3,7),(4,9),(4,10),(4,11),(5,9),(5,10),(5,11),(6,9),(6,10),(6,11),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11)],12)
=> ? = 6 + 1
[[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> [[1,1,1,1],[2,3,4],[3,4],[4]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ([(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
=> 5 = 4 + 1
[[0,1,0,0],[1,-1,0,1],[0,0,1,0],[0,1,0,0]]
=> [[1,1,1,2],[2,3,4],[3,4],[4]]
=> ([(0,5),(0,6),(1,8),(2,9),(3,8),(3,9),(4,1),(5,4),(6,7),(7,2),(7,3),(8,10),(9,10)],11)
=> ([(2,6),(2,10),(3,7),(3,8),(3,9),(4,7),(4,8),(4,9),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(7,10),(8,10),(9,10)],11)
=> 6 = 5 + 1
[[0,0,1,0],[1,0,-1,1],[0,0,1,0],[0,1,0,0]]
=> [[1,1,1,3],[2,3,4],[3,4],[4]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ([(4,9),(5,8),(6,7),(7,9),(8,9)],10)
=> 7 = 6 + 1
[[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
=> [[1,1,1,4],[2,3,4],[3,4],[4]]
=> ([(0,5),(0,10),(1,16),(2,15),(3,14),(4,13),(5,12),(6,2),(6,13),(7,4),(7,14),(8,1),(9,6),(10,11),(10,12),(11,3),(11,7),(12,9),(13,15),(14,8),(15,16)],17)
=> ([(2,14),(3,11),(3,12),(3,13),(3,16),(4,10),(4,11),(4,12),(4,13),(4,16),(5,7),(5,8),(5,9),(5,10),(5,14),(6,7),(6,8),(6,9),(6,10),(6,14),(6,16),(7,11),(7,12),(7,13),(7,15),(7,16),(8,11),(8,12),(8,13),(8,15),(8,16),(9,11),(9,12),(9,13),(9,15),(9,16),(10,11),(10,12),(10,13),(10,15),(10,16),(11,14),(11,15),(12,14),(12,15),(13,14),(13,15),(14,15),(14,16),(15,16)],17)
=> ? = 7 + 1
[[0,1,0,0],[0,0,0,1],[1,-1,1,0],[0,1,0,0]]
=> [[1,1,2,2],[2,3,4],[3,4],[4]]
=> ([(0,6),(0,7),(1,9),(2,12),(3,9),(3,12),(4,10),(5,1),(6,5),(7,8),(8,2),(8,3),(9,11),(11,10),(12,4),(12,11)],13)
=> ([(2,9),(3,10),(3,11),(3,12),(4,10),(4,11),(4,12),(5,6),(5,8),(5,9),(6,10),(6,11),(6,12),(7,8),(7,10),(7,11),(7,12),(8,10),(8,11),(8,12),(9,10),(9,11),(9,12)],13)
=> 7 = 6 + 1
[[0,0,1,0],[0,1,-1,1],[1,-1,1,0],[0,1,0,0]]
=> [[1,1,2,3],[2,3,4],[3,4],[4]]
=> ([(0,9),(0,11),(1,18),(2,17),(3,19),(4,13),(4,19),(5,12),(5,13),(6,16),(7,14),(8,5),(8,18),(9,10),(10,3),(10,4),(11,1),(11,8),(12,17),(13,15),(15,16),(16,14),(17,7),(18,2),(18,12),(19,6),(19,15)],20)
=> ([(2,10),(2,11),(2,13),(2,15),(2,17),(2,19),(3,8),(3,9),(3,12),(3,14),(3,16),(3,18),(4,8),(4,9),(4,12),(4,14),(4,16),(4,18),(4,19),(5,10),(5,11),(5,13),(5,15),(5,17),(5,18),(5,19),(6,8),(6,9),(6,12),(6,13),(6,14),(6,16),(6,17),(6,18),(6,19),(7,10),(7,11),(7,12),(7,13),(7,15),(7,16),(7,17),(7,18),(7,19),(8,10),(8,11),(8,13),(8,15),(8,17),(8,19),(9,10),(9,11),(9,13),(9,15),(9,17),(9,19),(10,12),(10,14),(10,16),(10,18),(11,12),(11,14),(11,16),(11,18),(12,13),(12,15),(12,17),(12,19),(13,14),(13,16),(13,18),(14,15),(14,16),(14,17),(14,19),(15,16),(15,17),(15,18),(16,17),(16,19),(17,18),(18,19)],20)
=> 8 = 7 + 1
[[0,0,0,1],[0,1,0,0],[1,-1,1,0],[0,1,0,0]]
=> [[1,1,2,4],[2,3,4],[3,4],[4]]
=> ([(0,12),(0,13),(1,16),(2,15),(3,23),(4,19),(5,17),(5,20),(6,4),(7,5),(7,15),(7,16),(8,10),(9,7),(10,2),(11,1),(11,23),(12,8),(12,22),(13,14),(13,22),(14,3),(14,11),(15,17),(15,21),(16,20),(16,21),(17,24),(19,18),(20,19),(20,24),(21,24),(22,9),(23,6),(24,18)],25)
=> ([(2,6),(2,21),(2,22),(2,23),(3,4),(3,18),(3,19),(3,20),(3,24),(4,5),(4,14),(4,15),(4,21),(4,22),(4,23),(5,12),(5,15),(5,16),(5,18),(5,19),(5,20),(5,24),(6,10),(6,13),(6,17),(6,18),(6,19),(6,20),(6,24),(7,10),(7,13),(7,17),(7,18),(7,19),(7,20),(7,21),(7,22),(7,23),(7,24),(8,10),(8,13),(8,17),(8,18),(8,19),(8,20),(8,21),(8,22),(8,23),(8,24),(9,10),(9,13),(9,17),(9,18),(9,19),(9,20),(9,21),(9,22),(9,23),(9,24),(10,14),(10,15),(10,16),(10,21),(10,22),(10,23),(11,14),(11,15),(11,16),(11,18),(11,19),(11,20),(11,21),(11,22),(11,23),(11,24),(12,14),(12,15),(12,18),(12,19),(12,20),(12,21),(12,22),(12,23),(12,24),(13,14),(13,15),(13,16),(13,21),(13,22),(13,23),(13,24),(14,16),(14,17),(14,18),(14,19),(14,20),(14,24),(15,17),(15,18),(15,19),(15,20),(15,24),(16,17),(16,18),(16,19),(16,20),(16,21),(16,22),(16,23),(16,24),(17,18),(17,19),(17,20),(17,21),(17,22),(17,23),(17,24),(18,21),(18,22),(18,23),(19,21),(19,22),(19,23),(20,21),(20,22),(20,23),(21,24),(22,24),(23,24)],25)
=> ? = 8 + 1
[[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> [[1,1,3,3],[2,3,4],[3,4],[4]]
=> ([(0,8),(2,11),(2,12),(3,10),(4,9),(5,4),(5,14),(6,3),(6,14),(7,1),(8,5),(8,6),(9,11),(9,13),(10,12),(10,13),(11,15),(12,15),(13,15),(14,2),(14,9),(14,10),(15,7)],16)
=> ([(5,14),(5,15),(6,7),(6,15),(7,14),(8,11),(8,12),(8,13),(9,10),(9,12),(9,13),(9,14),(10,11),(10,13),(10,15),(11,12),(11,14),(12,15),(13,14),(13,15),(14,15)],16)
=> ? = 8 + 1
[[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
=> [[1,1,3,4],[2,3,4],[3,4],[4]]
=> ([(0,13),(0,15),(1,17),(2,16),(2,17),(3,19),(4,16),(4,18),(5,22),(6,21),(7,20),(8,23),(8,29),(9,4),(9,28),(10,3),(10,28),(11,6),(12,7),(12,24),(13,14),(14,1),(14,2),(15,9),(15,10),(16,25),(17,12),(17,25),(18,26),(18,29),(19,23),(19,26),(20,27),(22,27),(23,30),(24,20),(24,22),(25,24),(26,30),(27,21),(28,8),(28,18),(28,19),(29,5),(29,30),(30,11)],31)
=> ([(2,11),(2,12),(2,13),(2,21),(2,25),(2,26),(2,30),(3,18),(3,19),(3,22),(3,23),(3,27),(3,28),(3,29),(4,10),(4,11),(4,12),(4,13),(4,21),(4,25),(4,26),(4,29),(4,30),(5,8),(5,18),(5,19),(5,22),(5,23),(5,27),(5,28),(5,29),(5,30),(6,7),(6,9),(6,10),(6,16),(6,18),(6,19),(6,20),(6,22),(6,23),(6,27),(6,28),(6,29),(6,30),(7,8),(7,17),(7,18),(7,19),(7,22),(7,23),(7,25),(7,26),(7,27),(7,28),(7,29),(7,30),(8,9),(8,10),(8,16),(8,18),(8,19),(8,20),(8,22),(8,23),(8,27),(8,28),(8,29),(9,11),(9,12),(9,13),(9,14),(9,15),(9,17),(9,21),(9,24),(9,25),(9,26),(9,29),(9,30),(10,11),(10,12),(10,13),(10,14),(10,15),(10,17),(10,21),(10,24),(10,25),(10,26),(10,30),(11,16),(11,18),(11,19),(11,20),(11,22),(11,23),(11,24),(11,27),(11,28),(11,29),(12,16),(12,18),(12,19),(12,20),(12,22),(12,23),(12,24),(12,27),(12,28),(12,29),(13,16),(13,18),(13,19),(13,20),(13,22),(13,23),(13,24),(13,27),(13,28),(13,29),(14,16),(14,18),(14,19),(14,20),(14,21),(14,22),(14,23),(14,25),(14,26),(14,27),(14,28),(14,29),(14,30),(15,16),(15,17),(15,18),(15,19),(15,20),(15,22),(15,23),(15,25),(15,26),(15,27),(15,28),(15,29),(15,30),(16,17),(16,21),(16,24),(16,25),(16,26),(16,27),(16,29),(16,30),(17,18),(17,19),(17,20),(17,21),(17,22),(17,23),(17,26),(17,27),(17,28),(17,29),(17,30),(18,21),(18,24),(18,25),(18,26),(18,30),(19,21),(19,24),(19,25),(19,26),(19,30),(20,21),(20,23),(20,24),(20,25),(20,26),(20,27),(20,28),(20,29),(20,30),(21,22),(21,23),(21,24),(21,25),(21,27),(21,28),(21,29),(22,24),(22,25),(22,26),(22,28),(22,30),(23,24),(23,25),(23,26),(23,30),(24,25),(24,26),(24,27),(24,28),(24,29),(24,30),(25,27),(25,28),(25,29),(26,27),(26,28),(26,29),(27,30),(28,30),(29,30)],31)
=> ? = 9 + 1
[[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> [[1,2,2,2],[2,3,3],[3,4],[4]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ([(6,7)],8)
=> ? = 6 + 1
[[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]]
=> [[1,2,2,3],[2,3,3],[3,4],[4]]
=> ([(0,5),(0,10),(1,16),(2,15),(3,14),(4,13),(5,12),(6,2),(6,13),(7,4),(7,14),(8,1),(9,6),(10,11),(10,12),(11,3),(11,7),(12,9),(13,15),(14,8),(15,16)],17)
=> ([(2,14),(3,11),(3,12),(3,13),(3,16),(4,10),(4,11),(4,12),(4,13),(4,16),(5,7),(5,8),(5,9),(5,10),(5,14),(6,7),(6,8),(6,9),(6,10),(6,14),(6,16),(7,11),(7,12),(7,13),(7,15),(7,16),(8,11),(8,12),(8,13),(8,15),(8,16),(9,11),(9,12),(9,13),(9,15),(9,16),(10,11),(10,12),(10,13),(10,15),(10,16),(11,14),(11,15),(12,14),(12,15),(13,14),(13,15),(14,15),(14,16),(15,16)],17)
=> ? = 7 + 1
[[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]]
=> [[1,2,2,2],[2,3,4],[3,4],[4]]
=> ([(0,7),(0,8),(1,12),(2,11),(3,10),(4,10),(4,11),(5,3),(6,1),(6,13),(7,9),(8,5),(9,2),(9,4),(10,14),(11,6),(11,14),(13,12),(14,13)],15)
=> ([(2,11),(3,10),(3,11),(4,12),(4,13),(4,14),(5,12),(5,13),(5,14),(6,8),(6,12),(6,13),(6,14),(7,9),(7,12),(7,13),(7,14),(8,9),(8,10),(8,11),(9,12),(9,13),(9,14),(10,12),(10,13),(10,14),(11,12),(11,13),(11,14)],15)
=> ? = 7 + 1
[[0,0,1,0],[0,1,-1,1],[0,0,1,0],[1,0,0,0]]
=> [[1,2,2,3],[2,3,4],[3,4],[4]]
=> ([(0,10),(0,12),(1,16),(2,17),(3,21),(4,22),(5,14),(6,13),(6,14),(7,13),(7,15),(8,7),(8,21),(9,2),(9,18),(10,11),(11,5),(11,6),(12,3),(12,8),(13,19),(14,9),(14,19),(15,22),(16,20),(17,20),(18,16),(18,17),(19,18),(21,4),(21,15),(22,1)],23)
=> ([(2,5),(2,22),(3,15),(3,16),(3,18),(3,19),(3,20),(3,21),(3,22),(4,8),(4,9),(4,12),(4,13),(4,14),(4,17),(4,22),(5,15),(5,16),(5,18),(5,19),(5,20),(5,21),(6,15),(6,16),(6,17),(6,18),(6,19),(6,20),(6,21),(6,22),(7,8),(7,9),(7,12),(7,13),(7,14),(7,17),(7,21),(7,22),(8,11),(8,15),(8,16),(8,18),(8,19),(8,20),(8,21),(9,11),(9,15),(9,16),(9,18),(9,19),(9,20),(9,21),(10,12),(10,14),(10,15),(10,16),(10,17),(10,18),(10,19),(10,20),(10,21),(10,22),(11,12),(11,13),(11,14),(11,17),(11,18),(11,20),(11,21),(11,22),(12,15),(12,16),(12,18),(12,19),(12,20),(12,21),(13,14),(13,15),(13,16),(13,18),(13,19),(13,20),(13,21),(14,15),(14,16),(14,18),(14,19),(14,20),(14,21),(15,17),(15,22),(16,17),(16,22),(17,18),(17,19),(17,20),(17,21),(18,22),(19,20),(19,22),(20,22),(21,22)],23)
=> ? = 8 + 1
[[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
=> [[1,2,2,4],[2,3,4],[3,4],[4]]
=> ([(0,14),(0,15),(1,19),(2,18),(3,29),(4,30),(5,22),(6,23),(7,24),(7,25),(8,9),(9,7),(9,18),(9,19),(10,5),(11,6),(12,2),(12,29),(13,1),(13,30),(14,16),(14,28),(15,17),(15,28),(16,3),(16,12),(17,4),(17,13),(18,24),(18,27),(19,25),(19,27),(20,26),(21,26),(22,20),(23,21),(24,22),(24,31),(25,23),(25,31),(27,31),(28,8),(29,10),(30,11),(31,20),(31,21)],32)
=> ([(2,3),(2,9),(2,27),(2,28),(2,29),(2,31),(3,8),(3,24),(3,25),(3,26),(3,30),(4,5),(4,18),(4,20),(4,23),(4,24),(4,25),(4,26),(4,30),(5,17),(5,19),(5,22),(5,27),(5,28),(5,29),(5,31),(6,8),(6,9),(6,24),(6,25),(6,26),(6,27),(6,28),(6,29),(6,30),(6,31),(7,8),(7,9),(7,13),(7,14),(7,21),(7,24),(7,25),(7,26),(7,27),(7,28),(7,29),(7,30),(7,31),(8,9),(8,13),(8,15),(8,17),(8,19),(8,22),(8,27),(8,28),(8,29),(8,31),(9,14),(9,16),(9,18),(9,20),(9,23),(9,24),(9,25),(9,26),(9,30),(10,17),(10,18),(10,19),(10,20),(10,22),(10,23),(10,24),(10,25),(10,26),(10,27),(10,28),(10,29),(10,30),(10,31),(11,17),(11,18),(11,19),(11,20),(11,22),(11,23),(11,24),(11,25),(11,26),(11,27),(11,28),(11,29),(11,30),(11,31),(12,17),(12,18),(12,19),(12,20),(12,22),(12,23),(12,24),(12,25),(12,26),(12,27),(12,28),(12,29),(12,30),(12,31),(13,14),(13,16),(13,18),(13,20),(13,23),(13,24),(13,25),(13,26),(13,27),(13,28),(13,29),(13,30),(13,31),(14,15),(14,17),(14,19),(14,22),(14,24),(14,25),(14,26),(14,27),(14,28),(14,29),(14,30),(14,31),(15,16),(15,18),(15,20),(15,21),(15,23),(15,24),(15,25),(15,26),(15,27),(15,28),(15,29),(15,30),(15,31),(16,17),(16,19),(16,21),(16,22),(16,24),(16,25),(16,26),(16,27),(16,28),(16,29),(16,30),(16,31),(17,18),(17,20),(17,21),(17,23),(17,24),(17,25),(17,26),(17,30),(18,19),(18,21),(18,22),(18,27),(18,28),(18,29),(18,31),(19,20),(19,21),(19,23),(19,24),(19,25),(19,26),(19,30),(19,31),(20,21),(20,22),(20,27),(20,28),(20,29),(20,30),(20,31),(21,22),(21,23),(21,24),(21,25),(21,26),(21,27),(21,28),(21,29),(21,30),(21,31),(22,23),(22,24),(22,25),(22,26),(22,27),(22,28),(22,29),(22,30),(22,31),(23,24),(23,25),(23,26),(23,27),(23,28),(23,29),(23,30),(23,31),(24,27),(24,28),(24,29),(24,31),(25,27),(25,28),(25,29),(25,31),(26,27),(26,28),(26,29),(26,31),(27,30),(28,30),(29,30),(30,31)],32)
=> ? = 9 + 1
[[0,0,1,0],[0,0,0,1],[0,1,0,0],[1,0,0,0]]
=> [[1,2,3,3],[2,3,4],[3,4],[4]]
=> ([(0,13),(0,15),(1,17),(2,16),(2,17),(3,19),(4,16),(4,18),(5,22),(6,21),(7,20),(8,23),(8,29),(9,4),(9,28),(10,3),(10,28),(11,6),(12,7),(12,24),(13,14),(14,1),(14,2),(15,9),(15,10),(16,25),(17,12),(17,25),(18,26),(18,29),(19,23),(19,26),(20,27),(22,27),(23,30),(24,20),(24,22),(25,24),(26,30),(27,21),(28,8),(28,18),(28,19),(29,5),(29,30),(30,11)],31)
=> ([(2,11),(2,12),(2,13),(2,21),(2,25),(2,26),(2,30),(3,18),(3,19),(3,22),(3,23),(3,27),(3,28),(3,29),(4,10),(4,11),(4,12),(4,13),(4,21),(4,25),(4,26),(4,29),(4,30),(5,8),(5,18),(5,19),(5,22),(5,23),(5,27),(5,28),(5,29),(5,30),(6,7),(6,9),(6,10),(6,16),(6,18),(6,19),(6,20),(6,22),(6,23),(6,27),(6,28),(6,29),(6,30),(7,8),(7,17),(7,18),(7,19),(7,22),(7,23),(7,25),(7,26),(7,27),(7,28),(7,29),(7,30),(8,9),(8,10),(8,16),(8,18),(8,19),(8,20),(8,22),(8,23),(8,27),(8,28),(8,29),(9,11),(9,12),(9,13),(9,14),(9,15),(9,17),(9,21),(9,24),(9,25),(9,26),(9,29),(9,30),(10,11),(10,12),(10,13),(10,14),(10,15),(10,17),(10,21),(10,24),(10,25),(10,26),(10,30),(11,16),(11,18),(11,19),(11,20),(11,22),(11,23),(11,24),(11,27),(11,28),(11,29),(12,16),(12,18),(12,19),(12,20),(12,22),(12,23),(12,24),(12,27),(12,28),(12,29),(13,16),(13,18),(13,19),(13,20),(13,22),(13,23),(13,24),(13,27),(13,28),(13,29),(14,16),(14,18),(14,19),(14,20),(14,21),(14,22),(14,23),(14,25),(14,26),(14,27),(14,28),(14,29),(14,30),(15,16),(15,17),(15,18),(15,19),(15,20),(15,22),(15,23),(15,25),(15,26),(15,27),(15,28),(15,29),(15,30),(16,17),(16,21),(16,24),(16,25),(16,26),(16,27),(16,29),(16,30),(17,18),(17,19),(17,20),(17,21),(17,22),(17,23),(17,26),(17,27),(17,28),(17,29),(17,30),(18,21),(18,24),(18,25),(18,26),(18,30),(19,21),(19,24),(19,25),(19,26),(19,30),(20,21),(20,23),(20,24),(20,25),(20,26),(20,27),(20,28),(20,29),(20,30),(21,22),(21,23),(21,24),(21,25),(21,27),(21,28),(21,29),(22,24),(22,25),(22,26),(22,28),(22,30),(23,24),(23,25),(23,26),(23,30),(24,25),(24,26),(24,27),(24,28),(24,29),(24,30),(25,27),(25,28),(25,29),(26,27),(26,28),(26,29),(27,30),(28,30),(29,30)],31)
=> ? = 9 + 1
[[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> [[1,2,3,4],[2,3,4],[3,4],[4]]
=> ([(0,5),(0,16),(0,17),(1,23),(2,19),(3,11),(3,21),(4,10),(4,20),(5,12),(5,13),(6,50),(7,51),(8,24),(8,58),(9,25),(9,59),(10,14),(10,52),(11,15),(11,53),(12,26),(12,60),(13,27),(13,60),(14,54),(15,55),(16,4),(16,29),(17,3),(17,29),(18,48),(18,49),(19,32),(19,33),(20,46),(20,52),(21,47),(21,53),(22,34),(22,35),(23,18),(23,54),(23,55),(24,40),(24,42),(25,41),(25,43),(26,46),(26,56),(27,47),(27,57),(28,63),(29,1),(30,62),(31,61),(32,61),(33,61),(34,6),(34,62),(35,7),(35,62),(36,58),(37,59),(38,32),(39,33),(40,44),(41,45),(42,38),(43,39),(44,31),(45,31),(46,36),(47,37),(48,40),(48,63),(49,41),(49,63),(50,38),(51,39),(52,8),(52,36),(53,9),(53,37),(54,28),(54,48),(55,28),(55,49),(56,30),(56,34),(57,30),(57,35),(58,42),(58,50),(59,43),(59,51),(60,22),(60,56),(60,57),(62,2),(63,44),(63,45)],64)
=> ([(2,3),(2,6),(2,7),(2,10),(2,11),(2,14),(2,15),(2,18),(2,19),(2,21),(2,23),(2,26),(2,27),(2,31),(2,32),(2,33),(2,36),(2,37),(2,40),(2,41),(2,44),(2,45),(2,48),(2,49),(2,52),(2,53),(2,55),(2,58),(2,59),(2,62),(2,63),(3,4),(3,5),(3,8),(3,9),(3,12),(3,13),(3,16),(3,17),(3,20),(3,22),(3,24),(3,25),(3,28),(3,29),(3,30),(3,34),(3,35),(3,38),(3,39),(3,42),(3,43),(3,46),(3,47),(3,50),(3,51),(3,54),(3,56),(3,57),(3,60),(3,61),(4,5),(4,7),(4,10),(4,11),(4,13),(4,15),(4,17),(4,19),(4,21),(4,23),(4,26),(4,27),(4,31),(4,32),(4,33),(4,36),(4,37),(4,40),(4,41),(4,43),(4,45),(4,47),(4,49),(4,51),(4,53),(4,55),(4,57),(4,59),(4,61),(4,62),(4,63),(5,6),(5,10),(5,11),(5,12),(5,14),(5,16),(5,18),(5,21),(5,23),(5,26),(5,27),(5,31),(5,32),(5,33),(5,36),(5,37),(5,40),(5,41),(5,42),(5,44),(5,46),(5,48),(5,50),(5,52),(5,55),(5,56),(5,58),(5,60),(5,62),(5,63),(6,7),(6,8),(6,9),(6,13),(6,15),(6,17),(6,19),(6,20),(6,22),(6,24),(6,25),(6,28),(6,29),(6,30),(6,34),(6,35),(6,38),(6,39),(6,43),(6,45),(6,47),(6,49),(6,51),(6,53),(6,54),(6,57),(6,59),(6,60),(6,61),(6,62),(7,8),(7,9),(7,12),(7,14),(7,16),(7,18),(7,20),(7,22),(7,24),(7,25),(7,28),(7,29),(7,30),(7,34),(7,35),(7,38),(7,39),(7,42),(7,44),(7,46),(7,48),(7,50),(7,52),(7,54),(7,56),(7,58),(7,60),(7,61),(7,63),(8,9),(8,10),(8,11),(8,14),(8,15),(8,18),(8,19),(8,21),(8,23),(8,25),(8,26),(8,27),(8,31),(8,32),(8,33),(8,36),(8,37),(8,40),(8,41),(8,42),(8,44),(8,45),(8,46),(8,48),(8,49),(8,50),(8,52),(8,53),(8,55),(8,56),(8,58),(8,59),(8,62),(8,63),(9,10),(9,11),(9,14),(9,15),(9,18),(9,19),(9,21),(9,23),(9,24),(9,26),(9,27),(9,31),(9,32),(9,33),(9,36),(9,37),(9,40),(9,41),(9,43),(9,44),(9,45),(9,47),(9,48),(9,49),(9,51),(9,52),(9,53),(9,55),(9,57),(9,58),(9,59),(9,62),(9,63),(10,11),(10,12),(10,13),(10,16),(10,17),(10,20),(10,22),(10,24),(10,25),(10,27),(10,28),(10,29),(10,30),(10,34),(10,35),(10,38),(10,39),(10,42),(10,43),(10,44),(10,46),(10,47),(10,48),(10,50),(10,51),(10,52),(10,54),(10,56),(10,57),(10,58),(10,60),(10,61),(11,12),(11,13),(11,16),(11,17),(11,20),(11,22),(11,24),(11,25),(11,26),(11,28),(11,29),(11,30),(11,34),(11,35),(11,38),(11,39),(11,42),(11,43),(11,45),(11,46),(11,47),(11,49),(11,50),(11,51),(11,53),(11,54),(11,56),(11,57),(11,59),(11,60),(11,61),(12,13),(12,15),(12,17),(12,19),(12,21),(12,23),(12,24),(12,26),(12,27),(12,28),(12,29),(12,30),(12,31),(12,32),(12,33),(12,34),(12,36),(12,37),(12,38),(12,40),(12,41),(12,43),(12,45),(12,47),(12,49),(12,51),(12,53),(12,54),(12,55),(12,57),(12,59),(12,61),(12,62),(12,63),(13,14),(13,16),(13,18),(13,21),(13,23),(13,25),(13,26),(13,27),(13,28),(13,29),(13,30),(13,31),(13,32),(13,33),(13,35),(13,36),(13,37),(13,39),(13,40),(13,41),(13,42),(13,44),(13,46),(13,48),(13,50),(13,52),(13,54),(13,55),(13,56),(13,58),(13,60),(13,62),(13,63),(14,15),(14,17),(14,19),(14,20),(14,22),(14,24),(14,25),(14,26),(14,28),(14,29),(14,30),(14,31),(14,32),(14,33),(14,34),(14,35),(14,36),(14,38),(14,39),(14,40),(14,43),(14,45),(14,47),(14,49),(14,51),(14,53),(14,54),(14,55),(14,57),(14,59),(14,60),(14,61),(14,62),(15,16),(15,18),(15,20),(15,22),(15,24),(15,25),(15,27),(15,28),(15,29),(15,30),(15,31),(15,32),(15,33),(15,34),(15,35),(15,37),(15,38),(15,39),(15,41),(15,42),(15,44),(15,46),(15,48),(15,50),(15,52),(15,54),(15,55),(15,56),(15,58),(15,60),(15,61),(15,63),(16,17),(16,19),(16,21),(16,23),(16,24),(16,26),(16,27),(16,28),(16,29),(16,30),(16,31),(16,32),(16,33),(16,34),(16,36),(16,37),(16,38),(16,40),(16,41),(16,43),(16,45),(16,46),(16,47),(16,49),(16,51),(16,53),(16,54),(16,55),(16,57),(16,58),(16,59),(16,61),(16,62),(16,63),(17,18),(17,21),(17,23),(17,25),(17,26),(17,27),(17,28),(17,29),(17,30),(17,31),(17,32),(17,33),(17,35),(17,36),(17,37),(17,39),(17,40),(17,41),(17,42),(17,44),(17,46),(17,47),(17,48),(17,50),(17,52),(17,54),(17,55),(17,56),(17,58),(17,59),(17,60),(17,62),(17,63),(18,19),(18,20),(18,22),(18,24),(18,25),(18,26),(18,28),(18,29),(18,30),(18,31),(18,32),(18,33),(18,34),(18,35),(18,36),(18,38),(18,39),(18,40),(18,43),(18,45),(18,47),(18,48),(18,49),(18,51),(18,53),(18,54),(18,55),(18,56),(18,57),(18,59),(18,60),(18,61),(18,62),(19,20),(19,22),(19,24),(19,25),(19,27),(19,28),(19,29),(19,30),(19,31),(19,32),(19,33),(19,34),(19,35),(19,37),(19,38),(19,39),(19,41),(19,42),(19,44),(19,46),(19,48),(19,49),(19,50),(19,52),(19,54),(19,55),(19,56),(19,57),(19,58),(19,60),(19,61),(19,63),(20,21),(20,23),(20,24),(20,25),(20,26),(20,27),(20,31),(20,32),(20,33),(20,36),(20,37),(20,40),(20,41),(20,42),(20,43),(20,44),(20,45),(20,46),(20,47),(20,48),(20,49),(20,50),(20,51),(20,52),(20,53),(20,55),(20,56),(20,57),(20,58),(20,59),(20,62),(20,63),(21,22),(21,24),(21,25),(21,26),(21,27),(21,28),(21,29),(21,30),(21,34),(21,35),(21,38),(21,39),(21,42),(21,43),(21,44),(21,45),(21,46),(21,47),(21,48),(21,49),(21,50),(21,51),(21,52),(21,53),(21,54),(21,56),(21,57),(21,58),(21,59),(21,60),(21,61),(22,23),(22,24),(22,25),(22,26),(22,27),(22,31),(22,32),(22,33),(22,34),(22,35),(22,36),(22,37),(22,40),(22,41),(22,42),(22,43),(22,44),(22,45),(22,46),(22,47),(22,48),(22,49),(22,50),(22,51),(22,52),(22,53),(22,54),(22,55),(22,56),(22,57),(22,58),(22,59),(22,62),(22,63),(23,24),(23,25),(23,26),(23,27),(23,28),(23,29),(23,30),(23,34),(23,35),(23,36),(23,37),(23,38),(23,39),(23,42),(23,43),(23,44),(23,45),(23,46),(23,47),(23,48),(23,49),(23,50),(23,51),(23,52),(23,53),(23,54),(23,55),(23,56),(23,57),(23,58),(23,59),(23,60),(23,61),(24,25),(24,26),(24,27),(24,31),(24,32),(24,33),(24,35),(24,36),(24,37),(24,39),(24,40),(24,41),(24,42),(24,44),(24,45),(24,46),(24,48),(24,49),(24,50),(24,52),(24,53),(24,55),(24,56),(24,58),(24,59),(24,60),(24,62),(24,63),(25,26),(25,27),(25,31),(25,32),(25,33),(25,34),(25,36),(25,37),(25,38),(25,40),(25,41),(25,43),(25,44),(25,45),(25,47),(25,48),(25,49),(25,51),(25,52),(25,53),(25,55),(25,57),(25,58),(25,59),(25,61),(25,62),(25,63),(26,27),(26,28),(26,29),(26,30),(26,34),(26,35),(26,37),(26,38),(26,39),(26,41),(26,42),(26,43),(26,44),(26,46),(26,47),(26,48),(26,50),(26,51),(26,52),(26,54),(26,56),(26,57),(26,58),(26,60),(26,61),(26,63),(27,28),(27,29),(27,30),(27,34),(27,35),(27,36),(27,38),(27,39),(27,40),(27,42),(27,43),(27,45),(27,46),(27,47),(27,49),(27,50),(27,51),(27,53),(27,54),(27,56),(27,57),(27,59),(27,60),(27,61),(27,62),(28,31),(28,32),(28,33),(28,36),(28,37),(28,40),(28,41),(28,42),(28,43),(28,44),(28,45),(28,46),(28,47),(28,48),(28,49),(28,50),(28,51),(28,52),(28,53),(28,55),(28,56),(28,57),(28,58),(28,59),(28,60),(28,61),(28,62),(28,63),(29,31),(29,32),(29,33),(29,36),(29,37),(29,40),(29,41),(29,42),(29,43),(29,44),(29,45),(29,46),(29,47),(29,48),(29,49),(29,50),(29,51),(29,52),(29,53),(29,55),(29,56),(29,57),(29,58),(29,59),(29,60),(29,61),(29,62),(29,63),(30,31),(30,32),(30,33),(30,36),(30,37),(30,40),(30,41),(30,42),(30,43),(30,44),(30,45),(30,46),(30,47),(30,48),(30,49),(30,50),(30,51),(30,52),(30,53),(30,55),(30,56),(30,57),(30,58),(30,59),(30,60),(30,61),(30,62),(30,63),(31,34),(31,35),(31,38),(31,39),(31,42),(31,43),(31,44),(31,45),(31,46),(31,47),(31,48),(31,49),(31,50),(31,51),(31,52),(31,53),(31,54),(31,56),(31,57),(31,58),(31,59),(31,60),(31,61),(31,62),(31,63),(32,34),(32,35),(32,38),(32,39),(32,42),(32,43),(32,44),(32,45),(32,46),(32,47),(32,48),(32,49),(32,50),(32,51),(32,52),(32,53),(32,54),(32,56),(32,57),(32,58),(32,59),(32,60),(32,61),(32,62),(32,63),(33,34),(33,35),(33,38),(33,39),(33,42),(33,43),(33,44),(33,45),(33,46),(33,47),(33,48),(33,49),(33,50),(33,51),(33,52),(33,53),(33,54),(33,56),(33,57),(33,58),(33,59),(33,60),(33,61),(33,62),(33,63),(34,35),(34,36),(34,37),(34,39),(34,40),(34,41),(34,42),(34,43),(34,44),(34,45),(34,46),(34,47),(34,48),(34,49),(34,50),(34,51),(34,52),(34,53),(34,55),(34,56),(34,57),(34,58),(34,59),(34,60),(34,62),(34,63),(35,36),(35,37),(35,38),(35,40),(35,41),(35,42),(35,43),(35,44),(35,45),(35,46),(35,47),(35,48),(35,49),(35,50),(35,51),(35,52),(35,53),(35,55),(35,56),(35,57),(35,58),(35,59),(35,61),(35,62),(35,63),(36,37),(36,38),(36,39),(36,41),(36,42),(36,43),(36,44),(36,45),(36,46),(36,47),(36,48),(36,49),(36,50),(36,51),(36,52),(36,53),(36,54),(36,56),(36,57),(36,58),(36,59),(36,60),(36,61),(36,63),(37,38),(37,39),(37,40),(37,42),(37,43),(37,44),(37,45),(37,46),(37,47),(37,48),(37,49),(37,50),(37,51),(37,52),(37,53),(37,54),(37,56),(37,57),(37,58),(37,59),(37,60),(37,61),(37,62),(38,39),(38,40),(38,41),(38,42),(38,43),(38,44),(38,45),(38,46),(38,47),(38,48),(38,49),(38,50),(38,51),(38,52),(38,53),(38,54),(38,55),(38,56),(38,57),(38,58),(38,59),(38,60),(38,62),(38,63),(39,40),(39,41),(39,42),(39,43),(39,44),(39,45),(39,46),(39,47),(39,48),(39,49),(39,50),(39,51),(39,52),(39,53),(39,54),(39,55),(39,56),(39,57),(39,58),(39,59),(39,61),(39,62),(39,63),(40,41),(40,42),(40,43),(40,44),(40,45),(40,46),(40,47),(40,48),(40,49),(40,50),(40,51),(40,52),(40,53),(40,54),(40,55),(40,56),(40,57),(40,58),(40,59),(40,60),(40,61),(40,63),(41,42),(41,43),(41,44),(41,45),(41,46),(41,47),(41,48),(41,49),(41,50),(41,51),(41,52),(41,53),(41,54),(41,55),(41,56),(41,57),(41,58),(41,59),(41,60),(41,61),(41,62),(42,43),(42,45),(42,47),(42,49),(42,51),(42,53),(42,54),(42,55),(42,57),(42,58),(42,59),(42,60),(42,61),(42,62),(42,63),(43,44),(43,46),(43,48),(43,50),(43,52),(43,54),(43,55),(43,56),(43,58),(43,59),(43,60),(43,61),(43,62),(43,63),(44,45),(44,47),(44,49),(44,51),(44,53),(44,54),(44,55),(44,56),(44,57),(44,59),(44,60),(44,61),(44,62),(44,63),(45,46),(45,48),(45,50),(45,52),(45,54),(45,55),(45,56),(45,57),(45,58),(45,60),(45,61),(45,62),(45,63),(46,47),(46,49),(46,51),(46,53),(46,54),(46,55),(46,57),(46,59),(46,60),(46,61),(46,62),(46,63),(47,48),(47,50),(47,52),(47,54),(47,55),(47,56),(47,58),(47,60),(47,61),(47,62),(47,63),(48,49),(48,51),(48,53),(48,54),(48,55),(48,57),(48,59),(48,60),(48,61),(48,62),(48,63),(49,50),(49,52),(49,54),(49,55),(49,56),(49,58),(49,60),(49,61),(49,62),(49,63),(50,51),(50,52),(50,53),(50,54),(50,55),(50,57),(50,58),(50,59),(50,60),(50,61),(50,62),(50,63),(51,52),(51,53),(51,54),(51,55),(51,56),(51,58),(51,59),(51,60),(51,61),(51,62),(51,63),(52,53),(52,54),(52,55),(52,56),(52,57),(52,59),(52,60),(52,61),(52,62),(52,63),(53,54),(53,55),(53,56),(53,57),(53,58),(53,60),(53,61),(53,62),(53,63),(54,55),(54,56),(54,57),(54,58),(54,59),(54,60),(54,61),(54,62),(54,63),(55,56),(55,57),(55,58),(55,59),(55,60),(55,61),(55,62),(55,63),(56,57),(56,58),(56,59),(56,60),(56,61),(56,62),(56,63),(57,58),(57,59),(57,60),(57,61),(57,62),(57,63),(58,59),(58,60),(58,61),(58,62),(58,63),(59,60),(59,61),(59,62),(59,63),(60,61),(60,62),(60,63),(61,62),(61,63),(62,63)],64)
=> ? = 10 + 1
[[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,2,2],[3,3,3],[4,4],[5]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,2],[2,2,2,2],[3,3,3],[4,4],[5]]
=> ([(0,1)],2)
=> ([],2)
=> 2 = 1 + 1
[[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,2,3],[3,3,3],[4,4],[5]]
=> ([(0,1)],2)
=> ([],2)
=> 2 = 1 + 1
[[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,2],[2,2,2,3],[3,3,3],[4,4],[5]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 3 = 2 + 1
[[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,3],[2,2,2,3],[3,3,3],[4,4],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 4 = 3 + 1
[[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,2,2],[2,2,2,3],[3,3,3],[4,4],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 4 = 3 + 1
[[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,2,3],[2,2,2,3],[3,3,3],[4,4],[5]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ([(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
=> 5 = 4 + 1
[[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,2,2],[3,3,4],[4,4],[5]]
=> ([(0,1)],2)
=> ([],2)
=> 2 = 1 + 1
[[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,2],[2,2,2,2],[3,3,4],[4,4],[5]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 3 = 2 + 1
[[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,2,3],[3,3,4],[4,4],[5]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 3 = 2 + 1
[[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,2],[2,2,2,3],[3,3,4],[4,4],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 4 = 3 + 1
[[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,3],[2,2,2,3],[3,3,4],[4,4],[5]]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(4,5)],6)
=> 5 = 4 + 1
[[0,1,0,0,0],[0,0,1,0,0],[1,0,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,2,2],[2,2,2,3],[3,3,4],[4,4],[5]]
=> ?
=> ?
=> ? = 4 + 1
[[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,4],[2,2,2,4],[3,3,4],[4,4],[5]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ([(6,7)],8)
=> ? = 6 + 1
[[0,0,1,0,0],[0,1,-1,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,2,3],[2,2,2,4],[3,3,4],[4,4],[5]]
=> ?
=> ?
=> ? = 6 + 1
[[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,2,4],[2,2,2,4],[3,3,4],[4,4],[5]]
=> ([(0,7),(0,8),(1,12),(2,11),(3,10),(4,10),(4,11),(5,3),(6,1),(6,13),(7,9),(8,5),(9,2),(9,4),(10,14),(11,6),(11,14),(13,12),(14,13)],15)
=> ([(2,11),(3,10),(3,11),(4,12),(4,13),(4,14),(5,12),(5,13),(5,14),(6,8),(6,12),(6,13),(6,14),(7,9),(7,12),(7,13),(7,14),(8,9),(8,10),(8,11),(9,12),(9,13),(9,14),(10,12),(10,13),(10,14),(11,12),(11,13),(11,14)],15)
=> ? = 7 + 1
[[0,1,0,0,0],[1,-1,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,2],[2,2,3,3],[3,3,4],[4,4],[5]]
=> ?
=> ?
=> ? = 4 + 1
[[0,1,0,0,0],[0,0,1,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,2,2],[2,2,3,3],[3,3,4],[4,4],[5]]
=> ?
=> ?
=> ? = 5 + 1
[[0,0,1,0,0],[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,2,3],[2,2,3,3],[3,3,4],[4,4],[5]]
=> ?
=> ?
=> ? = 6 + 1
[[0,0,1,0,0],[1,0,-1,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,3],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ?
=> ?
=> ? = 6 + 1
[[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,4],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ([(0,5),(0,10),(1,16),(2,15),(3,14),(4,13),(5,12),(6,2),(6,13),(7,4),(7,14),(8,1),(9,6),(10,11),(10,12),(11,3),(11,7),(12,9),(13,15),(14,8),(15,16)],17)
=> ([(2,14),(3,11),(3,12),(3,13),(3,16),(4,10),(4,11),(4,12),(4,13),(4,16),(5,7),(5,8),(5,9),(5,10),(5,14),(6,7),(6,8),(6,9),(6,10),(6,14),(6,16),(7,11),(7,12),(7,13),(7,15),(7,16),(8,11),(8,12),(8,13),(8,15),(8,16),(9,11),(9,12),(9,13),(9,15),(9,16),(10,11),(10,12),(10,13),(10,15),(10,16),(11,14),(11,15),(12,14),(12,15),(13,14),(13,15),(14,15),(14,16),(15,16)],17)
=> ? = 7 + 1
[[0,1,0,0,0],[0,0,0,1,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,2,2],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ?
=> ?
=> ? = 6 + 1
[[0,0,0,1,0],[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,2,4],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ?
=> ?
=> ? = 8 + 1
[[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,3,3],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ([(0,8),(2,11),(2,12),(3,10),(4,9),(5,4),(5,14),(6,3),(6,14),(7,1),(8,5),(8,6),(9,11),(9,13),(10,12),(10,13),(11,15),(12,15),(13,15),(14,2),(14,9),(14,10),(15,7)],16)
=> ([(5,14),(5,15),(6,7),(6,15),(7,14),(8,11),(8,12),(8,13),(9,10),(9,12),(9,13),(9,14),(10,11),(10,13),(10,15),(11,12),(11,14),(12,15),(13,14),(13,15),(14,15)],16)
=> ? = 8 + 1
[[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,3,4],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ([(0,13),(0,15),(1,17),(2,16),(2,17),(3,19),(4,16),(4,18),(5,22),(6,21),(7,20),(8,23),(8,29),(9,4),(9,28),(10,3),(10,28),(11,6),(12,7),(12,24),(13,14),(14,1),(14,2),(15,9),(15,10),(16,25),(17,12),(17,25),(18,26),(18,29),(19,23),(19,26),(20,27),(22,27),(23,30),(24,20),(24,22),(25,24),(26,30),(27,21),(28,8),(28,18),(28,19),(29,5),(29,30),(30,11)],31)
=> ([(2,11),(2,12),(2,13),(2,21),(2,25),(2,26),(2,30),(3,18),(3,19),(3,22),(3,23),(3,27),(3,28),(3,29),(4,10),(4,11),(4,12),(4,13),(4,21),(4,25),(4,26),(4,29),(4,30),(5,8),(5,18),(5,19),(5,22),(5,23),(5,27),(5,28),(5,29),(5,30),(6,7),(6,9),(6,10),(6,16),(6,18),(6,19),(6,20),(6,22),(6,23),(6,27),(6,28),(6,29),(6,30),(7,8),(7,17),(7,18),(7,19),(7,22),(7,23),(7,25),(7,26),(7,27),(7,28),(7,29),(7,30),(8,9),(8,10),(8,16),(8,18),(8,19),(8,20),(8,22),(8,23),(8,27),(8,28),(8,29),(9,11),(9,12),(9,13),(9,14),(9,15),(9,17),(9,21),(9,24),(9,25),(9,26),(9,29),(9,30),(10,11),(10,12),(10,13),(10,14),(10,15),(10,17),(10,21),(10,24),(10,25),(10,26),(10,30),(11,16),(11,18),(11,19),(11,20),(11,22),(11,23),(11,24),(11,27),(11,28),(11,29),(12,16),(12,18),(12,19),(12,20),(12,22),(12,23),(12,24),(12,27),(12,28),(12,29),(13,16),(13,18),(13,19),(13,20),(13,22),(13,23),(13,24),(13,27),(13,28),(13,29),(14,16),(14,18),(14,19),(14,20),(14,21),(14,22),(14,23),(14,25),(14,26),(14,27),(14,28),(14,29),(14,30),(15,16),(15,17),(15,18),(15,19),(15,20),(15,22),(15,23),(15,25),(15,26),(15,27),(15,28),(15,29),(15,30),(16,17),(16,21),(16,24),(16,25),(16,26),(16,27),(16,29),(16,30),(17,18),(17,19),(17,20),(17,21),(17,22),(17,23),(17,26),(17,27),(17,28),(17,29),(17,30),(18,21),(18,24),(18,25),(18,26),(18,30),(19,21),(19,24),(19,25),(19,26),(19,30),(20,21),(20,23),(20,24),(20,25),(20,26),(20,27),(20,28),(20,29),(20,30),(21,22),(21,23),(21,24),(21,25),(21,27),(21,28),(21,29),(22,24),(22,25),(22,26),(22,28),(22,30),(23,24),(23,25),(23,26),(23,30),(24,25),(24,26),(24,27),(24,28),(24,29),(24,30),(25,27),(25,28),(25,29),(26,27),(26,28),(26,29),(27,30),(28,30),(29,30)],31)
=> ? = 9 + 1
[[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,0,0,1]]
=> [[1,1,2,2,2],[2,2,3,3],[3,3,4],[4,4],[5]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ([(6,7)],8)
=> ? = 6 + 1
[[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,0,0,1]]
=> [[1,1,2,2,3],[2,2,3,3],[3,3,4],[4,4],[5]]
=> ([(0,5),(0,10),(1,16),(2,15),(3,14),(4,13),(5,12),(6,2),(6,13),(7,4),(7,14),(8,1),(9,6),(10,11),(10,12),(11,3),(11,7),(12,9),(13,15),(14,8),(15,16)],17)
=> ([(2,14),(3,11),(3,12),(3,13),(3,16),(4,10),(4,11),(4,12),(4,13),(4,16),(5,7),(5,8),(5,9),(5,10),(5,14),(6,7),(6,8),(6,9),(6,10),(6,14),(6,16),(7,11),(7,12),(7,13),(7,15),(7,16),(8,11),(8,12),(8,13),(8,15),(8,16),(9,11),(9,12),(9,13),(9,15),(9,16),(10,11),(10,12),(10,13),(10,15),(10,16),(11,14),(11,15),(12,14),(12,15),(13,14),(13,15),(14,15),(14,16),(15,16)],17)
=> ? = 7 + 1
[[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1]]
=> [[1,1,2,2,2],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ([(0,7),(0,8),(1,12),(2,11),(3,10),(4,10),(4,11),(5,3),(6,1),(6,13),(7,9),(8,5),(9,2),(9,4),(10,14),(11,6),(11,14),(13,12),(14,13)],15)
=> ([(2,11),(3,10),(3,11),(4,12),(4,13),(4,14),(5,12),(5,13),(5,14),(6,8),(6,12),(6,13),(6,14),(7,9),(7,12),(7,13),(7,14),(8,9),(8,10),(8,11),(9,12),(9,13),(9,14),(10,12),(10,13),(10,14),(11,12),(11,13),(11,14)],15)
=> ? = 7 + 1
[[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1]]
=> [[1,1,2,2,3],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ?
=> ?
=> ? = 8 + 1
[[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1]]
=> [[1,1,2,2,4],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ([(0,14),(0,15),(1,19),(2,18),(3,29),(4,30),(5,22),(6,23),(7,24),(7,25),(8,9),(9,7),(9,18),(9,19),(10,5),(11,6),(12,2),(12,29),(13,1),(13,30),(14,16),(14,28),(15,17),(15,28),(16,3),(16,12),(17,4),(17,13),(18,24),(18,27),(19,25),(19,27),(20,26),(21,26),(22,20),(23,21),(24,22),(24,31),(25,23),(25,31),(27,31),(28,8),(29,10),(30,11),(31,20),(31,21)],32)
=> ([(2,3),(2,9),(2,27),(2,28),(2,29),(2,31),(3,8),(3,24),(3,25),(3,26),(3,30),(4,5),(4,18),(4,20),(4,23),(4,24),(4,25),(4,26),(4,30),(5,17),(5,19),(5,22),(5,27),(5,28),(5,29),(5,31),(6,8),(6,9),(6,24),(6,25),(6,26),(6,27),(6,28),(6,29),(6,30),(6,31),(7,8),(7,9),(7,13),(7,14),(7,21),(7,24),(7,25),(7,26),(7,27),(7,28),(7,29),(7,30),(7,31),(8,9),(8,13),(8,15),(8,17),(8,19),(8,22),(8,27),(8,28),(8,29),(8,31),(9,14),(9,16),(9,18),(9,20),(9,23),(9,24),(9,25),(9,26),(9,30),(10,17),(10,18),(10,19),(10,20),(10,22),(10,23),(10,24),(10,25),(10,26),(10,27),(10,28),(10,29),(10,30),(10,31),(11,17),(11,18),(11,19),(11,20),(11,22),(11,23),(11,24),(11,25),(11,26),(11,27),(11,28),(11,29),(11,30),(11,31),(12,17),(12,18),(12,19),(12,20),(12,22),(12,23),(12,24),(12,25),(12,26),(12,27),(12,28),(12,29),(12,30),(12,31),(13,14),(13,16),(13,18),(13,20),(13,23),(13,24),(13,25),(13,26),(13,27),(13,28),(13,29),(13,30),(13,31),(14,15),(14,17),(14,19),(14,22),(14,24),(14,25),(14,26),(14,27),(14,28),(14,29),(14,30),(14,31),(15,16),(15,18),(15,20),(15,21),(15,23),(15,24),(15,25),(15,26),(15,27),(15,28),(15,29),(15,30),(15,31),(16,17),(16,19),(16,21),(16,22),(16,24),(16,25),(16,26),(16,27),(16,28),(16,29),(16,30),(16,31),(17,18),(17,20),(17,21),(17,23),(17,24),(17,25),(17,26),(17,30),(18,19),(18,21),(18,22),(18,27),(18,28),(18,29),(18,31),(19,20),(19,21),(19,23),(19,24),(19,25),(19,26),(19,30),(19,31),(20,21),(20,22),(20,27),(20,28),(20,29),(20,30),(20,31),(21,22),(21,23),(21,24),(21,25),(21,26),(21,27),(21,28),(21,29),(21,30),(21,31),(22,23),(22,24),(22,25),(22,26),(22,27),(22,28),(22,29),(22,30),(22,31),(23,24),(23,25),(23,26),(23,27),(23,28),(23,29),(23,30),(23,31),(24,27),(24,28),(24,29),(24,31),(25,27),(25,28),(25,29),(25,31),(26,27),(26,28),(26,29),(26,31),(27,30),(28,30),(29,30),(30,31)],32)
=> ? = 9 + 1
[[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1]]
=> [[1,1,2,3,3],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ([(0,13),(0,15),(1,17),(2,16),(2,17),(3,19),(4,16),(4,18),(5,22),(6,21),(7,20),(8,23),(8,29),(9,4),(9,28),(10,3),(10,28),(11,6),(12,7),(12,24),(13,14),(14,1),(14,2),(15,9),(15,10),(16,25),(17,12),(17,25),(18,26),(18,29),(19,23),(19,26),(20,27),(22,27),(23,30),(24,20),(24,22),(25,24),(26,30),(27,21),(28,8),(28,18),(28,19),(29,5),(29,30),(30,11)],31)
=> ([(2,11),(2,12),(2,13),(2,21),(2,25),(2,26),(2,30),(3,18),(3,19),(3,22),(3,23),(3,27),(3,28),(3,29),(4,10),(4,11),(4,12),(4,13),(4,21),(4,25),(4,26),(4,29),(4,30),(5,8),(5,18),(5,19),(5,22),(5,23),(5,27),(5,28),(5,29),(5,30),(6,7),(6,9),(6,10),(6,16),(6,18),(6,19),(6,20),(6,22),(6,23),(6,27),(6,28),(6,29),(6,30),(7,8),(7,17),(7,18),(7,19),(7,22),(7,23),(7,25),(7,26),(7,27),(7,28),(7,29),(7,30),(8,9),(8,10),(8,16),(8,18),(8,19),(8,20),(8,22),(8,23),(8,27),(8,28),(8,29),(9,11),(9,12),(9,13),(9,14),(9,15),(9,17),(9,21),(9,24),(9,25),(9,26),(9,29),(9,30),(10,11),(10,12),(10,13),(10,14),(10,15),(10,17),(10,21),(10,24),(10,25),(10,26),(10,30),(11,16),(11,18),(11,19),(11,20),(11,22),(11,23),(11,24),(11,27),(11,28),(11,29),(12,16),(12,18),(12,19),(12,20),(12,22),(12,23),(12,24),(12,27),(12,28),(12,29),(13,16),(13,18),(13,19),(13,20),(13,22),(13,23),(13,24),(13,27),(13,28),(13,29),(14,16),(14,18),(14,19),(14,20),(14,21),(14,22),(14,23),(14,25),(14,26),(14,27),(14,28),(14,29),(14,30),(15,16),(15,17),(15,18),(15,19),(15,20),(15,22),(15,23),(15,25),(15,26),(15,27),(15,28),(15,29),(15,30),(16,17),(16,21),(16,24),(16,25),(16,26),(16,27),(16,29),(16,30),(17,18),(17,19),(17,20),(17,21),(17,22),(17,23),(17,26),(17,27),(17,28),(17,29),(17,30),(18,21),(18,24),(18,25),(18,26),(18,30),(19,21),(19,24),(19,25),(19,26),(19,30),(20,21),(20,23),(20,24),(20,25),(20,26),(20,27),(20,28),(20,29),(20,30),(21,22),(21,23),(21,24),(21,25),(21,27),(21,28),(21,29),(22,24),(22,25),(22,26),(22,28),(22,30),(23,24),(23,25),(23,26),(23,30),(24,25),(24,26),(24,27),(24,28),(24,29),(24,30),(25,27),(25,28),(25,29),(26,27),(26,28),(26,29),(27,30),(28,30),(29,30)],31)
=> ? = 9 + 1
[[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1]]
=> [[1,1,2,3,4],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ([(0,5),(0,16),(0,17),(1,23),(2,19),(3,11),(3,21),(4,10),(4,20),(5,12),(5,13),(6,50),(7,51),(8,24),(8,58),(9,25),(9,59),(10,14),(10,52),(11,15),(11,53),(12,26),(12,60),(13,27),(13,60),(14,54),(15,55),(16,4),(16,29),(17,3),(17,29),(18,48),(18,49),(19,32),(19,33),(20,46),(20,52),(21,47),(21,53),(22,34),(22,35),(23,18),(23,54),(23,55),(24,40),(24,42),(25,41),(25,43),(26,46),(26,56),(27,47),(27,57),(28,63),(29,1),(30,62),(31,61),(32,61),(33,61),(34,6),(34,62),(35,7),(35,62),(36,58),(37,59),(38,32),(39,33),(40,44),(41,45),(42,38),(43,39),(44,31),(45,31),(46,36),(47,37),(48,40),(48,63),(49,41),(49,63),(50,38),(51,39),(52,8),(52,36),(53,9),(53,37),(54,28),(54,48),(55,28),(55,49),(56,30),(56,34),(57,30),(57,35),(58,42),(58,50),(59,43),(59,51),(60,22),(60,56),(60,57),(62,2),(63,44),(63,45)],64)
=> ([(2,3),(2,6),(2,7),(2,10),(2,11),(2,14),(2,15),(2,18),(2,19),(2,21),(2,23),(2,26),(2,27),(2,31),(2,32),(2,33),(2,36),(2,37),(2,40),(2,41),(2,44),(2,45),(2,48),(2,49),(2,52),(2,53),(2,55),(2,58),(2,59),(2,62),(2,63),(3,4),(3,5),(3,8),(3,9),(3,12),(3,13),(3,16),(3,17),(3,20),(3,22),(3,24),(3,25),(3,28),(3,29),(3,30),(3,34),(3,35),(3,38),(3,39),(3,42),(3,43),(3,46),(3,47),(3,50),(3,51),(3,54),(3,56),(3,57),(3,60),(3,61),(4,5),(4,7),(4,10),(4,11),(4,13),(4,15),(4,17),(4,19),(4,21),(4,23),(4,26),(4,27),(4,31),(4,32),(4,33),(4,36),(4,37),(4,40),(4,41),(4,43),(4,45),(4,47),(4,49),(4,51),(4,53),(4,55),(4,57),(4,59),(4,61),(4,62),(4,63),(5,6),(5,10),(5,11),(5,12),(5,14),(5,16),(5,18),(5,21),(5,23),(5,26),(5,27),(5,31),(5,32),(5,33),(5,36),(5,37),(5,40),(5,41),(5,42),(5,44),(5,46),(5,48),(5,50),(5,52),(5,55),(5,56),(5,58),(5,60),(5,62),(5,63),(6,7),(6,8),(6,9),(6,13),(6,15),(6,17),(6,19),(6,20),(6,22),(6,24),(6,25),(6,28),(6,29),(6,30),(6,34),(6,35),(6,38),(6,39),(6,43),(6,45),(6,47),(6,49),(6,51),(6,53),(6,54),(6,57),(6,59),(6,60),(6,61),(6,62),(7,8),(7,9),(7,12),(7,14),(7,16),(7,18),(7,20),(7,22),(7,24),(7,25),(7,28),(7,29),(7,30),(7,34),(7,35),(7,38),(7,39),(7,42),(7,44),(7,46),(7,48),(7,50),(7,52),(7,54),(7,56),(7,58),(7,60),(7,61),(7,63),(8,9),(8,10),(8,11),(8,14),(8,15),(8,18),(8,19),(8,21),(8,23),(8,25),(8,26),(8,27),(8,31),(8,32),(8,33),(8,36),(8,37),(8,40),(8,41),(8,42),(8,44),(8,45),(8,46),(8,48),(8,49),(8,50),(8,52),(8,53),(8,55),(8,56),(8,58),(8,59),(8,62),(8,63),(9,10),(9,11),(9,14),(9,15),(9,18),(9,19),(9,21),(9,23),(9,24),(9,26),(9,27),(9,31),(9,32),(9,33),(9,36),(9,37),(9,40),(9,41),(9,43),(9,44),(9,45),(9,47),(9,48),(9,49),(9,51),(9,52),(9,53),(9,55),(9,57),(9,58),(9,59),(9,62),(9,63),(10,11),(10,12),(10,13),(10,16),(10,17),(10,20),(10,22),(10,24),(10,25),(10,27),(10,28),(10,29),(10,30),(10,34),(10,35),(10,38),(10,39),(10,42),(10,43),(10,44),(10,46),(10,47),(10,48),(10,50),(10,51),(10,52),(10,54),(10,56),(10,57),(10,58),(10,60),(10,61),(11,12),(11,13),(11,16),(11,17),(11,20),(11,22),(11,24),(11,25),(11,26),(11,28),(11,29),(11,30),(11,34),(11,35),(11,38),(11,39),(11,42),(11,43),(11,45),(11,46),(11,47),(11,49),(11,50),(11,51),(11,53),(11,54),(11,56),(11,57),(11,59),(11,60),(11,61),(12,13),(12,15),(12,17),(12,19),(12,21),(12,23),(12,24),(12,26),(12,27),(12,28),(12,29),(12,30),(12,31),(12,32),(12,33),(12,34),(12,36),(12,37),(12,38),(12,40),(12,41),(12,43),(12,45),(12,47),(12,49),(12,51),(12,53),(12,54),(12,55),(12,57),(12,59),(12,61),(12,62),(12,63),(13,14),(13,16),(13,18),(13,21),(13,23),(13,25),(13,26),(13,27),(13,28),(13,29),(13,30),(13,31),(13,32),(13,33),(13,35),(13,36),(13,37),(13,39),(13,40),(13,41),(13,42),(13,44),(13,46),(13,48),(13,50),(13,52),(13,54),(13,55),(13,56),(13,58),(13,60),(13,62),(13,63),(14,15),(14,17),(14,19),(14,20),(14,22),(14,24),(14,25),(14,26),(14,28),(14,29),(14,30),(14,31),(14,32),(14,33),(14,34),(14,35),(14,36),(14,38),(14,39),(14,40),(14,43),(14,45),(14,47),(14,49),(14,51),(14,53),(14,54),(14,55),(14,57),(14,59),(14,60),(14,61),(14,62),(15,16),(15,18),(15,20),(15,22),(15,24),(15,25),(15,27),(15,28),(15,29),(15,30),(15,31),(15,32),(15,33),(15,34),(15,35),(15,37),(15,38),(15,39),(15,41),(15,42),(15,44),(15,46),(15,48),(15,50),(15,52),(15,54),(15,55),(15,56),(15,58),(15,60),(15,61),(15,63),(16,17),(16,19),(16,21),(16,23),(16,24),(16,26),(16,27),(16,28),(16,29),(16,30),(16,31),(16,32),(16,33),(16,34),(16,36),(16,37),(16,38),(16,40),(16,41),(16,43),(16,45),(16,46),(16,47),(16,49),(16,51),(16,53),(16,54),(16,55),(16,57),(16,58),(16,59),(16,61),(16,62),(16,63),(17,18),(17,21),(17,23),(17,25),(17,26),(17,27),(17,28),(17,29),(17,30),(17,31),(17,32),(17,33),(17,35),(17,36),(17,37),(17,39),(17,40),(17,41),(17,42),(17,44),(17,46),(17,47),(17,48),(17,50),(17,52),(17,54),(17,55),(17,56),(17,58),(17,59),(17,60),(17,62),(17,63),(18,19),(18,20),(18,22),(18,24),(18,25),(18,26),(18,28),(18,29),(18,30),(18,31),(18,32),(18,33),(18,34),(18,35),(18,36),(18,38),(18,39),(18,40),(18,43),(18,45),(18,47),(18,48),(18,49),(18,51),(18,53),(18,54),(18,55),(18,56),(18,57),(18,59),(18,60),(18,61),(18,62),(19,20),(19,22),(19,24),(19,25),(19,27),(19,28),(19,29),(19,30),(19,31),(19,32),(19,33),(19,34),(19,35),(19,37),(19,38),(19,39),(19,41),(19,42),(19,44),(19,46),(19,48),(19,49),(19,50),(19,52),(19,54),(19,55),(19,56),(19,57),(19,58),(19,60),(19,61),(19,63),(20,21),(20,23),(20,24),(20,25),(20,26),(20,27),(20,31),(20,32),(20,33),(20,36),(20,37),(20,40),(20,41),(20,42),(20,43),(20,44),(20,45),(20,46),(20,47),(20,48),(20,49),(20,50),(20,51),(20,52),(20,53),(20,55),(20,56),(20,57),(20,58),(20,59),(20,62),(20,63),(21,22),(21,24),(21,25),(21,26),(21,27),(21,28),(21,29),(21,30),(21,34),(21,35),(21,38),(21,39),(21,42),(21,43),(21,44),(21,45),(21,46),(21,47),(21,48),(21,49),(21,50),(21,51),(21,52),(21,53),(21,54),(21,56),(21,57),(21,58),(21,59),(21,60),(21,61),(22,23),(22,24),(22,25),(22,26),(22,27),(22,31),(22,32),(22,33),(22,34),(22,35),(22,36),(22,37),(22,40),(22,41),(22,42),(22,43),(22,44),(22,45),(22,46),(22,47),(22,48),(22,49),(22,50),(22,51),(22,52),(22,53),(22,54),(22,55),(22,56),(22,57),(22,58),(22,59),(22,62),(22,63),(23,24),(23,25),(23,26),(23,27),(23,28),(23,29),(23,30),(23,34),(23,35),(23,36),(23,37),(23,38),(23,39),(23,42),(23,43),(23,44),(23,45),(23,46),(23,47),(23,48),(23,49),(23,50),(23,51),(23,52),(23,53),(23,54),(23,55),(23,56),(23,57),(23,58),(23,59),(23,60),(23,61),(24,25),(24,26),(24,27),(24,31),(24,32),(24,33),(24,35),(24,36),(24,37),(24,39),(24,40),(24,41),(24,42),(24,44),(24,45),(24,46),(24,48),(24,49),(24,50),(24,52),(24,53),(24,55),(24,56),(24,58),(24,59),(24,60),(24,62),(24,63),(25,26),(25,27),(25,31),(25,32),(25,33),(25,34),(25,36),(25,37),(25,38),(25,40),(25,41),(25,43),(25,44),(25,45),(25,47),(25,48),(25,49),(25,51),(25,52),(25,53),(25,55),(25,57),(25,58),(25,59),(25,61),(25,62),(25,63),(26,27),(26,28),(26,29),(26,30),(26,34),(26,35),(26,37),(26,38),(26,39),(26,41),(26,42),(26,43),(26,44),(26,46),(26,47),(26,48),(26,50),(26,51),(26,52),(26,54),(26,56),(26,57),(26,58),(26,60),(26,61),(26,63),(27,28),(27,29),(27,30),(27,34),(27,35),(27,36),(27,38),(27,39),(27,40),(27,42),(27,43),(27,45),(27,46),(27,47),(27,49),(27,50),(27,51),(27,53),(27,54),(27,56),(27,57),(27,59),(27,60),(27,61),(27,62),(28,31),(28,32),(28,33),(28,36),(28,37),(28,40),(28,41),(28,42),(28,43),(28,44),(28,45),(28,46),(28,47),(28,48),(28,49),(28,50),(28,51),(28,52),(28,53),(28,55),(28,56),(28,57),(28,58),(28,59),(28,60),(28,61),(28,62),(28,63),(29,31),(29,32),(29,33),(29,36),(29,37),(29,40),(29,41),(29,42),(29,43),(29,44),(29,45),(29,46),(29,47),(29,48),(29,49),(29,50),(29,51),(29,52),(29,53),(29,55),(29,56),(29,57),(29,58),(29,59),(29,60),(29,61),(29,62),(29,63),(30,31),(30,32),(30,33),(30,36),(30,37),(30,40),(30,41),(30,42),(30,43),(30,44),(30,45),(30,46),(30,47),(30,48),(30,49),(30,50),(30,51),(30,52),(30,53),(30,55),(30,56),(30,57),(30,58),(30,59),(30,60),(30,61),(30,62),(30,63),(31,34),(31,35),(31,38),(31,39),(31,42),(31,43),(31,44),(31,45),(31,46),(31,47),(31,48),(31,49),(31,50),(31,51),(31,52),(31,53),(31,54),(31,56),(31,57),(31,58),(31,59),(31,60),(31,61),(31,62),(31,63),(32,34),(32,35),(32,38),(32,39),(32,42),(32,43),(32,44),(32,45),(32,46),(32,47),(32,48),(32,49),(32,50),(32,51),(32,52),(32,53),(32,54),(32,56),(32,57),(32,58),(32,59),(32,60),(32,61),(32,62),(32,63),(33,34),(33,35),(33,38),(33,39),(33,42),(33,43),(33,44),(33,45),(33,46),(33,47),(33,48),(33,49),(33,50),(33,51),(33,52),(33,53),(33,54),(33,56),(33,57),(33,58),(33,59),(33,60),(33,61),(33,62),(33,63),(34,35),(34,36),(34,37),(34,39),(34,40),(34,41),(34,42),(34,43),(34,44),(34,45),(34,46),(34,47),(34,48),(34,49),(34,50),(34,51),(34,52),(34,53),(34,55),(34,56),(34,57),(34,58),(34,59),(34,60),(34,62),(34,63),(35,36),(35,37),(35,38),(35,40),(35,41),(35,42),(35,43),(35,44),(35,45),(35,46),(35,47),(35,48),(35,49),(35,50),(35,51),(35,52),(35,53),(35,55),(35,56),(35,57),(35,58),(35,59),(35,61),(35,62),(35,63),(36,37),(36,38),(36,39),(36,41),(36,42),(36,43),(36,44),(36,45),(36,46),(36,47),(36,48),(36,49),(36,50),(36,51),(36,52),(36,53),(36,54),(36,56),(36,57),(36,58),(36,59),(36,60),(36,61),(36,63),(37,38),(37,39),(37,40),(37,42),(37,43),(37,44),(37,45),(37,46),(37,47),(37,48),(37,49),(37,50),(37,51),(37,52),(37,53),(37,54),(37,56),(37,57),(37,58),(37,59),(37,60),(37,61),(37,62),(38,39),(38,40),(38,41),(38,42),(38,43),(38,44),(38,45),(38,46),(38,47),(38,48),(38,49),(38,50),(38,51),(38,52),(38,53),(38,54),(38,55),(38,56),(38,57),(38,58),(38,59),(38,60),(38,62),(38,63),(39,40),(39,41),(39,42),(39,43),(39,44),(39,45),(39,46),(39,47),(39,48),(39,49),(39,50),(39,51),(39,52),(39,53),(39,54),(39,55),(39,56),(39,57),(39,58),(39,59),(39,61),(39,62),(39,63),(40,41),(40,42),(40,43),(40,44),(40,45),(40,46),(40,47),(40,48),(40,49),(40,50),(40,51),(40,52),(40,53),(40,54),(40,55),(40,56),(40,57),(40,58),(40,59),(40,60),(40,61),(40,63),(41,42),(41,43),(41,44),(41,45),(41,46),(41,47),(41,48),(41,49),(41,50),(41,51),(41,52),(41,53),(41,54),(41,55),(41,56),(41,57),(41,58),(41,59),(41,60),(41,61),(41,62),(42,43),(42,45),(42,47),(42,49),(42,51),(42,53),(42,54),(42,55),(42,57),(42,58),(42,59),(42,60),(42,61),(42,62),(42,63),(43,44),(43,46),(43,48),(43,50),(43,52),(43,54),(43,55),(43,56),(43,58),(43,59),(43,60),(43,61),(43,62),(43,63),(44,45),(44,47),(44,49),(44,51),(44,53),(44,54),(44,55),(44,56),(44,57),(44,59),(44,60),(44,61),(44,62),(44,63),(45,46),(45,48),(45,50),(45,52),(45,54),(45,55),(45,56),(45,57),(45,58),(45,60),(45,61),(45,62),(45,63),(46,47),(46,49),(46,51),(46,53),(46,54),(46,55),(46,57),(46,59),(46,60),(46,61),(46,62),(46,63),(47,48),(47,50),(47,52),(47,54),(47,55),(47,56),(47,58),(47,60),(47,61),(47,62),(47,63),(48,49),(48,51),(48,53),(48,54),(48,55),(48,57),(48,59),(48,60),(48,61),(48,62),(48,63),(49,50),(49,52),(49,54),(49,55),(49,56),(49,58),(49,60),(49,61),(49,62),(49,63),(50,51),(50,52),(50,53),(50,54),(50,55),(50,57),(50,58),(50,59),(50,60),(50,61),(50,62),(50,63),(51,52),(51,53),(51,54),(51,55),(51,56),(51,58),(51,59),(51,60),(51,61),(51,62),(51,63),(52,53),(52,54),(52,55),(52,56),(52,57),(52,59),(52,60),(52,61),(52,62),(52,63),(53,54),(53,55),(53,56),(53,57),(53,58),(53,60),(53,61),(53,62),(53,63),(54,55),(54,56),(54,57),(54,58),(54,59),(54,60),(54,61),(54,62),(54,63),(55,56),(55,57),(55,58),(55,59),(55,60),(55,61),(55,62),(55,63),(56,57),(56,58),(56,59),(56,60),(56,61),(56,62),(56,63),(57,58),(57,59),(57,60),(57,61),(57,62),(57,63),(58,59),(58,60),(58,61),(58,62),(58,63),(59,60),(59,61),(59,62),(59,63),(60,61),(60,62),(60,63),(61,62),(61,63),(62,63)],64)
=> ? = 10 + 1
[[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,2,3],[2,2,2,3],[3,3,3],[4,5],[5]]
=> ([(0,3),(0,6),(0,7),(1,8),(1,12),(2,8),(2,11),(3,9),(3,10),(4,2),(4,13),(5,1),(5,14),(6,4),(6,9),(7,5),(7,10),(8,15),(9,13),(10,14),(11,15),(12,15),(13,11),(14,12)],16)
=> ?
=> ? = 5 + 1
[[0,1,0,0,0],[0,0,1,0,0],[1,0,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[1,1,1,2,2],[2,2,2,3],[3,3,4],[4,5],[5]]
=> ?
=> ?
=> ? = 5 + 1
[[0,0,1,0,0],[0,1,0,0,0],[1,0,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[1,1,1,2,3],[2,2,2,3],[3,3,4],[4,5],[5]]
=> ([(0,3),(0,8),(1,10),(2,10),(3,9),(4,6),(5,4),(5,11),(6,2),(7,1),(8,5),(8,9),(9,11),(11,7)],12)
=> ?
=> ? = 6 + 1
[[0,0,1,0,0],[1,0,-1,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[1,1,1,1,3],[2,2,2,4],[3,3,4],[4,5],[5]]
=> ([(0,4),(0,6),(1,10),(2,9),(3,7),(4,8),(5,3),(5,11),(6,1),(6,8),(7,9),(8,5),(8,10),(10,11),(11,2),(11,7)],12)
=> ?
=> ? = 6 + 1
[[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[1,1,1,1,4],[2,2,2,4],[3,3,4],[4,5],[5]]
=> ([(0,4),(0,7),(2,10),(3,9),(4,8),(5,3),(5,11),(6,1),(7,5),(7,8),(8,11),(9,10),(10,6),(11,2),(11,9)],12)
=> ?
=> ? = 7 + 1
[[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[1,1,1,2,2],[2,2,2,4],[3,3,4],[4,5],[5]]
=> ?
=> ?
=> ? = 6 + 1
[[0,0,1,0,0],[0,1,-1,1,0],[1,0,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[1,1,1,2,3],[2,2,2,4],[3,3,4],[4,5],[5]]
=> ?
=> ?
=> ? = 7 + 1
[[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[1,1,1,2,4],[2,2,2,4],[3,3,4],[4,5],[5]]
=> ?
=> ?
=> ? = 8 + 1
[[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [[1,1,1,1,1],[2,2,3,3],[3,3,4],[4,5],[5]]
=> ?
=> ?
=> ? = 4 + 1
[[0,1,0,0,0],[1,-1,1,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [[1,1,1,1,2],[2,2,3,3],[3,3,4],[4,5],[5]]
=> ?
=> ?
=> ? = 5 + 1
[[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [[1,1,1,1,3],[2,2,3,3],[3,3,4],[4,5],[5]]
=> ?
=> ?
=> ? = 6 + 1
[[0,1,0,0,0],[0,0,1,0,0],[1,-1,0,1,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [[1,1,1,2,2],[2,2,3,3],[3,3,4],[4,5],[5]]
=> ?
=> ?
=> ? = 6 + 1
[[0,0,1,0,0],[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [[1,1,1,2,3],[2,2,3,3],[3,3,4],[4,5],[5]]
=> ?
=> ?
=> ? = 7 + 1
[[0,1,0,0,0],[1,-1,0,1,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [[1,1,1,1,2],[2,2,3,4],[3,3,4],[4,5],[5]]
=> ([(0,4),(0,7),(1,9),(2,10),(3,9),(3,10),(4,11),(5,8),(6,1),(7,5),(7,11),(8,2),(8,3),(9,12),(10,12),(11,6)],13)
=> ?
=> ? = 6 + 1
[[0,0,1,0,0],[1,0,-1,1,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [[1,1,1,1,3],[2,2,3,4],[3,3,4],[4,5],[5]]
=> ?
=> ?
=> ? = 7 + 1
[[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [[1,1,1,1,4],[2,2,3,4],[3,3,4],[4,5],[5]]
=> ?
=> ?
=> ? = 8 + 1
Description
The cardinality of a maximal independent set of vertices of a graph.
An independent set of a graph is a set of pairwise non-adjacent vertices. A maximum independent set is an independent set of maximum cardinality. This statistic is also called the independence number or stability number $\alpha(G)$ of $G$.
Matching statistic: St000384
Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000384: Integer partitions ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 10%
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000384: Integer partitions ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 10%
Values
[[1]]
=> [[1]]
=> ([],1)
=> [1]
=> 1 = 0 + 1
[[1,0],[0,1]]
=> [[1,1],[2]]
=> ([],1)
=> [1]
=> 1 = 0 + 1
[[0,1],[1,0]]
=> [[1,2],[2]]
=> ([(0,1)],2)
=> [2]
=> 2 = 1 + 1
[[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ([],1)
=> [1]
=> 1 = 0 + 1
[[0,1,0],[1,0,0],[0,0,1]]
=> [[1,1,2],[2,2],[3]]
=> ([(0,1)],2)
=> [2]
=> 2 = 1 + 1
[[1,0,0],[0,0,1],[0,1,0]]
=> [[1,1,1],[2,3],[3]]
=> ([(0,1)],2)
=> [2]
=> 2 = 1 + 1
[[0,1,0],[1,-1,1],[0,1,0]]
=> [[1,1,2],[2,3],[3]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 3 = 2 + 1
[[0,0,1],[1,0,0],[0,1,0]]
=> [[1,1,3],[2,3],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4 = 3 + 1
[[0,1,0],[0,0,1],[1,0,0]]
=> [[1,2,2],[2,3],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4 = 3 + 1
[[0,0,1],[0,1,0],[1,0,0]]
=> [[1,2,3],[2,3],[3]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> [5,3]
=> 5 = 4 + 1
[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,2],[3,3],[4]]
=> ([],1)
=> [1]
=> 1 = 0 + 1
[[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,2],[2,2,2],[3,3],[4]]
=> ([(0,1)],2)
=> [2]
=> 2 = 1 + 1
[[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,3],[3,3],[4]]
=> ([(0,1)],2)
=> [2]
=> 2 = 1 + 1
[[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
=> [[1,1,1,2],[2,2,3],[3,3],[4]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 3 = 2 + 1
[[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> [[1,1,1,3],[2,2,3],[3,3],[4]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4 = 3 + 1
[[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]]
=> [[1,1,2,2],[2,2,3],[3,3],[4]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4 = 3 + 1
[[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> [[1,1,2,3],[2,2,3],[3,3],[4]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> [5,3]
=> 5 = 4 + 1
[[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [[1,1,1,1],[2,2,2],[3,4],[4]]
=> ([(0,1)],2)
=> [2]
=> 2 = 1 + 1
[[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> [[1,1,1,2],[2,2,2],[3,4],[4]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> [3,1]
=> 3 = 2 + 1
[[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> [[1,1,1,1],[2,2,3],[3,4],[4]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 3 = 2 + 1
[[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> [[1,1,1,2],[2,2,3],[3,4],[4]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4 = 3 + 1
[[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]]
=> [[1,1,1,3],[2,2,3],[3,4],[4]]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> [5,1]
=> 5 = 4 + 1
[[0,1,0,0],[0,0,1,0],[1,0,-1,1],[0,0,1,0]]
=> [[1,1,2,2],[2,2,3],[3,4],[4]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 5 = 4 + 1
[[0,0,1,0],[0,1,0,0],[1,0,-1,1],[0,0,1,0]]
=> [[1,1,2,3],[2,2,3],[3,4],[4]]
=> ([(0,3),(0,7),(1,8),(2,8),(3,9),(4,5),(5,1),(6,2),(7,4),(7,9),(9,6)],10)
=> [6,4]
=> 6 = 5 + 1
[[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> [[1,1,1,1],[2,2,4],[3,4],[4]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4 = 3 + 1
[[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
=> [[1,1,1,2],[2,2,4],[3,4],[4]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [5,2]
=> 5 = 4 + 1
[[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
=> [[1,1,1,3],[2,2,4],[3,4],[4]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> [6,3]
=> 6 = 5 + 1
[[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [[1,1,1,4],[2,2,4],[3,4],[4]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> [7,1]
=> 7 = 6 + 1
[[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
=> [[1,1,2,2],[2,2,4],[3,4],[4]]
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> [6,3,1]
=> 6 = 5 + 1
[[0,0,1,0],[0,1,-1,1],[1,0,0,0],[0,0,1,0]]
=> [[1,1,2,3],[2,2,4],[3,4],[4]]
=> ([(0,6),(1,9),(1,10),(2,8),(3,7),(4,3),(4,12),(5,2),(5,12),(6,4),(6,5),(7,9),(7,11),(8,10),(8,11),(9,13),(10,13),(11,13),(12,1),(12,7),(12,8)],14)
=> [7,4,3]
=> ? = 6 + 1
[[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
=> [[1,1,2,4],[2,2,4],[3,4],[4]]
=> ([(0,7),(0,8),(1,12),(2,11),(3,10),(4,10),(4,11),(5,3),(6,1),(6,13),(7,9),(8,5),(9,2),(9,4),(10,14),(11,6),(11,14),(13,12),(14,13)],15)
=> [8,6,1]
=> ? = 7 + 1
[[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]]
=> [[1,1,1,1],[2,3,3],[3,4],[4]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4 = 3 + 1
[[0,1,0,0],[1,-1,1,0],[0,0,0,1],[0,1,0,0]]
=> [[1,1,1,2],[2,3,3],[3,4],[4]]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> [5,1]
=> 5 = 4 + 1
[[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
=> [[1,1,1,3],[2,3,3],[3,4],[4]]
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> [6,1]
=> 6 = 5 + 1
[[0,1,0,0],[0,0,1,0],[1,-1,0,1],[0,1,0,0]]
=> [[1,1,2,2],[2,3,3],[3,4],[4]]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> [6,1]
=> 6 = 5 + 1
[[0,0,1,0],[0,1,0,0],[1,-1,0,1],[0,1,0,0]]
=> [[1,1,2,3],[2,3,3],[3,4],[4]]
=> ([(0,3),(0,8),(1,10),(2,9),(3,11),(4,2),(5,4),(6,7),(7,1),(7,9),(8,5),(8,11),(9,10),(11,6)],12)
=> [7,5]
=> 7 = 6 + 1
[[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> [[1,1,1,1],[2,3,4],[3,4],[4]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> [5,3]
=> 5 = 4 + 1
[[0,1,0,0],[1,-1,0,1],[0,0,1,0],[0,1,0,0]]
=> [[1,1,1,2],[2,3,4],[3,4],[4]]
=> ([(0,5),(0,6),(1,8),(2,9),(3,8),(3,9),(4,1),(5,4),(6,7),(7,2),(7,3),(8,10),(9,10)],11)
=> [6,4,1]
=> 6 = 5 + 1
[[0,0,1,0],[1,0,-1,1],[0,0,1,0],[0,1,0,0]]
=> [[1,1,1,3],[2,3,4],[3,4],[4]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> [7,3]
=> 7 = 6 + 1
[[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
=> [[1,1,1,4],[2,3,4],[3,4],[4]]
=> ([(0,5),(0,10),(1,16),(2,15),(3,14),(4,13),(5,12),(6,2),(6,13),(7,4),(7,14),(8,1),(9,6),(10,11),(10,12),(11,3),(11,7),(12,9),(13,15),(14,8),(15,16)],17)
=> [8,6,3]
=> ? = 7 + 1
[[0,1,0,0],[0,0,0,1],[1,-1,1,0],[0,1,0,0]]
=> [[1,1,2,2],[2,3,4],[3,4],[4]]
=> ([(0,6),(0,7),(1,9),(2,12),(3,9),(3,12),(4,10),(5,1),(6,5),(7,8),(8,2),(8,3),(9,11),(11,10),(12,4),(12,11)],13)
=> [7,5,1]
=> ? = 6 + 1
[[0,0,1,0],[0,1,-1,1],[1,-1,1,0],[0,1,0,0]]
=> [[1,1,2,3],[2,3,4],[3,4],[4]]
=> ([(0,9),(0,11),(1,18),(2,17),(3,19),(4,13),(4,19),(5,12),(5,13),(6,16),(7,14),(8,5),(8,18),(9,10),(10,3),(10,4),(11,1),(11,8),(12,17),(13,15),(15,16),(16,14),(17,7),(18,2),(18,12),(19,6),(19,15)],20)
=> [8,6,4,2]
=> ? = 7 + 1
[[0,0,0,1],[0,1,0,0],[1,-1,1,0],[0,1,0,0]]
=> [[1,1,2,4],[2,3,4],[3,4],[4]]
=> ([(0,12),(0,13),(1,16),(2,15),(3,23),(4,19),(5,17),(5,20),(6,4),(7,5),(7,15),(7,16),(8,10),(9,7),(10,2),(11,1),(11,23),(12,8),(12,22),(13,14),(13,22),(14,3),(14,11),(15,17),(15,21),(16,20),(16,21),(17,24),(19,18),(20,19),(20,24),(21,24),(22,9),(23,6),(24,18)],25)
=> [9,7,5,4]
=> ? = 8 + 1
[[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> [[1,1,3,3],[2,3,4],[3,4],[4]]
=> ([(0,8),(2,11),(2,12),(3,10),(4,9),(5,4),(5,14),(6,3),(6,14),(7,1),(8,5),(8,6),(9,11),(9,13),(10,12),(10,13),(11,15),(12,15),(13,15),(14,2),(14,9),(14,10),(15,7)],16)
=> [9,4,3]
=> ? = 8 + 1
[[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
=> [[1,1,3,4],[2,3,4],[3,4],[4]]
=> ([(0,13),(0,15),(1,17),(2,16),(2,17),(3,19),(4,16),(4,18),(5,22),(6,21),(7,20),(8,23),(8,29),(9,4),(9,28),(10,3),(10,28),(11,6),(12,7),(12,24),(13,14),(14,1),(14,2),(15,9),(15,10),(16,25),(17,12),(17,25),(18,26),(18,29),(19,23),(19,26),(20,27),(22,27),(23,30),(24,20),(24,22),(25,24),(26,30),(27,21),(28,8),(28,18),(28,19),(29,5),(29,30),(30,11)],31)
=> [10,8,6,4,3]
=> ? = 9 + 1
[[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> [[1,2,2,2],[2,3,3],[3,4],[4]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> [7,1]
=> 7 = 6 + 1
[[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]]
=> [[1,2,2,3],[2,3,3],[3,4],[4]]
=> ([(0,5),(0,10),(1,16),(2,15),(3,14),(4,13),(5,12),(6,2),(6,13),(7,4),(7,14),(8,1),(9,6),(10,11),(10,12),(11,3),(11,7),(12,9),(13,15),(14,8),(15,16)],17)
=> [8,6,3]
=> ? = 7 + 1
[[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]]
=> [[1,2,2,2],[2,3,4],[3,4],[4]]
=> ([(0,7),(0,8),(1,12),(2,11),(3,10),(4,10),(4,11),(5,3),(6,1),(6,13),(7,9),(8,5),(9,2),(9,4),(10,14),(11,6),(11,14),(13,12),(14,13)],15)
=> [8,6,1]
=> ? = 7 + 1
[[0,0,1,0],[0,1,-1,1],[0,0,1,0],[1,0,0,0]]
=> [[1,2,2,3],[2,3,4],[3,4],[4]]
=> ([(0,10),(0,12),(1,16),(2,17),(3,21),(4,22),(5,14),(6,13),(6,14),(7,13),(7,15),(8,7),(8,21),(9,2),(9,18),(10,11),(11,5),(11,6),(12,3),(12,8),(13,19),(14,9),(14,19),(15,22),(16,20),(17,20),(18,16),(18,17),(19,18),(21,4),(21,15),(22,1)],23)
=> [9,7,5,2]
=> ? = 8 + 1
[[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
=> [[1,2,2,4],[2,3,4],[3,4],[4]]
=> ([(0,14),(0,15),(1,19),(2,18),(3,29),(4,30),(5,22),(6,23),(7,24),(7,25),(8,9),(9,7),(9,18),(9,19),(10,5),(11,6),(12,2),(12,29),(13,1),(13,30),(14,16),(14,28),(15,17),(15,28),(16,3),(16,12),(17,4),(17,13),(18,24),(18,27),(19,25),(19,27),(20,26),(21,26),(22,20),(23,21),(24,22),(24,31),(25,23),(25,31),(27,31),(28,8),(29,10),(30,11),(31,20),(31,21)],32)
=> [10,8,6,4,4]
=> ? = 9 + 1
[[0,0,1,0],[0,0,0,1],[0,1,0,0],[1,0,0,0]]
=> [[1,2,3,3],[2,3,4],[3,4],[4]]
=> ([(0,13),(0,15),(1,17),(2,16),(2,17),(3,19),(4,16),(4,18),(5,22),(6,21),(7,20),(8,23),(8,29),(9,4),(9,28),(10,3),(10,28),(11,6),(12,7),(12,24),(13,14),(14,1),(14,2),(15,9),(15,10),(16,25),(17,12),(17,25),(18,26),(18,29),(19,23),(19,26),(20,27),(22,27),(23,30),(24,20),(24,22),(25,24),(26,30),(27,21),(28,8),(28,18),(28,19),(29,5),(29,30),(30,11)],31)
=> [10,8,6,4,3]
=> ? = 9 + 1
[[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> [[1,2,3,4],[2,3,4],[3,4],[4]]
=> ([(0,5),(0,16),(0,17),(1,23),(2,19),(3,11),(3,21),(4,10),(4,20),(5,12),(5,13),(6,50),(7,51),(8,24),(8,58),(9,25),(9,59),(10,14),(10,52),(11,15),(11,53),(12,26),(12,60),(13,27),(13,60),(14,54),(15,55),(16,4),(16,29),(17,3),(17,29),(18,48),(18,49),(19,32),(19,33),(20,46),(20,52),(21,47),(21,53),(22,34),(22,35),(23,18),(23,54),(23,55),(24,40),(24,42),(25,41),(25,43),(26,46),(26,56),(27,47),(27,57),(28,63),(29,1),(30,62),(31,61),(32,61),(33,61),(34,6),(34,62),(35,7),(35,62),(36,58),(37,59),(38,32),(39,33),(40,44),(41,45),(42,38),(43,39),(44,31),(45,31),(46,36),(47,37),(48,40),(48,63),(49,41),(49,63),(50,38),(51,39),(52,8),(52,36),(53,9),(53,37),(54,28),(54,48),(55,28),(55,49),(56,30),(56,34),(57,30),(57,35),(58,42),(58,50),(59,43),(59,51),(60,22),(60,56),(60,57),(62,2),(63,44),(63,45)],64)
=> [11,9,9,7,7,5,5,5,3,3]
=> ? = 10 + 1
[[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,2,2],[3,3,3],[4,4],[5]]
=> ([],1)
=> [1]
=> 1 = 0 + 1
[[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,2],[2,2,2,2],[3,3,3],[4,4],[5]]
=> ([(0,1)],2)
=> [2]
=> 2 = 1 + 1
[[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,2,3],[3,3,3],[4,4],[5]]
=> ([(0,1)],2)
=> [2]
=> 2 = 1 + 1
[[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,2],[2,2,2,3],[3,3,3],[4,4],[5]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 3 = 2 + 1
[[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,3],[2,2,2,3],[3,3,3],[4,4],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4 = 3 + 1
[[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,2,2],[2,2,2,3],[3,3,3],[4,4],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4 = 3 + 1
[[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,2,3],[2,2,2,3],[3,3,3],[4,4],[5]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> [5,3]
=> 5 = 4 + 1
[[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,2,2],[3,3,4],[4,4],[5]]
=> ([(0,1)],2)
=> [2]
=> 2 = 1 + 1
[[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,2],[2,2,2,2],[3,3,4],[4,4],[5]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> [3,1]
=> 3 = 2 + 1
[[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,2,3],[3,3,4],[4,4],[5]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 3 = 2 + 1
[[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,2],[2,2,2,3],[3,3,4],[4,4],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4 = 3 + 1
[[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,3],[2,2,2,3],[3,3,4],[4,4],[5]]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> [5,1]
=> 5 = 4 + 1
[[0,1,0,0,0],[0,0,1,0,0],[1,0,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,2,2],[2,2,2,3],[3,3,4],[4,4],[5]]
=> ?
=> ?
=> ? = 4 + 1
[[0,0,1,0,0],[0,1,-1,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,2,3],[2,2,2,4],[3,3,4],[4,4],[5]]
=> ?
=> ?
=> ? = 6 + 1
[[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,2,4],[2,2,2,4],[3,3,4],[4,4],[5]]
=> ([(0,7),(0,8),(1,12),(2,11),(3,10),(4,10),(4,11),(5,3),(6,1),(6,13),(7,9),(8,5),(9,2),(9,4),(10,14),(11,6),(11,14),(13,12),(14,13)],15)
=> [8,6,1]
=> ? = 7 + 1
[[0,1,0,0,0],[1,-1,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,2],[2,2,3,3],[3,3,4],[4,4],[5]]
=> ?
=> ?
=> ? = 4 + 1
[[0,1,0,0,0],[0,0,1,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,2,2],[2,2,3,3],[3,3,4],[4,4],[5]]
=> ?
=> ?
=> ? = 5 + 1
[[0,0,1,0,0],[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,2,3],[2,2,3,3],[3,3,4],[4,4],[5]]
=> ?
=> ?
=> ? = 6 + 1
[[0,0,1,0,0],[1,0,-1,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,3],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ?
=> ?
=> ? = 6 + 1
[[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,4],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ([(0,5),(0,10),(1,16),(2,15),(3,14),(4,13),(5,12),(6,2),(6,13),(7,4),(7,14),(8,1),(9,6),(10,11),(10,12),(11,3),(11,7),(12,9),(13,15),(14,8),(15,16)],17)
=> [8,6,3]
=> ? = 7 + 1
[[0,1,0,0,0],[0,0,0,1,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,2,2],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ?
=> ?
=> ? = 6 + 1
[[0,0,1,0,0],[0,1,-1,1,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,2,3],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ([(0,9),(0,11),(1,18),(2,17),(3,19),(4,13),(4,19),(5,12),(5,13),(6,16),(7,14),(8,5),(8,18),(9,10),(10,3),(10,4),(11,1),(11,8),(12,17),(13,15),(15,16),(16,14),(17,7),(18,2),(18,12),(19,6),(19,15)],20)
=> [8,6,4,2]
=> ? = 7 + 1
[[0,0,0,1,0],[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,2,4],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ?
=> ?
=> ? = 8 + 1
[[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,3,3],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ([(0,8),(2,11),(2,12),(3,10),(4,9),(5,4),(5,14),(6,3),(6,14),(7,1),(8,5),(8,6),(9,11),(9,13),(10,12),(10,13),(11,15),(12,15),(13,15),(14,2),(14,9),(14,10),(15,7)],16)
=> [9,4,3]
=> ? = 8 + 1
[[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,3,4],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ([(0,13),(0,15),(1,17),(2,16),(2,17),(3,19),(4,16),(4,18),(5,22),(6,21),(7,20),(8,23),(8,29),(9,4),(9,28),(10,3),(10,28),(11,6),(12,7),(12,24),(13,14),(14,1),(14,2),(15,9),(15,10),(16,25),(17,12),(17,25),(18,26),(18,29),(19,23),(19,26),(20,27),(22,27),(23,30),(24,20),(24,22),(25,24),(26,30),(27,21),(28,8),(28,18),(28,19),(29,5),(29,30),(30,11)],31)
=> [10,8,6,4,3]
=> ? = 9 + 1
[[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,0,0,1]]
=> [[1,1,2,2,3],[2,2,3,3],[3,3,4],[4,4],[5]]
=> ([(0,5),(0,10),(1,16),(2,15),(3,14),(4,13),(5,12),(6,2),(6,13),(7,4),(7,14),(8,1),(9,6),(10,11),(10,12),(11,3),(11,7),(12,9),(13,15),(14,8),(15,16)],17)
=> [8,6,3]
=> ? = 7 + 1
[[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1]]
=> [[1,1,2,2,2],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ([(0,7),(0,8),(1,12),(2,11),(3,10),(4,10),(4,11),(5,3),(6,1),(6,13),(7,9),(8,5),(9,2),(9,4),(10,14),(11,6),(11,14),(13,12),(14,13)],15)
=> [8,6,1]
=> ? = 7 + 1
[[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1]]
=> [[1,1,2,2,3],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ?
=> ?
=> ? = 8 + 1
[[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1]]
=> [[1,1,2,2,4],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ([(0,14),(0,15),(1,19),(2,18),(3,29),(4,30),(5,22),(6,23),(7,24),(7,25),(8,9),(9,7),(9,18),(9,19),(10,5),(11,6),(12,2),(12,29),(13,1),(13,30),(14,16),(14,28),(15,17),(15,28),(16,3),(16,12),(17,4),(17,13),(18,24),(18,27),(19,25),(19,27),(20,26),(21,26),(22,20),(23,21),(24,22),(24,31),(25,23),(25,31),(27,31),(28,8),(29,10),(30,11),(31,20),(31,21)],32)
=> [10,8,6,4,4]
=> ? = 9 + 1
[[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1]]
=> [[1,1,2,3,3],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ([(0,13),(0,15),(1,17),(2,16),(2,17),(3,19),(4,16),(4,18),(5,22),(6,21),(7,20),(8,23),(8,29),(9,4),(9,28),(10,3),(10,28),(11,6),(12,7),(12,24),(13,14),(14,1),(14,2),(15,9),(15,10),(16,25),(17,12),(17,25),(18,26),(18,29),(19,23),(19,26),(20,27),(22,27),(23,30),(24,20),(24,22),(25,24),(26,30),(27,21),(28,8),(28,18),(28,19),(29,5),(29,30),(30,11)],31)
=> [10,8,6,4,3]
=> ? = 9 + 1
[[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1]]
=> [[1,1,2,3,4],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ([(0,5),(0,16),(0,17),(1,23),(2,19),(3,11),(3,21),(4,10),(4,20),(5,12),(5,13),(6,50),(7,51),(8,24),(8,58),(9,25),(9,59),(10,14),(10,52),(11,15),(11,53),(12,26),(12,60),(13,27),(13,60),(14,54),(15,55),(16,4),(16,29),(17,3),(17,29),(18,48),(18,49),(19,32),(19,33),(20,46),(20,52),(21,47),(21,53),(22,34),(22,35),(23,18),(23,54),(23,55),(24,40),(24,42),(25,41),(25,43),(26,46),(26,56),(27,47),(27,57),(28,63),(29,1),(30,62),(31,61),(32,61),(33,61),(34,6),(34,62),(35,7),(35,62),(36,58),(37,59),(38,32),(39,33),(40,44),(41,45),(42,38),(43,39),(44,31),(45,31),(46,36),(47,37),(48,40),(48,63),(49,41),(49,63),(50,38),(51,39),(52,8),(52,36),(53,9),(53,37),(54,28),(54,48),(55,28),(55,49),(56,30),(56,34),(57,30),(57,35),(58,42),(58,50),(59,43),(59,51),(60,22),(60,56),(60,57),(62,2),(63,44),(63,45)],64)
=> [11,9,9,7,7,5,5,5,3,3]
=> ? = 10 + 1
[[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,2,3],[2,2,2,3],[3,3,3],[4,5],[5]]
=> ([(0,3),(0,6),(0,7),(1,8),(1,12),(2,8),(2,11),(3,9),(3,10),(4,2),(4,13),(5,1),(5,14),(6,4),(6,9),(7,5),(7,10),(8,15),(9,13),(10,14),(11,15),(12,15),(13,11),(14,12)],16)
=> ?
=> ? = 5 + 1
[[0,1,0,0,0],[0,0,1,0,0],[1,0,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[1,1,1,2,2],[2,2,2,3],[3,3,4],[4,5],[5]]
=> ?
=> ?
=> ? = 5 + 1
[[0,0,1,0,0],[0,1,0,0,0],[1,0,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[1,1,1,2,3],[2,2,2,3],[3,3,4],[4,5],[5]]
=> ([(0,3),(0,8),(1,10),(2,10),(3,9),(4,6),(5,4),(5,11),(6,2),(7,1),(8,5),(8,9),(9,11),(11,7)],12)
=> ?
=> ? = 6 + 1
[[0,0,1,0,0],[1,0,-1,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[1,1,1,1,3],[2,2,2,4],[3,3,4],[4,5],[5]]
=> ([(0,4),(0,6),(1,10),(2,9),(3,7),(4,8),(5,3),(5,11),(6,1),(6,8),(7,9),(8,5),(8,10),(10,11),(11,2),(11,7)],12)
=> ?
=> ? = 6 + 1
[[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[1,1,1,1,4],[2,2,2,4],[3,3,4],[4,5],[5]]
=> ([(0,4),(0,7),(2,10),(3,9),(4,8),(5,3),(5,11),(6,1),(7,5),(7,8),(8,11),(9,10),(10,6),(11,2),(11,9)],12)
=> ?
=> ? = 7 + 1
[[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[1,1,1,2,2],[2,2,2,4],[3,3,4],[4,5],[5]]
=> ?
=> ?
=> ? = 6 + 1
[[0,0,1,0,0],[0,1,-1,1,0],[1,0,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[1,1,1,2,3],[2,2,2,4],[3,3,4],[4,5],[5]]
=> ?
=> ?
=> ? = 7 + 1
[[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[1,1,1,2,4],[2,2,2,4],[3,3,4],[4,5],[5]]
=> ?
=> ?
=> ? = 8 + 1
[[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [[1,1,1,1,1],[2,2,3,3],[3,3,4],[4,5],[5]]
=> ?
=> ?
=> ? = 4 + 1
[[0,1,0,0,0],[1,-1,1,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [[1,1,1,1,2],[2,2,3,3],[3,3,4],[4,5],[5]]
=> ?
=> ?
=> ? = 5 + 1
[[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [[1,1,1,1,3],[2,2,3,3],[3,3,4],[4,5],[5]]
=> ?
=> ?
=> ? = 6 + 1
[[0,1,0,0,0],[0,0,1,0,0],[1,-1,0,1,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [[1,1,1,2,2],[2,2,3,3],[3,3,4],[4,5],[5]]
=> ?
=> ?
=> ? = 6 + 1
[[0,0,1,0,0],[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [[1,1,1,2,3],[2,2,3,3],[3,3,4],[4,5],[5]]
=> ?
=> ?
=> ? = 7 + 1
[[0,1,0,0,0],[1,-1,0,1,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [[1,1,1,1,2],[2,2,3,4],[3,3,4],[4,5],[5]]
=> ([(0,4),(0,7),(1,9),(2,10),(3,9),(3,10),(4,11),(5,8),(6,1),(7,5),(7,11),(8,2),(8,3),(9,12),(10,12),(11,6)],13)
=> ?
=> ? = 6 + 1
[[0,0,1,0,0],[1,0,-1,1,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [[1,1,1,1,3],[2,2,3,4],[3,3,4],[4,5],[5]]
=> ?
=> ?
=> ? = 7 + 1
[[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [[1,1,1,1,4],[2,2,3,4],[3,3,4],[4,5],[5]]
=> ?
=> ?
=> ? = 8 + 1
[[0,1,0,0,0],[0,0,0,1,0],[1,-1,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [[1,1,1,2,2],[2,2,3,4],[3,3,4],[4,5],[5]]
=> ?
=> ?
=> ? = 7 + 1
Description
The maximal part of the shifted composition of an integer partition.
A partition $\lambda = (\lambda_1,\ldots,\lambda_k)$ is shifted into a composition by adding $i-1$ to the $i$-th part.
The statistic is then $\operatorname{max}_i\{ \lambda_i + i - 1 \}$.
See also [[St000380]].
Matching statistic: St000784
Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000784: Integer partitions ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 10%
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000784: Integer partitions ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 10%
Values
[[1]]
=> [[1]]
=> ([],1)
=> [1]
=> 1 = 0 + 1
[[1,0],[0,1]]
=> [[1,1],[2]]
=> ([],1)
=> [1]
=> 1 = 0 + 1
[[0,1],[1,0]]
=> [[1,2],[2]]
=> ([(0,1)],2)
=> [2]
=> 2 = 1 + 1
[[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ([],1)
=> [1]
=> 1 = 0 + 1
[[0,1,0],[1,0,0],[0,0,1]]
=> [[1,1,2],[2,2],[3]]
=> ([(0,1)],2)
=> [2]
=> 2 = 1 + 1
[[1,0,0],[0,0,1],[0,1,0]]
=> [[1,1,1],[2,3],[3]]
=> ([(0,1)],2)
=> [2]
=> 2 = 1 + 1
[[0,1,0],[1,-1,1],[0,1,0]]
=> [[1,1,2],[2,3],[3]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 3 = 2 + 1
[[0,0,1],[1,0,0],[0,1,0]]
=> [[1,1,3],[2,3],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4 = 3 + 1
[[0,1,0],[0,0,1],[1,0,0]]
=> [[1,2,2],[2,3],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4 = 3 + 1
[[0,0,1],[0,1,0],[1,0,0]]
=> [[1,2,3],[2,3],[3]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> [5,3]
=> 5 = 4 + 1
[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,2],[3,3],[4]]
=> ([],1)
=> [1]
=> 1 = 0 + 1
[[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,2],[2,2,2],[3,3],[4]]
=> ([(0,1)],2)
=> [2]
=> 2 = 1 + 1
[[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,3],[3,3],[4]]
=> ([(0,1)],2)
=> [2]
=> 2 = 1 + 1
[[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
=> [[1,1,1,2],[2,2,3],[3,3],[4]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 3 = 2 + 1
[[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> [[1,1,1,3],[2,2,3],[3,3],[4]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4 = 3 + 1
[[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]]
=> [[1,1,2,2],[2,2,3],[3,3],[4]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4 = 3 + 1
[[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> [[1,1,2,3],[2,2,3],[3,3],[4]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> [5,3]
=> 5 = 4 + 1
[[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [[1,1,1,1],[2,2,2],[3,4],[4]]
=> ([(0,1)],2)
=> [2]
=> 2 = 1 + 1
[[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> [[1,1,1,2],[2,2,2],[3,4],[4]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> [3,1]
=> 3 = 2 + 1
[[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> [[1,1,1,1],[2,2,3],[3,4],[4]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 3 = 2 + 1
[[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> [[1,1,1,2],[2,2,3],[3,4],[4]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4 = 3 + 1
[[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]]
=> [[1,1,1,3],[2,2,3],[3,4],[4]]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> [5,1]
=> 5 = 4 + 1
[[0,1,0,0],[0,0,1,0],[1,0,-1,1],[0,0,1,0]]
=> [[1,1,2,2],[2,2,3],[3,4],[4]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 5 = 4 + 1
[[0,0,1,0],[0,1,0,0],[1,0,-1,1],[0,0,1,0]]
=> [[1,1,2,3],[2,2,3],[3,4],[4]]
=> ([(0,3),(0,7),(1,8),(2,8),(3,9),(4,5),(5,1),(6,2),(7,4),(7,9),(9,6)],10)
=> [6,4]
=> 6 = 5 + 1
[[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> [[1,1,1,1],[2,2,4],[3,4],[4]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4 = 3 + 1
[[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
=> [[1,1,1,2],[2,2,4],[3,4],[4]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [5,2]
=> 5 = 4 + 1
[[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
=> [[1,1,1,3],[2,2,4],[3,4],[4]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> [6,3]
=> 6 = 5 + 1
[[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [[1,1,1,4],[2,2,4],[3,4],[4]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> [7,1]
=> 7 = 6 + 1
[[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
=> [[1,1,2,2],[2,2,4],[3,4],[4]]
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> [6,3,1]
=> 6 = 5 + 1
[[0,0,1,0],[0,1,-1,1],[1,0,0,0],[0,0,1,0]]
=> [[1,1,2,3],[2,2,4],[3,4],[4]]
=> ([(0,6),(1,9),(1,10),(2,8),(3,7),(4,3),(4,12),(5,2),(5,12),(6,4),(6,5),(7,9),(7,11),(8,10),(8,11),(9,13),(10,13),(11,13),(12,1),(12,7),(12,8)],14)
=> [7,4,3]
=> ? = 6 + 1
[[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
=> [[1,1,2,4],[2,2,4],[3,4],[4]]
=> ([(0,7),(0,8),(1,12),(2,11),(3,10),(4,10),(4,11),(5,3),(6,1),(6,13),(7,9),(8,5),(9,2),(9,4),(10,14),(11,6),(11,14),(13,12),(14,13)],15)
=> [8,6,1]
=> ? = 7 + 1
[[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]]
=> [[1,1,1,1],[2,3,3],[3,4],[4]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4 = 3 + 1
[[0,1,0,0],[1,-1,1,0],[0,0,0,1],[0,1,0,0]]
=> [[1,1,1,2],[2,3,3],[3,4],[4]]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> [5,1]
=> 5 = 4 + 1
[[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
=> [[1,1,1,3],[2,3,3],[3,4],[4]]
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> [6,1]
=> 6 = 5 + 1
[[0,1,0,0],[0,0,1,0],[1,-1,0,1],[0,1,0,0]]
=> [[1,1,2,2],[2,3,3],[3,4],[4]]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> [6,1]
=> 6 = 5 + 1
[[0,0,1,0],[0,1,0,0],[1,-1,0,1],[0,1,0,0]]
=> [[1,1,2,3],[2,3,3],[3,4],[4]]
=> ([(0,3),(0,8),(1,10),(2,9),(3,11),(4,2),(5,4),(6,7),(7,1),(7,9),(8,5),(8,11),(9,10),(11,6)],12)
=> [7,5]
=> 7 = 6 + 1
[[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> [[1,1,1,1],[2,3,4],[3,4],[4]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> [5,3]
=> 5 = 4 + 1
[[0,1,0,0],[1,-1,0,1],[0,0,1,0],[0,1,0,0]]
=> [[1,1,1,2],[2,3,4],[3,4],[4]]
=> ([(0,5),(0,6),(1,8),(2,9),(3,8),(3,9),(4,1),(5,4),(6,7),(7,2),(7,3),(8,10),(9,10)],11)
=> [6,4,1]
=> 6 = 5 + 1
[[0,0,1,0],[1,0,-1,1],[0,0,1,0],[0,1,0,0]]
=> [[1,1,1,3],[2,3,4],[3,4],[4]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> [7,3]
=> 7 = 6 + 1
[[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
=> [[1,1,1,4],[2,3,4],[3,4],[4]]
=> ([(0,5),(0,10),(1,16),(2,15),(3,14),(4,13),(5,12),(6,2),(6,13),(7,4),(7,14),(8,1),(9,6),(10,11),(10,12),(11,3),(11,7),(12,9),(13,15),(14,8),(15,16)],17)
=> [8,6,3]
=> ? = 7 + 1
[[0,1,0,0],[0,0,0,1],[1,-1,1,0],[0,1,0,0]]
=> [[1,1,2,2],[2,3,4],[3,4],[4]]
=> ([(0,6),(0,7),(1,9),(2,12),(3,9),(3,12),(4,10),(5,1),(6,5),(7,8),(8,2),(8,3),(9,11),(11,10),(12,4),(12,11)],13)
=> [7,5,1]
=> ? = 6 + 1
[[0,0,1,0],[0,1,-1,1],[1,-1,1,0],[0,1,0,0]]
=> [[1,1,2,3],[2,3,4],[3,4],[4]]
=> ([(0,9),(0,11),(1,18),(2,17),(3,19),(4,13),(4,19),(5,12),(5,13),(6,16),(7,14),(8,5),(8,18),(9,10),(10,3),(10,4),(11,1),(11,8),(12,17),(13,15),(15,16),(16,14),(17,7),(18,2),(18,12),(19,6),(19,15)],20)
=> [8,6,4,2]
=> ? = 7 + 1
[[0,0,0,1],[0,1,0,0],[1,-1,1,0],[0,1,0,0]]
=> [[1,1,2,4],[2,3,4],[3,4],[4]]
=> ([(0,12),(0,13),(1,16),(2,15),(3,23),(4,19),(5,17),(5,20),(6,4),(7,5),(7,15),(7,16),(8,10),(9,7),(10,2),(11,1),(11,23),(12,8),(12,22),(13,14),(13,22),(14,3),(14,11),(15,17),(15,21),(16,20),(16,21),(17,24),(19,18),(20,19),(20,24),(21,24),(22,9),(23,6),(24,18)],25)
=> [9,7,5,4]
=> ? = 8 + 1
[[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> [[1,1,3,3],[2,3,4],[3,4],[4]]
=> ([(0,8),(2,11),(2,12),(3,10),(4,9),(5,4),(5,14),(6,3),(6,14),(7,1),(8,5),(8,6),(9,11),(9,13),(10,12),(10,13),(11,15),(12,15),(13,15),(14,2),(14,9),(14,10),(15,7)],16)
=> [9,4,3]
=> ? = 8 + 1
[[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
=> [[1,1,3,4],[2,3,4],[3,4],[4]]
=> ([(0,13),(0,15),(1,17),(2,16),(2,17),(3,19),(4,16),(4,18),(5,22),(6,21),(7,20),(8,23),(8,29),(9,4),(9,28),(10,3),(10,28),(11,6),(12,7),(12,24),(13,14),(14,1),(14,2),(15,9),(15,10),(16,25),(17,12),(17,25),(18,26),(18,29),(19,23),(19,26),(20,27),(22,27),(23,30),(24,20),(24,22),(25,24),(26,30),(27,21),(28,8),(28,18),(28,19),(29,5),(29,30),(30,11)],31)
=> [10,8,6,4,3]
=> ? = 9 + 1
[[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> [[1,2,2,2],[2,3,3],[3,4],[4]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> [7,1]
=> 7 = 6 + 1
[[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]]
=> [[1,2,2,3],[2,3,3],[3,4],[4]]
=> ([(0,5),(0,10),(1,16),(2,15),(3,14),(4,13),(5,12),(6,2),(6,13),(7,4),(7,14),(8,1),(9,6),(10,11),(10,12),(11,3),(11,7),(12,9),(13,15),(14,8),(15,16)],17)
=> [8,6,3]
=> ? = 7 + 1
[[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]]
=> [[1,2,2,2],[2,3,4],[3,4],[4]]
=> ([(0,7),(0,8),(1,12),(2,11),(3,10),(4,10),(4,11),(5,3),(6,1),(6,13),(7,9),(8,5),(9,2),(9,4),(10,14),(11,6),(11,14),(13,12),(14,13)],15)
=> [8,6,1]
=> ? = 7 + 1
[[0,0,1,0],[0,1,-1,1],[0,0,1,0],[1,0,0,0]]
=> [[1,2,2,3],[2,3,4],[3,4],[4]]
=> ([(0,10),(0,12),(1,16),(2,17),(3,21),(4,22),(5,14),(6,13),(6,14),(7,13),(7,15),(8,7),(8,21),(9,2),(9,18),(10,11),(11,5),(11,6),(12,3),(12,8),(13,19),(14,9),(14,19),(15,22),(16,20),(17,20),(18,16),(18,17),(19,18),(21,4),(21,15),(22,1)],23)
=> [9,7,5,2]
=> ? = 8 + 1
[[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
=> [[1,2,2,4],[2,3,4],[3,4],[4]]
=> ([(0,14),(0,15),(1,19),(2,18),(3,29),(4,30),(5,22),(6,23),(7,24),(7,25),(8,9),(9,7),(9,18),(9,19),(10,5),(11,6),(12,2),(12,29),(13,1),(13,30),(14,16),(14,28),(15,17),(15,28),(16,3),(16,12),(17,4),(17,13),(18,24),(18,27),(19,25),(19,27),(20,26),(21,26),(22,20),(23,21),(24,22),(24,31),(25,23),(25,31),(27,31),(28,8),(29,10),(30,11),(31,20),(31,21)],32)
=> [10,8,6,4,4]
=> ? = 9 + 1
[[0,0,1,0],[0,0,0,1],[0,1,0,0],[1,0,0,0]]
=> [[1,2,3,3],[2,3,4],[3,4],[4]]
=> ([(0,13),(0,15),(1,17),(2,16),(2,17),(3,19),(4,16),(4,18),(5,22),(6,21),(7,20),(8,23),(8,29),(9,4),(9,28),(10,3),(10,28),(11,6),(12,7),(12,24),(13,14),(14,1),(14,2),(15,9),(15,10),(16,25),(17,12),(17,25),(18,26),(18,29),(19,23),(19,26),(20,27),(22,27),(23,30),(24,20),(24,22),(25,24),(26,30),(27,21),(28,8),(28,18),(28,19),(29,5),(29,30),(30,11)],31)
=> [10,8,6,4,3]
=> ? = 9 + 1
[[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> [[1,2,3,4],[2,3,4],[3,4],[4]]
=> ([(0,5),(0,16),(0,17),(1,23),(2,19),(3,11),(3,21),(4,10),(4,20),(5,12),(5,13),(6,50),(7,51),(8,24),(8,58),(9,25),(9,59),(10,14),(10,52),(11,15),(11,53),(12,26),(12,60),(13,27),(13,60),(14,54),(15,55),(16,4),(16,29),(17,3),(17,29),(18,48),(18,49),(19,32),(19,33),(20,46),(20,52),(21,47),(21,53),(22,34),(22,35),(23,18),(23,54),(23,55),(24,40),(24,42),(25,41),(25,43),(26,46),(26,56),(27,47),(27,57),(28,63),(29,1),(30,62),(31,61),(32,61),(33,61),(34,6),(34,62),(35,7),(35,62),(36,58),(37,59),(38,32),(39,33),(40,44),(41,45),(42,38),(43,39),(44,31),(45,31),(46,36),(47,37),(48,40),(48,63),(49,41),(49,63),(50,38),(51,39),(52,8),(52,36),(53,9),(53,37),(54,28),(54,48),(55,28),(55,49),(56,30),(56,34),(57,30),(57,35),(58,42),(58,50),(59,43),(59,51),(60,22),(60,56),(60,57),(62,2),(63,44),(63,45)],64)
=> [11,9,9,7,7,5,5,5,3,3]
=> ? = 10 + 1
[[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,2,2],[3,3,3],[4,4],[5]]
=> ([],1)
=> [1]
=> 1 = 0 + 1
[[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,2],[2,2,2,2],[3,3,3],[4,4],[5]]
=> ([(0,1)],2)
=> [2]
=> 2 = 1 + 1
[[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,2,3],[3,3,3],[4,4],[5]]
=> ([(0,1)],2)
=> [2]
=> 2 = 1 + 1
[[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,2],[2,2,2,3],[3,3,3],[4,4],[5]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 3 = 2 + 1
[[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,3],[2,2,2,3],[3,3,3],[4,4],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4 = 3 + 1
[[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,2,2],[2,2,2,3],[3,3,3],[4,4],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4 = 3 + 1
[[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,2,3],[2,2,2,3],[3,3,3],[4,4],[5]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> [5,3]
=> 5 = 4 + 1
[[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,2,2],[3,3,4],[4,4],[5]]
=> ([(0,1)],2)
=> [2]
=> 2 = 1 + 1
[[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,2],[2,2,2,2],[3,3,4],[4,4],[5]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> [3,1]
=> 3 = 2 + 1
[[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,2,3],[3,3,4],[4,4],[5]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 3 = 2 + 1
[[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,2],[2,2,2,3],[3,3,4],[4,4],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4 = 3 + 1
[[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,3],[2,2,2,3],[3,3,4],[4,4],[5]]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> [5,1]
=> 5 = 4 + 1
[[0,1,0,0,0],[0,0,1,0,0],[1,0,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,2,2],[2,2,2,3],[3,3,4],[4,4],[5]]
=> ?
=> ?
=> ? = 4 + 1
[[0,0,1,0,0],[0,1,-1,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,2,3],[2,2,2,4],[3,3,4],[4,4],[5]]
=> ?
=> ?
=> ? = 6 + 1
[[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,2,4],[2,2,2,4],[3,3,4],[4,4],[5]]
=> ([(0,7),(0,8),(1,12),(2,11),(3,10),(4,10),(4,11),(5,3),(6,1),(6,13),(7,9),(8,5),(9,2),(9,4),(10,14),(11,6),(11,14),(13,12),(14,13)],15)
=> [8,6,1]
=> ? = 7 + 1
[[0,1,0,0,0],[1,-1,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,2],[2,2,3,3],[3,3,4],[4,4],[5]]
=> ?
=> ?
=> ? = 4 + 1
[[0,1,0,0,0],[0,0,1,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,2,2],[2,2,3,3],[3,3,4],[4,4],[5]]
=> ?
=> ?
=> ? = 5 + 1
[[0,0,1,0,0],[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,2,3],[2,2,3,3],[3,3,4],[4,4],[5]]
=> ?
=> ?
=> ? = 6 + 1
[[0,0,1,0,0],[1,0,-1,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,3],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ?
=> ?
=> ? = 6 + 1
[[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,4],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ([(0,5),(0,10),(1,16),(2,15),(3,14),(4,13),(5,12),(6,2),(6,13),(7,4),(7,14),(8,1),(9,6),(10,11),(10,12),(11,3),(11,7),(12,9),(13,15),(14,8),(15,16)],17)
=> [8,6,3]
=> ? = 7 + 1
[[0,1,0,0,0],[0,0,0,1,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,2,2],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ?
=> ?
=> ? = 6 + 1
[[0,0,1,0,0],[0,1,-1,1,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,2,3],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ([(0,9),(0,11),(1,18),(2,17),(3,19),(4,13),(4,19),(5,12),(5,13),(6,16),(7,14),(8,5),(8,18),(9,10),(10,3),(10,4),(11,1),(11,8),(12,17),(13,15),(15,16),(16,14),(17,7),(18,2),(18,12),(19,6),(19,15)],20)
=> [8,6,4,2]
=> ? = 7 + 1
[[0,0,0,1,0],[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,2,4],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ?
=> ?
=> ? = 8 + 1
[[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,3,3],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ([(0,8),(2,11),(2,12),(3,10),(4,9),(5,4),(5,14),(6,3),(6,14),(7,1),(8,5),(8,6),(9,11),(9,13),(10,12),(10,13),(11,15),(12,15),(13,15),(14,2),(14,9),(14,10),(15,7)],16)
=> [9,4,3]
=> ? = 8 + 1
[[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,3,4],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ([(0,13),(0,15),(1,17),(2,16),(2,17),(3,19),(4,16),(4,18),(5,22),(6,21),(7,20),(8,23),(8,29),(9,4),(9,28),(10,3),(10,28),(11,6),(12,7),(12,24),(13,14),(14,1),(14,2),(15,9),(15,10),(16,25),(17,12),(17,25),(18,26),(18,29),(19,23),(19,26),(20,27),(22,27),(23,30),(24,20),(24,22),(25,24),(26,30),(27,21),(28,8),(28,18),(28,19),(29,5),(29,30),(30,11)],31)
=> [10,8,6,4,3]
=> ? = 9 + 1
[[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,0,0,1]]
=> [[1,1,2,2,3],[2,2,3,3],[3,3,4],[4,4],[5]]
=> ([(0,5),(0,10),(1,16),(2,15),(3,14),(4,13),(5,12),(6,2),(6,13),(7,4),(7,14),(8,1),(9,6),(10,11),(10,12),(11,3),(11,7),(12,9),(13,15),(14,8),(15,16)],17)
=> [8,6,3]
=> ? = 7 + 1
[[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1]]
=> [[1,1,2,2,2],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ([(0,7),(0,8),(1,12),(2,11),(3,10),(4,10),(4,11),(5,3),(6,1),(6,13),(7,9),(8,5),(9,2),(9,4),(10,14),(11,6),(11,14),(13,12),(14,13)],15)
=> [8,6,1]
=> ? = 7 + 1
[[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1]]
=> [[1,1,2,2,3],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ?
=> ?
=> ? = 8 + 1
[[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1]]
=> [[1,1,2,2,4],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ([(0,14),(0,15),(1,19),(2,18),(3,29),(4,30),(5,22),(6,23),(7,24),(7,25),(8,9),(9,7),(9,18),(9,19),(10,5),(11,6),(12,2),(12,29),(13,1),(13,30),(14,16),(14,28),(15,17),(15,28),(16,3),(16,12),(17,4),(17,13),(18,24),(18,27),(19,25),(19,27),(20,26),(21,26),(22,20),(23,21),(24,22),(24,31),(25,23),(25,31),(27,31),(28,8),(29,10),(30,11),(31,20),(31,21)],32)
=> [10,8,6,4,4]
=> ? = 9 + 1
[[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1]]
=> [[1,1,2,3,3],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ([(0,13),(0,15),(1,17),(2,16),(2,17),(3,19),(4,16),(4,18),(5,22),(6,21),(7,20),(8,23),(8,29),(9,4),(9,28),(10,3),(10,28),(11,6),(12,7),(12,24),(13,14),(14,1),(14,2),(15,9),(15,10),(16,25),(17,12),(17,25),(18,26),(18,29),(19,23),(19,26),(20,27),(22,27),(23,30),(24,20),(24,22),(25,24),(26,30),(27,21),(28,8),(28,18),(28,19),(29,5),(29,30),(30,11)],31)
=> [10,8,6,4,3]
=> ? = 9 + 1
[[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1]]
=> [[1,1,2,3,4],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ([(0,5),(0,16),(0,17),(1,23),(2,19),(3,11),(3,21),(4,10),(4,20),(5,12),(5,13),(6,50),(7,51),(8,24),(8,58),(9,25),(9,59),(10,14),(10,52),(11,15),(11,53),(12,26),(12,60),(13,27),(13,60),(14,54),(15,55),(16,4),(16,29),(17,3),(17,29),(18,48),(18,49),(19,32),(19,33),(20,46),(20,52),(21,47),(21,53),(22,34),(22,35),(23,18),(23,54),(23,55),(24,40),(24,42),(25,41),(25,43),(26,46),(26,56),(27,47),(27,57),(28,63),(29,1),(30,62),(31,61),(32,61),(33,61),(34,6),(34,62),(35,7),(35,62),(36,58),(37,59),(38,32),(39,33),(40,44),(41,45),(42,38),(43,39),(44,31),(45,31),(46,36),(47,37),(48,40),(48,63),(49,41),(49,63),(50,38),(51,39),(52,8),(52,36),(53,9),(53,37),(54,28),(54,48),(55,28),(55,49),(56,30),(56,34),(57,30),(57,35),(58,42),(58,50),(59,43),(59,51),(60,22),(60,56),(60,57),(62,2),(63,44),(63,45)],64)
=> [11,9,9,7,7,5,5,5,3,3]
=> ? = 10 + 1
[[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,2,3],[2,2,2,3],[3,3,3],[4,5],[5]]
=> ([(0,3),(0,6),(0,7),(1,8),(1,12),(2,8),(2,11),(3,9),(3,10),(4,2),(4,13),(5,1),(5,14),(6,4),(6,9),(7,5),(7,10),(8,15),(9,13),(10,14),(11,15),(12,15),(13,11),(14,12)],16)
=> ?
=> ? = 5 + 1
[[0,1,0,0,0],[0,0,1,0,0],[1,0,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[1,1,1,2,2],[2,2,2,3],[3,3,4],[4,5],[5]]
=> ?
=> ?
=> ? = 5 + 1
[[0,0,1,0,0],[0,1,0,0,0],[1,0,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[1,1,1,2,3],[2,2,2,3],[3,3,4],[4,5],[5]]
=> ([(0,3),(0,8),(1,10),(2,10),(3,9),(4,6),(5,4),(5,11),(6,2),(7,1),(8,5),(8,9),(9,11),(11,7)],12)
=> ?
=> ? = 6 + 1
[[0,0,1,0,0],[1,0,-1,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[1,1,1,1,3],[2,2,2,4],[3,3,4],[4,5],[5]]
=> ([(0,4),(0,6),(1,10),(2,9),(3,7),(4,8),(5,3),(5,11),(6,1),(6,8),(7,9),(8,5),(8,10),(10,11),(11,2),(11,7)],12)
=> ?
=> ? = 6 + 1
[[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[1,1,1,1,4],[2,2,2,4],[3,3,4],[4,5],[5]]
=> ([(0,4),(0,7),(2,10),(3,9),(4,8),(5,3),(5,11),(6,1),(7,5),(7,8),(8,11),(9,10),(10,6),(11,2),(11,9)],12)
=> ?
=> ? = 7 + 1
[[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[1,1,1,2,2],[2,2,2,4],[3,3,4],[4,5],[5]]
=> ?
=> ?
=> ? = 6 + 1
[[0,0,1,0,0],[0,1,-1,1,0],[1,0,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[1,1,1,2,3],[2,2,2,4],[3,3,4],[4,5],[5]]
=> ?
=> ?
=> ? = 7 + 1
[[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[1,1,1,2,4],[2,2,2,4],[3,3,4],[4,5],[5]]
=> ?
=> ?
=> ? = 8 + 1
[[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [[1,1,1,1,1],[2,2,3,3],[3,3,4],[4,5],[5]]
=> ?
=> ?
=> ? = 4 + 1
[[0,1,0,0,0],[1,-1,1,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [[1,1,1,1,2],[2,2,3,3],[3,3,4],[4,5],[5]]
=> ?
=> ?
=> ? = 5 + 1
[[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [[1,1,1,1,3],[2,2,3,3],[3,3,4],[4,5],[5]]
=> ?
=> ?
=> ? = 6 + 1
[[0,1,0,0,0],[0,0,1,0,0],[1,-1,0,1,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [[1,1,1,2,2],[2,2,3,3],[3,3,4],[4,5],[5]]
=> ?
=> ?
=> ? = 6 + 1
[[0,0,1,0,0],[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [[1,1,1,2,3],[2,2,3,3],[3,3,4],[4,5],[5]]
=> ?
=> ?
=> ? = 7 + 1
[[0,1,0,0,0],[1,-1,0,1,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [[1,1,1,1,2],[2,2,3,4],[3,3,4],[4,5],[5]]
=> ([(0,4),(0,7),(1,9),(2,10),(3,9),(3,10),(4,11),(5,8),(6,1),(7,5),(7,11),(8,2),(8,3),(9,12),(10,12),(11,6)],13)
=> ?
=> ? = 6 + 1
[[0,0,1,0,0],[1,0,-1,1,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [[1,1,1,1,3],[2,2,3,4],[3,3,4],[4,5],[5]]
=> ?
=> ?
=> ? = 7 + 1
[[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [[1,1,1,1,4],[2,2,3,4],[3,3,4],[4,5],[5]]
=> ?
=> ?
=> ? = 8 + 1
[[0,1,0,0,0],[0,0,0,1,0],[1,-1,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [[1,1,1,2,2],[2,2,3,4],[3,3,4],[4,5],[5]]
=> ?
=> ?
=> ? = 7 + 1
Description
The maximum of the length and the largest part of the integer partition.
This is the side length of the smallest square the Ferrers diagram of the partition fits into. It is also the minimal number of colours required to colour the cells of the Ferrers diagram such that no two cells in a column or in a row have the same colour, see [1].
See also [[St001214]].
Matching statistic: St000259
Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00074: Posets —to graph⟶ Graphs
St000259: Graphs ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 9%
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00074: Posets —to graph⟶ Graphs
St000259: Graphs ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 9%
Values
[[1]]
=> [[1]]
=> ([],1)
=> ([],1)
=> 0
[[1,0],[0,1]]
=> [[1,1],[2]]
=> ([],1)
=> ([],1)
=> 0
[[0,1],[1,0]]
=> [[1,2],[2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ([],1)
=> ([],1)
=> 0
[[0,1,0],[1,0,0],[0,0,1]]
=> [[1,1,2],[2,2],[3]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[1,0,0],[0,0,1],[0,1,0]]
=> [[1,1,1],[2,3],[3]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[0,1,0],[1,-1,1],[0,1,0]]
=> [[1,1,2],[2,3],[3]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 2
[[0,0,1],[1,0,0],[0,1,0]]
=> [[1,1,3],[2,3],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 3
[[0,1,0],[0,0,1],[1,0,0]]
=> [[1,2,2],[2,3],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 3
[[0,0,1],[0,1,0],[1,0,0]]
=> [[1,2,3],[2,3],[3]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ([(0,6),(0,7),(1,2),(1,3),(2,5),(3,4),(4,6),(5,7)],8)
=> ? = 4
[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,2],[3,3],[4]]
=> ([],1)
=> ([],1)
=> 0
[[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,2],[2,2,2],[3,3],[4]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,3],[3,3],[4]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
=> [[1,1,1,2],[2,2,3],[3,3],[4]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 2
[[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> [[1,1,1,3],[2,2,3],[3,3],[4]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 3
[[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]]
=> [[1,1,2,2],[2,2,3],[3,3],[4]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 3
[[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> [[1,1,2,3],[2,2,3],[3,3],[4]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ([(0,6),(0,7),(1,2),(1,3),(2,5),(3,4),(4,6),(5,7)],8)
=> ? = 4
[[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [[1,1,1,1],[2,2,2],[3,4],[4]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> [[1,1,1,2],[2,2,2],[3,4],[4]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> [[1,1,1,1],[2,2,3],[3,4],[4]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 2
[[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> [[1,1,1,2],[2,2,3],[3,4],[4]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 3
[[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]]
=> [[1,1,1,3],[2,2,3],[3,4],[4]]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> 4
[[0,1,0,0],[0,0,1,0],[1,0,-1,1],[0,0,1,0]]
=> [[1,1,2,2],[2,2,3],[3,4],[4]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 4
[[0,0,1,0],[0,1,0,0],[1,0,-1,1],[0,0,1,0]]
=> [[1,1,2,3],[2,2,3],[3,4],[4]]
=> ([(0,3),(0,7),(1,8),(2,8),(3,9),(4,5),(5,1),(6,2),(7,4),(7,9),(9,6)],10)
=> ([(0,1),(0,3),(1,2),(2,4),(3,5),(4,8),(5,9),(6,7),(6,8),(7,9),(8,9)],10)
=> ? = 5
[[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> [[1,1,1,1],[2,2,4],[3,4],[4]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 3
[[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
=> [[1,1,1,2],[2,2,4],[3,4],[4]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(0,6),(1,2),(1,4),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
=> 4
[[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
=> [[1,1,1,3],[2,2,4],[3,4],[4]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ([(0,7),(1,3),(1,8),(2,7),(2,8),(3,5),(4,5),(4,6),(5,8),(6,7),(6,8)],9)
=> ? = 5
[[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [[1,1,1,4],[2,2,4],[3,4],[4]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ([(0,5),(1,4),(2,6),(2,7),(3,6),(3,7),(4,6),(5,7)],8)
=> ? = 6
[[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
=> [[1,1,2,2],[2,2,4],[3,4],[4]]
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> ([(0,8),(1,5),(1,7),(2,4),(2,6),(3,6),(3,7),(4,8),(4,9),(5,8),(5,9),(6,9),(7,9)],10)
=> ? = 5
[[0,0,1,0],[0,1,-1,1],[1,0,0,0],[0,0,1,0]]
=> [[1,1,2,3],[2,2,4],[3,4],[4]]
=> ([(0,6),(1,9),(1,10),(2,8),(3,7),(4,3),(4,12),(5,2),(5,12),(6,4),(6,5),(7,9),(7,11),(8,10),(8,11),(9,13),(10,13),(11,13),(12,1),(12,7),(12,8)],14)
=> ([(0,10),(1,9),(1,12),(2,8),(2,11),(3,5),(3,6),(3,7),(4,5),(4,6),(4,13),(5,11),(6,12),(7,11),(7,12),(8,10),(8,13),(9,10),(9,13),(11,13),(12,13)],14)
=> ? = 6
[[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
=> [[1,1,2,4],[2,2,4],[3,4],[4]]
=> ([(0,7),(0,8),(1,12),(2,11),(3,10),(4,10),(4,11),(5,3),(6,1),(6,13),(7,9),(8,5),(9,2),(9,4),(10,14),(11,6),(11,14),(13,12),(14,13)],15)
=> ([(0,2),(0,6),(1,2),(1,5),(3,4),(3,11),(4,9),(5,10),(6,8),(7,8),(7,14),(8,13),(9,11),(9,14),(10,12),(10,13),(11,12),(12,14),(13,14)],15)
=> ? = 7
[[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]]
=> [[1,1,1,1],[2,3,3],[3,4],[4]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 3
[[0,1,0,0],[1,-1,1,0],[0,0,0,1],[0,1,0,0]]
=> [[1,1,1,2],[2,3,3],[3,4],[4]]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> 4
[[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
=> [[1,1,1,3],[2,3,3],[3,4],[4]]
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ([(0,5),(1,2),(1,3),(2,6),(3,6),(4,5),(4,6)],7)
=> 5
[[0,1,0,0],[0,0,1,0],[1,-1,0,1],[0,1,0,0]]
=> [[1,1,2,2],[2,3,3],[3,4],[4]]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5)],7)
=> 5
[[0,0,1,0],[0,1,0,0],[1,-1,0,1],[0,1,0,0]]
=> [[1,1,2,3],[2,3,3],[3,4],[4]]
=> ([(0,3),(0,8),(1,10),(2,9),(3,11),(4,2),(5,4),(6,7),(7,1),(7,9),(8,5),(8,11),(9,10),(11,6)],12)
=> ([(0,5),(0,6),(1,4),(1,9),(2,3),(2,8),(3,10),(4,11),(5,8),(6,9),(7,10),(7,11),(8,10),(9,11)],12)
=> ? = 6
[[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> [[1,1,1,1],[2,3,4],[3,4],[4]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ([(0,6),(0,7),(1,2),(1,3),(2,5),(3,4),(4,6),(5,7)],8)
=> ? = 4
[[0,1,0,0],[1,-1,0,1],[0,0,1,0],[0,1,0,0]]
=> [[1,1,1,2],[2,3,4],[3,4],[4]]
=> ([(0,5),(0,6),(1,8),(2,9),(3,8),(3,9),(4,1),(5,4),(6,7),(7,2),(7,3),(8,10),(9,10)],11)
=> ([(0,1),(0,2),(1,4),(2,3),(3,8),(4,9),(5,7),(5,8),(6,7),(6,9),(7,10),(8,10),(9,10)],11)
=> ? = 5
[[0,0,1,0],[1,0,-1,1],[0,0,1,0],[0,1,0,0]]
=> [[1,1,1,3],[2,3,4],[3,4],[4]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ([(0,8),(1,7),(2,5),(2,6),(3,7),(3,9),(4,8),(4,9),(5,7),(5,9),(6,8),(6,9)],10)
=> ? = 6
[[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
=> [[1,1,1,4],[2,3,4],[3,4],[4]]
=> ([(0,5),(0,10),(1,16),(2,15),(3,14),(4,13),(5,12),(6,2),(6,13),(7,4),(7,14),(8,1),(9,6),(10,11),(10,12),(11,3),(11,7),(12,9),(13,15),(14,8),(15,16)],17)
=> ([(0,3),(0,4),(1,2),(1,9),(2,13),(3,10),(4,11),(5,10),(5,12),(6,9),(6,12),(7,11),(7,16),(8,14),(8,15),(9,13),(10,14),(11,15),(12,14),(13,16),(15,16)],17)
=> ? = 7
[[0,1,0,0],[0,0,0,1],[1,-1,1,0],[0,1,0,0]]
=> [[1,1,2,2],[2,3,4],[3,4],[4]]
=> ([(0,6),(0,7),(1,9),(2,12),(3,9),(3,12),(4,10),(5,1),(6,5),(7,8),(8,2),(8,3),(9,11),(11,10),(12,4),(12,11)],13)
=> ([(0,1),(0,2),(1,4),(2,6),(3,5),(3,12),(4,8),(5,9),(6,10),(7,8),(7,12),(8,11),(9,10),(9,12),(10,11),(11,12)],13)
=> ? = 6
[[0,0,1,0],[0,1,-1,1],[1,-1,1,0],[0,1,0,0]]
=> [[1,1,2,3],[2,3,4],[3,4],[4]]
=> ([(0,9),(0,11),(1,18),(2,17),(3,19),(4,13),(4,19),(5,12),(5,13),(6,16),(7,14),(8,5),(8,18),(9,10),(10,3),(10,4),(11,1),(11,8),(12,17),(13,15),(15,16),(16,14),(17,7),(18,2),(18,12),(19,6),(19,15)],20)
=> ([(0,15),(0,18),(1,14),(1,18),(2,16),(2,19),(3,17),(3,19),(4,6),(4,14),(5,7),(5,15),(6,16),(7,17),(8,9),(8,12),(8,13),(9,10),(9,11),(10,14),(10,18),(11,15),(11,18),(12,16),(12,19),(13,17),(13,19)],20)
=> ? = 7
[[0,0,0,1],[0,1,0,0],[1,-1,1,0],[0,1,0,0]]
=> [[1,1,2,4],[2,3,4],[3,4],[4]]
=> ([(0,12),(0,13),(1,16),(2,15),(3,23),(4,19),(5,17),(5,20),(6,4),(7,5),(7,15),(7,16),(8,10),(9,7),(10,2),(11,1),(11,23),(12,8),(12,22),(13,14),(13,22),(14,3),(14,11),(15,17),(15,21),(16,20),(16,21),(17,24),(19,18),(20,19),(20,24),(21,24),(22,9),(23,6),(24,18)],25)
=> ([(0,1),(0,5),(1,22),(2,4),(2,12),(3,16),(3,20),(4,10),(5,13),(6,17),(6,23),(7,10),(7,21),(8,12),(8,18),(9,13),(9,19),(10,24),(11,20),(11,21),(11,22),(12,16),(13,17),(14,15),(14,21),(14,22),(15,23),(15,24),(16,18),(17,19),(18,19),(20,23),(20,24),(21,24),(22,23)],25)
=> ? = 8
[[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> [[1,1,3,3],[2,3,4],[3,4],[4]]
=> ([(0,8),(2,11),(2,12),(3,10),(4,9),(5,4),(5,14),(6,3),(6,14),(7,1),(8,5),(8,6),(9,11),(9,13),(10,12),(10,13),(11,15),(12,15),(13,15),(14,2),(14,9),(14,10),(15,7)],16)
=> ([(0,11),(1,4),(2,10),(2,14),(3,9),(3,13),(4,12),(5,6),(5,7),(5,15),(6,12),(6,13),(7,12),(7,14),(8,12),(8,13),(8,14),(9,11),(9,15),(10,11),(10,15),(13,15),(14,15)],16)
=> ? = 8
[[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
=> [[1,1,3,4],[2,3,4],[3,4],[4]]
=> ([(0,13),(0,15),(1,17),(2,16),(2,17),(3,19),(4,16),(4,18),(5,22),(6,21),(7,20),(8,23),(8,29),(9,4),(9,28),(10,3),(10,28),(11,6),(12,7),(12,24),(13,14),(14,1),(14,2),(15,9),(15,10),(16,25),(17,12),(17,25),(18,26),(18,29),(19,23),(19,26),(20,27),(22,27),(23,30),(24,20),(24,22),(25,24),(26,30),(27,21),(28,8),(28,18),(28,19),(29,5),(29,30),(30,11)],31)
=> ([(0,20),(0,28),(1,19),(1,27),(2,22),(2,29),(3,21),(3,23),(4,8),(4,11),(5,6),(5,12),(6,19),(7,8),(7,24),(9,10),(9,16),(9,17),(10,19),(10,27),(11,21),(11,22),(12,13),(12,20),(13,16),(13,30),(14,15),(14,29),(14,30),(15,24),(15,28),(16,26),(17,25),(17,27),(18,24),(18,26),(18,28),(20,30),(21,25),(22,25),(23,25),(23,27),(24,29),(26,29),(26,30),(28,30)],31)
=> ? = 9
[[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> [[1,2,2,2],[2,3,3],[3,4],[4]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ([(0,5),(1,4),(2,6),(2,7),(3,6),(3,7),(4,6),(5,7)],8)
=> ? = 6
[[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]]
=> [[1,2,2,3],[2,3,3],[3,4],[4]]
=> ([(0,5),(0,10),(1,16),(2,15),(3,14),(4,13),(5,12),(6,2),(6,13),(7,4),(7,14),(8,1),(9,6),(10,11),(10,12),(11,3),(11,7),(12,9),(13,15),(14,8),(15,16)],17)
=> ([(0,3),(0,4),(1,2),(1,9),(2,13),(3,10),(4,11),(5,10),(5,12),(6,9),(6,12),(7,11),(7,16),(8,14),(8,15),(9,13),(10,14),(11,15),(12,14),(13,16),(15,16)],17)
=> ? = 7
[[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]]
=> [[1,2,2,2],[2,3,4],[3,4],[4]]
=> ([(0,7),(0,8),(1,12),(2,11),(3,10),(4,10),(4,11),(5,3),(6,1),(6,13),(7,9),(8,5),(9,2),(9,4),(10,14),(11,6),(11,14),(13,12),(14,13)],15)
=> ([(0,2),(0,6),(1,2),(1,5),(3,4),(3,11),(4,9),(5,10),(6,8),(7,8),(7,14),(8,13),(9,11),(9,14),(10,12),(10,13),(11,12),(12,14),(13,14)],15)
=> ? = 7
[[0,0,1,0],[0,1,-1,1],[0,0,1,0],[1,0,0,0]]
=> [[1,2,2,3],[2,3,4],[3,4],[4]]
=> ([(0,10),(0,12),(1,16),(2,17),(3,21),(4,22),(5,14),(6,13),(6,14),(7,13),(7,15),(8,7),(8,21),(9,2),(9,18),(10,11),(11,5),(11,6),(12,3),(12,8),(13,19),(14,9),(14,19),(15,22),(16,20),(17,20),(18,16),(18,17),(19,18),(21,4),(21,15),(22,1)],23)
=> ([(0,17),(0,22),(1,17),(1,18),(2,16),(2,19),(3,18),(3,19),(4,15),(4,22),(5,14),(5,21),(6,7),(6,14),(7,15),(8,9),(8,10),(8,11),(9,12),(9,13),(10,17),(10,22),(11,15),(11,22),(12,14),(12,21),(13,20),(13,21),(16,20),(16,21),(18,20),(19,20)],23)
=> ? = 8
[[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
=> [[1,2,2,4],[2,3,4],[3,4],[4]]
=> ([(0,14),(0,15),(1,19),(2,18),(3,29),(4,30),(5,22),(6,23),(7,24),(7,25),(8,9),(9,7),(9,18),(9,19),(10,5),(11,6),(12,2),(12,29),(13,1),(13,30),(14,16),(14,28),(15,17),(15,28),(16,3),(16,12),(17,4),(17,13),(18,24),(18,27),(19,25),(19,27),(20,26),(21,26),(22,20),(23,21),(24,22),(24,31),(25,23),(25,31),(27,31),(28,8),(29,10),(30,11),(31,20),(31,21)],32)
=> ([(0,15),(0,16),(1,18),(1,23),(2,17),(2,22),(3,24),(3,25),(4,20),(4,28),(5,21),(5,29),(6,19),(6,30),(7,9),(7,22),(8,10),(8,23),(9,11),(10,12),(11,24),(11,26),(12,25),(12,27),(13,26),(13,27),(13,30),(14,28),(14,29),(14,31),(15,17),(15,19),(16,18),(16,19),(17,20),(18,21),(20,22),(21,23),(24,31),(25,31),(26,28),(26,31),(27,29),(27,31),(28,30),(29,30)],32)
=> ? = 9
[[0,0,1,0],[0,0,0,1],[0,1,0,0],[1,0,0,0]]
=> [[1,2,3,3],[2,3,4],[3,4],[4]]
=> ([(0,13),(0,15),(1,17),(2,16),(2,17),(3,19),(4,16),(4,18),(5,22),(6,21),(7,20),(8,23),(8,29),(9,4),(9,28),(10,3),(10,28),(11,6),(12,7),(12,24),(13,14),(14,1),(14,2),(15,9),(15,10),(16,25),(17,12),(17,25),(18,26),(18,29),(19,23),(19,26),(20,27),(22,27),(23,30),(24,20),(24,22),(25,24),(26,30),(27,21),(28,8),(28,18),(28,19),(29,5),(29,30),(30,11)],31)
=> ([(0,20),(0,28),(1,19),(1,27),(2,22),(2,29),(3,21),(3,23),(4,8),(4,11),(5,6),(5,12),(6,19),(7,8),(7,24),(9,10),(9,16),(9,17),(10,19),(10,27),(11,21),(11,22),(12,13),(12,20),(13,16),(13,30),(14,15),(14,29),(14,30),(15,24),(15,28),(16,26),(17,25),(17,27),(18,24),(18,26),(18,28),(20,30),(21,25),(22,25),(23,25),(23,27),(24,29),(26,29),(26,30),(28,30)],31)
=> ? = 9
[[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> [[1,2,3,4],[2,3,4],[3,4],[4]]
=> ([(0,5),(0,16),(0,17),(1,23),(2,19),(3,11),(3,21),(4,10),(4,20),(5,12),(5,13),(6,50),(7,51),(8,24),(8,58),(9,25),(9,59),(10,14),(10,52),(11,15),(11,53),(12,26),(12,60),(13,27),(13,60),(14,54),(15,55),(16,4),(16,29),(17,3),(17,29),(18,48),(18,49),(19,32),(19,33),(20,46),(20,52),(21,47),(21,53),(22,34),(22,35),(23,18),(23,54),(23,55),(24,40),(24,42),(25,41),(25,43),(26,46),(26,56),(27,47),(27,57),(28,63),(29,1),(30,62),(31,61),(32,61),(33,61),(34,6),(34,62),(35,7),(35,62),(36,58),(37,59),(38,32),(39,33),(40,44),(41,45),(42,38),(43,39),(44,31),(45,31),(46,36),(47,37),(48,40),(48,63),(49,41),(49,63),(50,38),(51,39),(52,8),(52,36),(53,9),(53,37),(54,28),(54,48),(55,28),(55,49),(56,30),(56,34),(57,30),(57,35),(58,42),(58,50),(59,43),(59,51),(60,22),(60,56),(60,57),(62,2),(63,44),(63,45)],64)
=> ([(0,43),(0,57),(1,42),(1,56),(2,45),(2,59),(3,44),(3,58),(4,46),(4,60),(5,47),(5,61),(6,8),(6,10),(6,11),(7,9),(7,12),(7,13),(8,22),(8,23),(9,24),(9,25),(10,18),(10,42),(11,19),(11,42),(12,20),(12,43),(13,21),(13,43),(14,28),(14,34),(14,36),(15,29),(15,35),(15,37),(16,30),(16,32),(16,38),(17,31),(17,33),(17,39),(18,28),(18,44),(19,29),(19,45),(20,30),(20,46),(21,31),(21,47),(22,34),(22,63),(23,35),(23,63),(24,32),(24,62),(25,33),(25,62),(26,58),(26,59),(26,62),(27,60),(27,61),(27,63),(28,48),(29,49),(30,50),(31,51),(32,52),(33,53),(34,54),(35,55),(36,48),(36,50),(37,49),(37,51),(38,48),(38,50),(39,49),(39,51),(40,52),(40,53),(40,56),(41,54),(41,55),(41,57),(44,48),(45,49),(46,50),(47,51),(52,58),(52,62),(53,59),(53,62),(54,60),(54,63),(55,61),(55,63),(56,58),(56,59),(57,60),(57,61)],64)
=> ? = 10
[[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,2,2],[3,3,3],[4,4],[5]]
=> ([],1)
=> ([],1)
=> 0
[[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,2],[2,2,2,2],[3,3,3],[4,4],[5]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,2,3],[3,3,3],[4,4],[5]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,2],[2,2,2,3],[3,3,3],[4,4],[5]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 2
[[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,3],[2,2,2,3],[3,3,3],[4,4],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 3
[[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,2,2],[2,2,2,3],[3,3,3],[4,4],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 3
[[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,2,3],[2,2,2,3],[3,3,3],[4,4],[5]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ([(0,6),(0,7),(1,2),(1,3),(2,5),(3,4),(4,6),(5,7)],8)
=> ? = 4
[[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,2,2],[3,3,4],[4,4],[5]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,2],[2,2,2,2],[3,3,4],[4,4],[5]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,2,3],[3,3,4],[4,4],[5]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 2
[[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,2],[2,2,2,3],[3,3,4],[4,4],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 3
[[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,3],[2,2,2,3],[3,3,4],[4,4],[5]]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> 4
[[0,1,0,0,0],[0,0,1,0,0],[1,0,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,2,2],[2,2,2,3],[3,3,4],[4,4],[5]]
=> ?
=> ?
=> ? = 4
[[0,0,1,0,0],[0,1,0,0,0],[1,0,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,2,3],[2,2,2,3],[3,3,4],[4,4],[5]]
=> ([(0,3),(0,7),(1,8),(2,8),(3,9),(4,5),(5,1),(6,2),(7,4),(7,9),(9,6)],10)
=> ([(0,1),(0,3),(1,2),(2,4),(3,5),(4,8),(5,9),(6,7),(6,8),(7,9),(8,9)],10)
=> ? = 5
[[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,2,4],[3,3,4],[4,4],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 3
[[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,2],[2,2,2,4],[3,3,4],[4,4],[5]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(0,6),(1,2),(1,4),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
=> 4
[[0,0,1,0,0],[1,0,-1,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,3],[2,2,2,4],[3,3,4],[4,4],[5]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ([(0,7),(1,3),(1,8),(2,7),(2,8),(3,5),(4,5),(4,6),(5,8),(6,7),(6,8)],9)
=> ? = 5
[[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,4],[2,2,2,4],[3,3,4],[4,4],[5]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ([(0,5),(1,4),(2,6),(2,7),(3,6),(3,7),(4,6),(5,7)],8)
=> ? = 6
[[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,2,2],[2,2,2,4],[3,3,4],[4,4],[5]]
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> ([(0,8),(1,5),(1,7),(2,4),(2,6),(3,6),(3,7),(4,8),(4,9),(5,8),(5,9),(6,9),(7,9)],10)
=> ? = 5
[[0,0,1,0,0],[0,1,-1,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,2,3],[2,2,2,4],[3,3,4],[4,4],[5]]
=> ?
=> ?
=> ? = 6
[[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,2,4],[2,2,2,4],[3,3,4],[4,4],[5]]
=> ([(0,7),(0,8),(1,12),(2,11),(3,10),(4,10),(4,11),(5,3),(6,1),(6,13),(7,9),(8,5),(9,2),(9,4),(10,14),(11,6),(11,14),(13,12),(14,13)],15)
=> ([(0,2),(0,6),(1,2),(1,5),(3,4),(3,11),(4,9),(5,10),(6,8),(7,8),(7,14),(8,13),(9,11),(9,14),(10,12),(10,13),(11,12),(12,14),(13,14)],15)
=> ? = 7
[[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,3,3],[3,3,4],[4,4],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 3
[[0,1,0,0,0],[1,-1,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,2],[2,2,3,3],[3,3,4],[4,4],[5]]
=> ?
=> ?
=> ? = 4
[[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,3],[2,2,3,3],[3,3,4],[4,4],[5]]
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ([(0,5),(1,2),(1,3),(2,6),(3,6),(4,5),(4,6)],7)
=> 5
[[0,1,0,0,0],[0,0,1,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,2,2],[2,2,3,3],[3,3,4],[4,4],[5]]
=> ?
=> ?
=> ? = 5
[[0,0,1,0,0],[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,2,3],[2,2,3,3],[3,3,4],[4,4],[5]]
=> ?
=> ?
=> ? = 6
[[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ([(0,6),(0,7),(1,2),(1,3),(2,5),(3,4),(4,6),(5,7)],8)
=> ? = 4
[[0,1,0,0,0],[1,-1,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,2],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ([(0,5),(0,6),(1,8),(2,9),(3,8),(3,9),(4,1),(5,4),(6,7),(7,2),(7,3),(8,10),(9,10)],11)
=> ([(0,1),(0,2),(1,4),(2,3),(3,8),(4,9),(5,7),(5,8),(6,7),(6,9),(7,10),(8,10),(9,10)],11)
=> ? = 5
[[0,0,1,0,0],[1,0,-1,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,3],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ?
=> ?
=> ? = 6
[[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,4],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ([(0,5),(0,10),(1,16),(2,15),(3,14),(4,13),(5,12),(6,2),(6,13),(7,4),(7,14),(8,1),(9,6),(10,11),(10,12),(11,3),(11,7),(12,9),(13,15),(14,8),(15,16)],17)
=> ([(0,3),(0,4),(1,2),(1,9),(2,13),(3,10),(4,11),(5,10),(5,12),(6,9),(6,12),(7,11),(7,16),(8,14),(8,15),(9,13),(10,14),(11,15),(12,14),(13,16),(15,16)],17)
=> ? = 7
[[0,1,0,0,0],[0,0,0,1,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,2,2],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ?
=> ?
=> ? = 6
[[0,0,1,0,0],[0,1,-1,1,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,2,3],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ([(0,9),(0,11),(1,18),(2,17),(3,19),(4,13),(4,19),(5,12),(5,13),(6,16),(7,14),(8,5),(8,18),(9,10),(10,3),(10,4),(11,1),(11,8),(12,17),(13,15),(15,16),(16,14),(17,7),(18,2),(18,12),(19,6),(19,15)],20)
=> ([(0,15),(0,18),(1,14),(1,18),(2,16),(2,19),(3,17),(3,19),(4,6),(4,14),(5,7),(5,15),(6,16),(7,17),(8,9),(8,12),(8,13),(9,10),(9,11),(10,14),(10,18),(11,15),(11,18),(12,16),(12,19),(13,17),(13,19)],20)
=> ? = 7
[[0,0,0,1,0],[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,2,4],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ?
=> ?
=> ? = 8
[[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,3,3],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ([(0,8),(2,11),(2,12),(3,10),(4,9),(5,4),(5,14),(6,3),(6,14),(7,1),(8,5),(8,6),(9,11),(9,13),(10,12),(10,13),(11,15),(12,15),(13,15),(14,2),(14,9),(14,10),(15,7)],16)
=> ([(0,11),(1,4),(2,10),(2,14),(3,9),(3,13),(4,12),(5,6),(5,7),(5,15),(6,12),(6,13),(7,12),(7,14),(8,12),(8,13),(8,14),(9,11),(9,15),(10,11),(10,15),(13,15),(14,15)],16)
=> ? = 8
[[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,3,4],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ([(0,13),(0,15),(1,17),(2,16),(2,17),(3,19),(4,16),(4,18),(5,22),(6,21),(7,20),(8,23),(8,29),(9,4),(9,28),(10,3),(10,28),(11,6),(12,7),(12,24),(13,14),(14,1),(14,2),(15,9),(15,10),(16,25),(17,12),(17,25),(18,26),(18,29),(19,23),(19,26),(20,27),(22,27),(23,30),(24,20),(24,22),(25,24),(26,30),(27,21),(28,8),(28,18),(28,19),(29,5),(29,30),(30,11)],31)
=> ([(0,20),(0,28),(1,19),(1,27),(2,22),(2,29),(3,21),(3,23),(4,8),(4,11),(5,6),(5,12),(6,19),(7,8),(7,24),(9,10),(9,16),(9,17),(10,19),(10,27),(11,21),(11,22),(12,13),(12,20),(13,16),(13,30),(14,15),(14,29),(14,30),(15,24),(15,28),(16,26),(17,25),(17,27),(18,24),(18,26),(18,28),(20,30),(21,25),(22,25),(23,25),(23,27),(24,29),(26,29),(26,30),(28,30)],31)
=> ? = 9
[[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,0,0,1]]
=> [[1,1,2,2,2],[2,2,3,3],[3,3,4],[4,4],[5]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ([(0,5),(1,4),(2,6),(2,7),(3,6),(3,7),(4,6),(5,7)],8)
=> ? = 6
[[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,0,0,1]]
=> [[1,1,2,2,3],[2,2,3,3],[3,3,4],[4,4],[5]]
=> ([(0,5),(0,10),(1,16),(2,15),(3,14),(4,13),(5,12),(6,2),(6,13),(7,4),(7,14),(8,1),(9,6),(10,11),(10,12),(11,3),(11,7),(12,9),(13,15),(14,8),(15,16)],17)
=> ([(0,3),(0,4),(1,2),(1,9),(2,13),(3,10),(4,11),(5,10),(5,12),(6,9),(6,12),(7,11),(7,16),(8,14),(8,15),(9,13),(10,14),(11,15),(12,14),(13,16),(15,16)],17)
=> ? = 7
[[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1]]
=> [[1,1,2,2,2],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ([(0,7),(0,8),(1,12),(2,11),(3,10),(4,10),(4,11),(5,3),(6,1),(6,13),(7,9),(8,5),(9,2),(9,4),(10,14),(11,6),(11,14),(13,12),(14,13)],15)
=> ([(0,2),(0,6),(1,2),(1,5),(3,4),(3,11),(4,9),(5,10),(6,8),(7,8),(7,14),(8,13),(9,11),(9,14),(10,12),(10,13),(11,12),(12,14),(13,14)],15)
=> ? = 7
[[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1]]
=> [[1,1,2,2,3],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ?
=> ?
=> ? = 8
[[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1]]
=> [[1,1,2,2,4],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ([(0,14),(0,15),(1,19),(2,18),(3,29),(4,30),(5,22),(6,23),(7,24),(7,25),(8,9),(9,7),(9,18),(9,19),(10,5),(11,6),(12,2),(12,29),(13,1),(13,30),(14,16),(14,28),(15,17),(15,28),(16,3),(16,12),(17,4),(17,13),(18,24),(18,27),(19,25),(19,27),(20,26),(21,26),(22,20),(23,21),(24,22),(24,31),(25,23),(25,31),(27,31),(28,8),(29,10),(30,11),(31,20),(31,21)],32)
=> ([(0,15),(0,16),(1,18),(1,23),(2,17),(2,22),(3,24),(3,25),(4,20),(4,28),(5,21),(5,29),(6,19),(6,30),(7,9),(7,22),(8,10),(8,23),(9,11),(10,12),(11,24),(11,26),(12,25),(12,27),(13,26),(13,27),(13,30),(14,28),(14,29),(14,31),(15,17),(15,19),(16,18),(16,19),(17,20),(18,21),(20,22),(21,23),(24,31),(25,31),(26,28),(26,31),(27,29),(27,31),(28,30),(29,30)],32)
=> ? = 9
[[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,1,1],[2,2,2,2],[3,3,3],[4,5],[5]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,1,2],[2,2,2,2],[3,3,3],[4,5],[5]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,1,1],[2,2,2,3],[3,3,3],[4,5],[5]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,1,2],[2,2,2,3],[3,3,3],[4,5],[5]]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
=> 3
[[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[1,1,1,1,1],[2,2,2,2],[3,3,4],[4,5],[5]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 2
[[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[1,1,1,1,2],[2,2,2,2],[3,3,4],[4,5],[5]]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
=> 3
[[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[1,1,1,1,1],[2,2,2,3],[3,3,4],[4,5],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 3
[[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[1,1,1,1,2],[2,2,2,3],[3,3,4],[4,5],[5]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 4
Description
The diameter of a connected graph.
This is the greatest distance between any pair of vertices.
Matching statistic: St001340
Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St001340: Graphs ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 9%
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St001340: Graphs ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 9%
Values
[[1]]
=> [[1]]
=> ([],1)
=> ([],1)
=> 0
[[1,0],[0,1]]
=> [[1,1],[2]]
=> ([],1)
=> ([],1)
=> 0
[[0,1],[1,0]]
=> [[1,2],[2]]
=> ([(0,1)],2)
=> ([],2)
=> 1
[[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ([],1)
=> ([],1)
=> 0
[[0,1,0],[1,0,0],[0,0,1]]
=> [[1,1,2],[2,2],[3]]
=> ([(0,1)],2)
=> ([],2)
=> 1
[[1,0,0],[0,0,1],[0,1,0]]
=> [[1,1,1],[2,3],[3]]
=> ([(0,1)],2)
=> ([],2)
=> 1
[[0,1,0],[1,-1,1],[0,1,0]]
=> [[1,1,2],[2,3],[3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 2
[[0,0,1],[1,0,0],[0,1,0]]
=> [[1,1,3],[2,3],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 3
[[0,1,0],[0,0,1],[1,0,0]]
=> [[1,2,2],[2,3],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 3
[[0,0,1],[0,1,0],[1,0,0]]
=> [[1,2,3],[2,3],[3]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ([(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
=> ? = 4
[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,2],[3,3],[4]]
=> ([],1)
=> ([],1)
=> 0
[[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,2],[2,2,2],[3,3],[4]]
=> ([(0,1)],2)
=> ([],2)
=> 1
[[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,3],[3,3],[4]]
=> ([(0,1)],2)
=> ([],2)
=> 1
[[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
=> [[1,1,1,2],[2,2,3],[3,3],[4]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 2
[[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> [[1,1,1,3],[2,2,3],[3,3],[4]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 3
[[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]]
=> [[1,1,2,2],[2,2,3],[3,3],[4]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 3
[[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> [[1,1,2,3],[2,2,3],[3,3],[4]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ([(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
=> ? = 4
[[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [[1,1,1,1],[2,2,2],[3,4],[4]]
=> ([(0,1)],2)
=> ([],2)
=> 1
[[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> [[1,1,1,2],[2,2,2],[3,4],[4]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 2
[[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> [[1,1,1,1],[2,2,3],[3,4],[4]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 2
[[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> [[1,1,1,2],[2,2,3],[3,4],[4]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 3
[[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]]
=> [[1,1,1,3],[2,2,3],[3,4],[4]]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(4,5)],6)
=> 4
[[0,1,0,0],[0,0,1,0],[1,0,-1,1],[0,0,1,0]]
=> [[1,1,2,2],[2,2,3],[3,4],[4]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 4
[[0,0,1,0],[0,1,0,0],[1,0,-1,1],[0,0,1,0]]
=> [[1,1,2,3],[2,2,3],[3,4],[4]]
=> ([(0,3),(0,7),(1,8),(2,8),(3,9),(4,5),(5,1),(6,2),(7,4),(7,9),(9,6)],10)
=> ([(2,9),(3,6),(3,7),(3,8),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,9),(7,9),(8,9)],10)
=> ? = 5
[[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> [[1,1,1,1],[2,2,4],[3,4],[4]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 3
[[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
=> [[1,1,1,2],[2,2,4],[3,4],[4]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> 4
[[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
=> [[1,1,1,3],[2,2,4],[3,4],[4]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
=> ? = 5
[[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [[1,1,1,4],[2,2,4],[3,4],[4]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ([(6,7)],8)
=> ? = 6
[[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
=> [[1,1,2,2],[2,2,4],[3,4],[4]]
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> ([(3,6),(3,9),(4,5),(4,9),(5,8),(6,8),(7,8),(7,9),(8,9)],10)
=> ? = 5
[[0,0,1,0],[0,1,-1,1],[1,0,0,0],[0,0,1,0]]
=> [[1,1,2,3],[2,2,4],[3,4],[4]]
=> ([(0,6),(1,9),(1,10),(2,8),(3,7),(4,3),(4,12),(5,2),(5,12),(6,4),(6,5),(7,9),(7,11),(8,10),(8,11),(9,13),(10,13),(11,13),(12,1),(12,7),(12,8)],14)
=> ([(3,12),(3,13),(4,5),(4,13),(5,12),(6,9),(6,10),(6,11),(7,8),(7,10),(7,11),(7,12),(8,9),(8,11),(8,13),(9,10),(9,12),(10,13),(11,12),(11,13),(12,13)],14)
=> ? = 6
[[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
=> [[1,1,2,4],[2,2,4],[3,4],[4]]
=> ([(0,7),(0,8),(1,12),(2,11),(3,10),(4,10),(4,11),(5,3),(6,1),(6,13),(7,9),(8,5),(9,2),(9,4),(10,14),(11,6),(11,14),(13,12),(14,13)],15)
=> ([(2,11),(3,10),(3,11),(4,12),(4,13),(4,14),(5,12),(5,13),(5,14),(6,8),(6,12),(6,13),(6,14),(7,9),(7,12),(7,13),(7,14),(8,9),(8,10),(8,11),(9,12),(9,13),(9,14),(10,12),(10,13),(10,14),(11,12),(11,13),(11,14)],15)
=> ? = 7
[[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]]
=> [[1,1,1,1],[2,3,3],[3,4],[4]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 3
[[0,1,0,0],[1,-1,1,0],[0,0,0,1],[0,1,0,0]]
=> [[1,1,1,2],[2,3,3],[3,4],[4]]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(4,5)],6)
=> 4
[[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
=> [[1,1,1,3],[2,3,3],[3,4],[4]]
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ([(5,6)],7)
=> 5
[[0,1,0,0],[0,0,1,0],[1,-1,0,1],[0,1,0,0]]
=> [[1,1,2,2],[2,3,3],[3,4],[4]]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ([(5,6)],7)
=> 5
[[0,0,1,0],[0,1,0,0],[1,-1,0,1],[0,1,0,0]]
=> [[1,1,2,3],[2,3,3],[3,4],[4]]
=> ([(0,3),(0,8),(1,10),(2,9),(3,11),(4,2),(5,4),(6,7),(7,1),(7,9),(8,5),(8,11),(9,10),(11,6)],12)
=> ([(2,8),(3,7),(4,9),(4,10),(4,11),(5,9),(5,10),(5,11),(6,9),(6,10),(6,11),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11)],12)
=> ? = 6
[[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> [[1,1,1,1],[2,3,4],[3,4],[4]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ([(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
=> ? = 4
[[0,1,0,0],[1,-1,0,1],[0,0,1,0],[0,1,0,0]]
=> [[1,1,1,2],[2,3,4],[3,4],[4]]
=> ([(0,5),(0,6),(1,8),(2,9),(3,8),(3,9),(4,1),(5,4),(6,7),(7,2),(7,3),(8,10),(9,10)],11)
=> ([(2,6),(2,10),(3,7),(3,8),(3,9),(4,7),(4,8),(4,9),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(7,10),(8,10),(9,10)],11)
=> ? = 5
[[0,0,1,0],[1,0,-1,1],[0,0,1,0],[0,1,0,0]]
=> [[1,1,1,3],[2,3,4],[3,4],[4]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ([(4,9),(5,8),(6,7),(7,9),(8,9)],10)
=> ? = 6
[[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
=> [[1,1,1,4],[2,3,4],[3,4],[4]]
=> ([(0,5),(0,10),(1,16),(2,15),(3,14),(4,13),(5,12),(6,2),(6,13),(7,4),(7,14),(8,1),(9,6),(10,11),(10,12),(11,3),(11,7),(12,9),(13,15),(14,8),(15,16)],17)
=> ([(2,14),(3,11),(3,12),(3,13),(3,16),(4,10),(4,11),(4,12),(4,13),(4,16),(5,7),(5,8),(5,9),(5,10),(5,14),(6,7),(6,8),(6,9),(6,10),(6,14),(6,16),(7,11),(7,12),(7,13),(7,15),(7,16),(8,11),(8,12),(8,13),(8,15),(8,16),(9,11),(9,12),(9,13),(9,15),(9,16),(10,11),(10,12),(10,13),(10,15),(10,16),(11,14),(11,15),(12,14),(12,15),(13,14),(13,15),(14,15),(14,16),(15,16)],17)
=> ? = 7
[[0,1,0,0],[0,0,0,1],[1,-1,1,0],[0,1,0,0]]
=> [[1,1,2,2],[2,3,4],[3,4],[4]]
=> ([(0,6),(0,7),(1,9),(2,12),(3,9),(3,12),(4,10),(5,1),(6,5),(7,8),(8,2),(8,3),(9,11),(11,10),(12,4),(12,11)],13)
=> ([(2,9),(3,10),(3,11),(3,12),(4,10),(4,11),(4,12),(5,6),(5,8),(5,9),(6,10),(6,11),(6,12),(7,8),(7,10),(7,11),(7,12),(8,10),(8,11),(8,12),(9,10),(9,11),(9,12)],13)
=> ? = 6
[[0,0,1,0],[0,1,-1,1],[1,-1,1,0],[0,1,0,0]]
=> [[1,1,2,3],[2,3,4],[3,4],[4]]
=> ([(0,9),(0,11),(1,18),(2,17),(3,19),(4,13),(4,19),(5,12),(5,13),(6,16),(7,14),(8,5),(8,18),(9,10),(10,3),(10,4),(11,1),(11,8),(12,17),(13,15),(15,16),(16,14),(17,7),(18,2),(18,12),(19,6),(19,15)],20)
=> ([(2,10),(2,11),(2,13),(2,15),(2,17),(2,19),(3,8),(3,9),(3,12),(3,14),(3,16),(3,18),(4,8),(4,9),(4,12),(4,14),(4,16),(4,18),(4,19),(5,10),(5,11),(5,13),(5,15),(5,17),(5,18),(5,19),(6,8),(6,9),(6,12),(6,13),(6,14),(6,16),(6,17),(6,18),(6,19),(7,10),(7,11),(7,12),(7,13),(7,15),(7,16),(7,17),(7,18),(7,19),(8,10),(8,11),(8,13),(8,15),(8,17),(8,19),(9,10),(9,11),(9,13),(9,15),(9,17),(9,19),(10,12),(10,14),(10,16),(10,18),(11,12),(11,14),(11,16),(11,18),(12,13),(12,15),(12,17),(12,19),(13,14),(13,16),(13,18),(14,15),(14,16),(14,17),(14,19),(15,16),(15,17),(15,18),(16,17),(16,19),(17,18),(18,19)],20)
=> ? = 7
[[0,0,0,1],[0,1,0,0],[1,-1,1,0],[0,1,0,0]]
=> [[1,1,2,4],[2,3,4],[3,4],[4]]
=> ([(0,12),(0,13),(1,16),(2,15),(3,23),(4,19),(5,17),(5,20),(6,4),(7,5),(7,15),(7,16),(8,10),(9,7),(10,2),(11,1),(11,23),(12,8),(12,22),(13,14),(13,22),(14,3),(14,11),(15,17),(15,21),(16,20),(16,21),(17,24),(19,18),(20,19),(20,24),(21,24),(22,9),(23,6),(24,18)],25)
=> ([(2,6),(2,21),(2,22),(2,23),(3,4),(3,18),(3,19),(3,20),(3,24),(4,5),(4,14),(4,15),(4,21),(4,22),(4,23),(5,12),(5,15),(5,16),(5,18),(5,19),(5,20),(5,24),(6,10),(6,13),(6,17),(6,18),(6,19),(6,20),(6,24),(7,10),(7,13),(7,17),(7,18),(7,19),(7,20),(7,21),(7,22),(7,23),(7,24),(8,10),(8,13),(8,17),(8,18),(8,19),(8,20),(8,21),(8,22),(8,23),(8,24),(9,10),(9,13),(9,17),(9,18),(9,19),(9,20),(9,21),(9,22),(9,23),(9,24),(10,14),(10,15),(10,16),(10,21),(10,22),(10,23),(11,14),(11,15),(11,16),(11,18),(11,19),(11,20),(11,21),(11,22),(11,23),(11,24),(12,14),(12,15),(12,18),(12,19),(12,20),(12,21),(12,22),(12,23),(12,24),(13,14),(13,15),(13,16),(13,21),(13,22),(13,23),(13,24),(14,16),(14,17),(14,18),(14,19),(14,20),(14,24),(15,17),(15,18),(15,19),(15,20),(15,24),(16,17),(16,18),(16,19),(16,20),(16,21),(16,22),(16,23),(16,24),(17,18),(17,19),(17,20),(17,21),(17,22),(17,23),(17,24),(18,21),(18,22),(18,23),(19,21),(19,22),(19,23),(20,21),(20,22),(20,23),(21,24),(22,24),(23,24)],25)
=> ? = 8
[[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> [[1,1,3,3],[2,3,4],[3,4],[4]]
=> ([(0,8),(2,11),(2,12),(3,10),(4,9),(5,4),(5,14),(6,3),(6,14),(7,1),(8,5),(8,6),(9,11),(9,13),(10,12),(10,13),(11,15),(12,15),(13,15),(14,2),(14,9),(14,10),(15,7)],16)
=> ([(5,14),(5,15),(6,7),(6,15),(7,14),(8,11),(8,12),(8,13),(9,10),(9,12),(9,13),(9,14),(10,11),(10,13),(10,15),(11,12),(11,14),(12,15),(13,14),(13,15),(14,15)],16)
=> ? = 8
[[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
=> [[1,1,3,4],[2,3,4],[3,4],[4]]
=> ([(0,13),(0,15),(1,17),(2,16),(2,17),(3,19),(4,16),(4,18),(5,22),(6,21),(7,20),(8,23),(8,29),(9,4),(9,28),(10,3),(10,28),(11,6),(12,7),(12,24),(13,14),(14,1),(14,2),(15,9),(15,10),(16,25),(17,12),(17,25),(18,26),(18,29),(19,23),(19,26),(20,27),(22,27),(23,30),(24,20),(24,22),(25,24),(26,30),(27,21),(28,8),(28,18),(28,19),(29,5),(29,30),(30,11)],31)
=> ([(2,11),(2,12),(2,13),(2,21),(2,25),(2,26),(2,30),(3,18),(3,19),(3,22),(3,23),(3,27),(3,28),(3,29),(4,10),(4,11),(4,12),(4,13),(4,21),(4,25),(4,26),(4,29),(4,30),(5,8),(5,18),(5,19),(5,22),(5,23),(5,27),(5,28),(5,29),(5,30),(6,7),(6,9),(6,10),(6,16),(6,18),(6,19),(6,20),(6,22),(6,23),(6,27),(6,28),(6,29),(6,30),(7,8),(7,17),(7,18),(7,19),(7,22),(7,23),(7,25),(7,26),(7,27),(7,28),(7,29),(7,30),(8,9),(8,10),(8,16),(8,18),(8,19),(8,20),(8,22),(8,23),(8,27),(8,28),(8,29),(9,11),(9,12),(9,13),(9,14),(9,15),(9,17),(9,21),(9,24),(9,25),(9,26),(9,29),(9,30),(10,11),(10,12),(10,13),(10,14),(10,15),(10,17),(10,21),(10,24),(10,25),(10,26),(10,30),(11,16),(11,18),(11,19),(11,20),(11,22),(11,23),(11,24),(11,27),(11,28),(11,29),(12,16),(12,18),(12,19),(12,20),(12,22),(12,23),(12,24),(12,27),(12,28),(12,29),(13,16),(13,18),(13,19),(13,20),(13,22),(13,23),(13,24),(13,27),(13,28),(13,29),(14,16),(14,18),(14,19),(14,20),(14,21),(14,22),(14,23),(14,25),(14,26),(14,27),(14,28),(14,29),(14,30),(15,16),(15,17),(15,18),(15,19),(15,20),(15,22),(15,23),(15,25),(15,26),(15,27),(15,28),(15,29),(15,30),(16,17),(16,21),(16,24),(16,25),(16,26),(16,27),(16,29),(16,30),(17,18),(17,19),(17,20),(17,21),(17,22),(17,23),(17,26),(17,27),(17,28),(17,29),(17,30),(18,21),(18,24),(18,25),(18,26),(18,30),(19,21),(19,24),(19,25),(19,26),(19,30),(20,21),(20,23),(20,24),(20,25),(20,26),(20,27),(20,28),(20,29),(20,30),(21,22),(21,23),(21,24),(21,25),(21,27),(21,28),(21,29),(22,24),(22,25),(22,26),(22,28),(22,30),(23,24),(23,25),(23,26),(23,30),(24,25),(24,26),(24,27),(24,28),(24,29),(24,30),(25,27),(25,28),(25,29),(26,27),(26,28),(26,29),(27,30),(28,30),(29,30)],31)
=> ? = 9
[[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> [[1,2,2,2],[2,3,3],[3,4],[4]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ([(6,7)],8)
=> ? = 6
[[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]]
=> [[1,2,2,3],[2,3,3],[3,4],[4]]
=> ([(0,5),(0,10),(1,16),(2,15),(3,14),(4,13),(5,12),(6,2),(6,13),(7,4),(7,14),(8,1),(9,6),(10,11),(10,12),(11,3),(11,7),(12,9),(13,15),(14,8),(15,16)],17)
=> ([(2,14),(3,11),(3,12),(3,13),(3,16),(4,10),(4,11),(4,12),(4,13),(4,16),(5,7),(5,8),(5,9),(5,10),(5,14),(6,7),(6,8),(6,9),(6,10),(6,14),(6,16),(7,11),(7,12),(7,13),(7,15),(7,16),(8,11),(8,12),(8,13),(8,15),(8,16),(9,11),(9,12),(9,13),(9,15),(9,16),(10,11),(10,12),(10,13),(10,15),(10,16),(11,14),(11,15),(12,14),(12,15),(13,14),(13,15),(14,15),(14,16),(15,16)],17)
=> ? = 7
[[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]]
=> [[1,2,2,2],[2,3,4],[3,4],[4]]
=> ([(0,7),(0,8),(1,12),(2,11),(3,10),(4,10),(4,11),(5,3),(6,1),(6,13),(7,9),(8,5),(9,2),(9,4),(10,14),(11,6),(11,14),(13,12),(14,13)],15)
=> ([(2,11),(3,10),(3,11),(4,12),(4,13),(4,14),(5,12),(5,13),(5,14),(6,8),(6,12),(6,13),(6,14),(7,9),(7,12),(7,13),(7,14),(8,9),(8,10),(8,11),(9,12),(9,13),(9,14),(10,12),(10,13),(10,14),(11,12),(11,13),(11,14)],15)
=> ? = 7
[[0,0,1,0],[0,1,-1,1],[0,0,1,0],[1,0,0,0]]
=> [[1,2,2,3],[2,3,4],[3,4],[4]]
=> ([(0,10),(0,12),(1,16),(2,17),(3,21),(4,22),(5,14),(6,13),(6,14),(7,13),(7,15),(8,7),(8,21),(9,2),(9,18),(10,11),(11,5),(11,6),(12,3),(12,8),(13,19),(14,9),(14,19),(15,22),(16,20),(17,20),(18,16),(18,17),(19,18),(21,4),(21,15),(22,1)],23)
=> ([(2,5),(2,22),(3,15),(3,16),(3,18),(3,19),(3,20),(3,21),(3,22),(4,8),(4,9),(4,12),(4,13),(4,14),(4,17),(4,22),(5,15),(5,16),(5,18),(5,19),(5,20),(5,21),(6,15),(6,16),(6,17),(6,18),(6,19),(6,20),(6,21),(6,22),(7,8),(7,9),(7,12),(7,13),(7,14),(7,17),(7,21),(7,22),(8,11),(8,15),(8,16),(8,18),(8,19),(8,20),(8,21),(9,11),(9,15),(9,16),(9,18),(9,19),(9,20),(9,21),(10,12),(10,14),(10,15),(10,16),(10,17),(10,18),(10,19),(10,20),(10,21),(10,22),(11,12),(11,13),(11,14),(11,17),(11,18),(11,20),(11,21),(11,22),(12,15),(12,16),(12,18),(12,19),(12,20),(12,21),(13,14),(13,15),(13,16),(13,18),(13,19),(13,20),(13,21),(14,15),(14,16),(14,18),(14,19),(14,20),(14,21),(15,17),(15,22),(16,17),(16,22),(17,18),(17,19),(17,20),(17,21),(18,22),(19,20),(19,22),(20,22),(21,22)],23)
=> ? = 8
[[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
=> [[1,2,2,4],[2,3,4],[3,4],[4]]
=> ([(0,14),(0,15),(1,19),(2,18),(3,29),(4,30),(5,22),(6,23),(7,24),(7,25),(8,9),(9,7),(9,18),(9,19),(10,5),(11,6),(12,2),(12,29),(13,1),(13,30),(14,16),(14,28),(15,17),(15,28),(16,3),(16,12),(17,4),(17,13),(18,24),(18,27),(19,25),(19,27),(20,26),(21,26),(22,20),(23,21),(24,22),(24,31),(25,23),(25,31),(27,31),(28,8),(29,10),(30,11),(31,20),(31,21)],32)
=> ([(2,3),(2,9),(2,27),(2,28),(2,29),(2,31),(3,8),(3,24),(3,25),(3,26),(3,30),(4,5),(4,18),(4,20),(4,23),(4,24),(4,25),(4,26),(4,30),(5,17),(5,19),(5,22),(5,27),(5,28),(5,29),(5,31),(6,8),(6,9),(6,24),(6,25),(6,26),(6,27),(6,28),(6,29),(6,30),(6,31),(7,8),(7,9),(7,13),(7,14),(7,21),(7,24),(7,25),(7,26),(7,27),(7,28),(7,29),(7,30),(7,31),(8,9),(8,13),(8,15),(8,17),(8,19),(8,22),(8,27),(8,28),(8,29),(8,31),(9,14),(9,16),(9,18),(9,20),(9,23),(9,24),(9,25),(9,26),(9,30),(10,17),(10,18),(10,19),(10,20),(10,22),(10,23),(10,24),(10,25),(10,26),(10,27),(10,28),(10,29),(10,30),(10,31),(11,17),(11,18),(11,19),(11,20),(11,22),(11,23),(11,24),(11,25),(11,26),(11,27),(11,28),(11,29),(11,30),(11,31),(12,17),(12,18),(12,19),(12,20),(12,22),(12,23),(12,24),(12,25),(12,26),(12,27),(12,28),(12,29),(12,30),(12,31),(13,14),(13,16),(13,18),(13,20),(13,23),(13,24),(13,25),(13,26),(13,27),(13,28),(13,29),(13,30),(13,31),(14,15),(14,17),(14,19),(14,22),(14,24),(14,25),(14,26),(14,27),(14,28),(14,29),(14,30),(14,31),(15,16),(15,18),(15,20),(15,21),(15,23),(15,24),(15,25),(15,26),(15,27),(15,28),(15,29),(15,30),(15,31),(16,17),(16,19),(16,21),(16,22),(16,24),(16,25),(16,26),(16,27),(16,28),(16,29),(16,30),(16,31),(17,18),(17,20),(17,21),(17,23),(17,24),(17,25),(17,26),(17,30),(18,19),(18,21),(18,22),(18,27),(18,28),(18,29),(18,31),(19,20),(19,21),(19,23),(19,24),(19,25),(19,26),(19,30),(19,31),(20,21),(20,22),(20,27),(20,28),(20,29),(20,30),(20,31),(21,22),(21,23),(21,24),(21,25),(21,26),(21,27),(21,28),(21,29),(21,30),(21,31),(22,23),(22,24),(22,25),(22,26),(22,27),(22,28),(22,29),(22,30),(22,31),(23,24),(23,25),(23,26),(23,27),(23,28),(23,29),(23,30),(23,31),(24,27),(24,28),(24,29),(24,31),(25,27),(25,28),(25,29),(25,31),(26,27),(26,28),(26,29),(26,31),(27,30),(28,30),(29,30),(30,31)],32)
=> ? = 9
[[0,0,1,0],[0,0,0,1],[0,1,0,0],[1,0,0,0]]
=> [[1,2,3,3],[2,3,4],[3,4],[4]]
=> ([(0,13),(0,15),(1,17),(2,16),(2,17),(3,19),(4,16),(4,18),(5,22),(6,21),(7,20),(8,23),(8,29),(9,4),(9,28),(10,3),(10,28),(11,6),(12,7),(12,24),(13,14),(14,1),(14,2),(15,9),(15,10),(16,25),(17,12),(17,25),(18,26),(18,29),(19,23),(19,26),(20,27),(22,27),(23,30),(24,20),(24,22),(25,24),(26,30),(27,21),(28,8),(28,18),(28,19),(29,5),(29,30),(30,11)],31)
=> ([(2,11),(2,12),(2,13),(2,21),(2,25),(2,26),(2,30),(3,18),(3,19),(3,22),(3,23),(3,27),(3,28),(3,29),(4,10),(4,11),(4,12),(4,13),(4,21),(4,25),(4,26),(4,29),(4,30),(5,8),(5,18),(5,19),(5,22),(5,23),(5,27),(5,28),(5,29),(5,30),(6,7),(6,9),(6,10),(6,16),(6,18),(6,19),(6,20),(6,22),(6,23),(6,27),(6,28),(6,29),(6,30),(7,8),(7,17),(7,18),(7,19),(7,22),(7,23),(7,25),(7,26),(7,27),(7,28),(7,29),(7,30),(8,9),(8,10),(8,16),(8,18),(8,19),(8,20),(8,22),(8,23),(8,27),(8,28),(8,29),(9,11),(9,12),(9,13),(9,14),(9,15),(9,17),(9,21),(9,24),(9,25),(9,26),(9,29),(9,30),(10,11),(10,12),(10,13),(10,14),(10,15),(10,17),(10,21),(10,24),(10,25),(10,26),(10,30),(11,16),(11,18),(11,19),(11,20),(11,22),(11,23),(11,24),(11,27),(11,28),(11,29),(12,16),(12,18),(12,19),(12,20),(12,22),(12,23),(12,24),(12,27),(12,28),(12,29),(13,16),(13,18),(13,19),(13,20),(13,22),(13,23),(13,24),(13,27),(13,28),(13,29),(14,16),(14,18),(14,19),(14,20),(14,21),(14,22),(14,23),(14,25),(14,26),(14,27),(14,28),(14,29),(14,30),(15,16),(15,17),(15,18),(15,19),(15,20),(15,22),(15,23),(15,25),(15,26),(15,27),(15,28),(15,29),(15,30),(16,17),(16,21),(16,24),(16,25),(16,26),(16,27),(16,29),(16,30),(17,18),(17,19),(17,20),(17,21),(17,22),(17,23),(17,26),(17,27),(17,28),(17,29),(17,30),(18,21),(18,24),(18,25),(18,26),(18,30),(19,21),(19,24),(19,25),(19,26),(19,30),(20,21),(20,23),(20,24),(20,25),(20,26),(20,27),(20,28),(20,29),(20,30),(21,22),(21,23),(21,24),(21,25),(21,27),(21,28),(21,29),(22,24),(22,25),(22,26),(22,28),(22,30),(23,24),(23,25),(23,26),(23,30),(24,25),(24,26),(24,27),(24,28),(24,29),(24,30),(25,27),(25,28),(25,29),(26,27),(26,28),(26,29),(27,30),(28,30),(29,30)],31)
=> ? = 9
[[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> [[1,2,3,4],[2,3,4],[3,4],[4]]
=> ([(0,5),(0,16),(0,17),(1,23),(2,19),(3,11),(3,21),(4,10),(4,20),(5,12),(5,13),(6,50),(7,51),(8,24),(8,58),(9,25),(9,59),(10,14),(10,52),(11,15),(11,53),(12,26),(12,60),(13,27),(13,60),(14,54),(15,55),(16,4),(16,29),(17,3),(17,29),(18,48),(18,49),(19,32),(19,33),(20,46),(20,52),(21,47),(21,53),(22,34),(22,35),(23,18),(23,54),(23,55),(24,40),(24,42),(25,41),(25,43),(26,46),(26,56),(27,47),(27,57),(28,63),(29,1),(30,62),(31,61),(32,61),(33,61),(34,6),(34,62),(35,7),(35,62),(36,58),(37,59),(38,32),(39,33),(40,44),(41,45),(42,38),(43,39),(44,31),(45,31),(46,36),(47,37),(48,40),(48,63),(49,41),(49,63),(50,38),(51,39),(52,8),(52,36),(53,9),(53,37),(54,28),(54,48),(55,28),(55,49),(56,30),(56,34),(57,30),(57,35),(58,42),(58,50),(59,43),(59,51),(60,22),(60,56),(60,57),(62,2),(63,44),(63,45)],64)
=> ([(2,3),(2,6),(2,7),(2,10),(2,11),(2,14),(2,15),(2,18),(2,19),(2,21),(2,23),(2,26),(2,27),(2,31),(2,32),(2,33),(2,36),(2,37),(2,40),(2,41),(2,44),(2,45),(2,48),(2,49),(2,52),(2,53),(2,55),(2,58),(2,59),(2,62),(2,63),(3,4),(3,5),(3,8),(3,9),(3,12),(3,13),(3,16),(3,17),(3,20),(3,22),(3,24),(3,25),(3,28),(3,29),(3,30),(3,34),(3,35),(3,38),(3,39),(3,42),(3,43),(3,46),(3,47),(3,50),(3,51),(3,54),(3,56),(3,57),(3,60),(3,61),(4,5),(4,7),(4,10),(4,11),(4,13),(4,15),(4,17),(4,19),(4,21),(4,23),(4,26),(4,27),(4,31),(4,32),(4,33),(4,36),(4,37),(4,40),(4,41),(4,43),(4,45),(4,47),(4,49),(4,51),(4,53),(4,55),(4,57),(4,59),(4,61),(4,62),(4,63),(5,6),(5,10),(5,11),(5,12),(5,14),(5,16),(5,18),(5,21),(5,23),(5,26),(5,27),(5,31),(5,32),(5,33),(5,36),(5,37),(5,40),(5,41),(5,42),(5,44),(5,46),(5,48),(5,50),(5,52),(5,55),(5,56),(5,58),(5,60),(5,62),(5,63),(6,7),(6,8),(6,9),(6,13),(6,15),(6,17),(6,19),(6,20),(6,22),(6,24),(6,25),(6,28),(6,29),(6,30),(6,34),(6,35),(6,38),(6,39),(6,43),(6,45),(6,47),(6,49),(6,51),(6,53),(6,54),(6,57),(6,59),(6,60),(6,61),(6,62),(7,8),(7,9),(7,12),(7,14),(7,16),(7,18),(7,20),(7,22),(7,24),(7,25),(7,28),(7,29),(7,30),(7,34),(7,35),(7,38),(7,39),(7,42),(7,44),(7,46),(7,48),(7,50),(7,52),(7,54),(7,56),(7,58),(7,60),(7,61),(7,63),(8,9),(8,10),(8,11),(8,14),(8,15),(8,18),(8,19),(8,21),(8,23),(8,25),(8,26),(8,27),(8,31),(8,32),(8,33),(8,36),(8,37),(8,40),(8,41),(8,42),(8,44),(8,45),(8,46),(8,48),(8,49),(8,50),(8,52),(8,53),(8,55),(8,56),(8,58),(8,59),(8,62),(8,63),(9,10),(9,11),(9,14),(9,15),(9,18),(9,19),(9,21),(9,23),(9,24),(9,26),(9,27),(9,31),(9,32),(9,33),(9,36),(9,37),(9,40),(9,41),(9,43),(9,44),(9,45),(9,47),(9,48),(9,49),(9,51),(9,52),(9,53),(9,55),(9,57),(9,58),(9,59),(9,62),(9,63),(10,11),(10,12),(10,13),(10,16),(10,17),(10,20),(10,22),(10,24),(10,25),(10,27),(10,28),(10,29),(10,30),(10,34),(10,35),(10,38),(10,39),(10,42),(10,43),(10,44),(10,46),(10,47),(10,48),(10,50),(10,51),(10,52),(10,54),(10,56),(10,57),(10,58),(10,60),(10,61),(11,12),(11,13),(11,16),(11,17),(11,20),(11,22),(11,24),(11,25),(11,26),(11,28),(11,29),(11,30),(11,34),(11,35),(11,38),(11,39),(11,42),(11,43),(11,45),(11,46),(11,47),(11,49),(11,50),(11,51),(11,53),(11,54),(11,56),(11,57),(11,59),(11,60),(11,61),(12,13),(12,15),(12,17),(12,19),(12,21),(12,23),(12,24),(12,26),(12,27),(12,28),(12,29),(12,30),(12,31),(12,32),(12,33),(12,34),(12,36),(12,37),(12,38),(12,40),(12,41),(12,43),(12,45),(12,47),(12,49),(12,51),(12,53),(12,54),(12,55),(12,57),(12,59),(12,61),(12,62),(12,63),(13,14),(13,16),(13,18),(13,21),(13,23),(13,25),(13,26),(13,27),(13,28),(13,29),(13,30),(13,31),(13,32),(13,33),(13,35),(13,36),(13,37),(13,39),(13,40),(13,41),(13,42),(13,44),(13,46),(13,48),(13,50),(13,52),(13,54),(13,55),(13,56),(13,58),(13,60),(13,62),(13,63),(14,15),(14,17),(14,19),(14,20),(14,22),(14,24),(14,25),(14,26),(14,28),(14,29),(14,30),(14,31),(14,32),(14,33),(14,34),(14,35),(14,36),(14,38),(14,39),(14,40),(14,43),(14,45),(14,47),(14,49),(14,51),(14,53),(14,54),(14,55),(14,57),(14,59),(14,60),(14,61),(14,62),(15,16),(15,18),(15,20),(15,22),(15,24),(15,25),(15,27),(15,28),(15,29),(15,30),(15,31),(15,32),(15,33),(15,34),(15,35),(15,37),(15,38),(15,39),(15,41),(15,42),(15,44),(15,46),(15,48),(15,50),(15,52),(15,54),(15,55),(15,56),(15,58),(15,60),(15,61),(15,63),(16,17),(16,19),(16,21),(16,23),(16,24),(16,26),(16,27),(16,28),(16,29),(16,30),(16,31),(16,32),(16,33),(16,34),(16,36),(16,37),(16,38),(16,40),(16,41),(16,43),(16,45),(16,46),(16,47),(16,49),(16,51),(16,53),(16,54),(16,55),(16,57),(16,58),(16,59),(16,61),(16,62),(16,63),(17,18),(17,21),(17,23),(17,25),(17,26),(17,27),(17,28),(17,29),(17,30),(17,31),(17,32),(17,33),(17,35),(17,36),(17,37),(17,39),(17,40),(17,41),(17,42),(17,44),(17,46),(17,47),(17,48),(17,50),(17,52),(17,54),(17,55),(17,56),(17,58),(17,59),(17,60),(17,62),(17,63),(18,19),(18,20),(18,22),(18,24),(18,25),(18,26),(18,28),(18,29),(18,30),(18,31),(18,32),(18,33),(18,34),(18,35),(18,36),(18,38),(18,39),(18,40),(18,43),(18,45),(18,47),(18,48),(18,49),(18,51),(18,53),(18,54),(18,55),(18,56),(18,57),(18,59),(18,60),(18,61),(18,62),(19,20),(19,22),(19,24),(19,25),(19,27),(19,28),(19,29),(19,30),(19,31),(19,32),(19,33),(19,34),(19,35),(19,37),(19,38),(19,39),(19,41),(19,42),(19,44),(19,46),(19,48),(19,49),(19,50),(19,52),(19,54),(19,55),(19,56),(19,57),(19,58),(19,60),(19,61),(19,63),(20,21),(20,23),(20,24),(20,25),(20,26),(20,27),(20,31),(20,32),(20,33),(20,36),(20,37),(20,40),(20,41),(20,42),(20,43),(20,44),(20,45),(20,46),(20,47),(20,48),(20,49),(20,50),(20,51),(20,52),(20,53),(20,55),(20,56),(20,57),(20,58),(20,59),(20,62),(20,63),(21,22),(21,24),(21,25),(21,26),(21,27),(21,28),(21,29),(21,30),(21,34),(21,35),(21,38),(21,39),(21,42),(21,43),(21,44),(21,45),(21,46),(21,47),(21,48),(21,49),(21,50),(21,51),(21,52),(21,53),(21,54),(21,56),(21,57),(21,58),(21,59),(21,60),(21,61),(22,23),(22,24),(22,25),(22,26),(22,27),(22,31),(22,32),(22,33),(22,34),(22,35),(22,36),(22,37),(22,40),(22,41),(22,42),(22,43),(22,44),(22,45),(22,46),(22,47),(22,48),(22,49),(22,50),(22,51),(22,52),(22,53),(22,54),(22,55),(22,56),(22,57),(22,58),(22,59),(22,62),(22,63),(23,24),(23,25),(23,26),(23,27),(23,28),(23,29),(23,30),(23,34),(23,35),(23,36),(23,37),(23,38),(23,39),(23,42),(23,43),(23,44),(23,45),(23,46),(23,47),(23,48),(23,49),(23,50),(23,51),(23,52),(23,53),(23,54),(23,55),(23,56),(23,57),(23,58),(23,59),(23,60),(23,61),(24,25),(24,26),(24,27),(24,31),(24,32),(24,33),(24,35),(24,36),(24,37),(24,39),(24,40),(24,41),(24,42),(24,44),(24,45),(24,46),(24,48),(24,49),(24,50),(24,52),(24,53),(24,55),(24,56),(24,58),(24,59),(24,60),(24,62),(24,63),(25,26),(25,27),(25,31),(25,32),(25,33),(25,34),(25,36),(25,37),(25,38),(25,40),(25,41),(25,43),(25,44),(25,45),(25,47),(25,48),(25,49),(25,51),(25,52),(25,53),(25,55),(25,57),(25,58),(25,59),(25,61),(25,62),(25,63),(26,27),(26,28),(26,29),(26,30),(26,34),(26,35),(26,37),(26,38),(26,39),(26,41),(26,42),(26,43),(26,44),(26,46),(26,47),(26,48),(26,50),(26,51),(26,52),(26,54),(26,56),(26,57),(26,58),(26,60),(26,61),(26,63),(27,28),(27,29),(27,30),(27,34),(27,35),(27,36),(27,38),(27,39),(27,40),(27,42),(27,43),(27,45),(27,46),(27,47),(27,49),(27,50),(27,51),(27,53),(27,54),(27,56),(27,57),(27,59),(27,60),(27,61),(27,62),(28,31),(28,32),(28,33),(28,36),(28,37),(28,40),(28,41),(28,42),(28,43),(28,44),(28,45),(28,46),(28,47),(28,48),(28,49),(28,50),(28,51),(28,52),(28,53),(28,55),(28,56),(28,57),(28,58),(28,59),(28,60),(28,61),(28,62),(28,63),(29,31),(29,32),(29,33),(29,36),(29,37),(29,40),(29,41),(29,42),(29,43),(29,44),(29,45),(29,46),(29,47),(29,48),(29,49),(29,50),(29,51),(29,52),(29,53),(29,55),(29,56),(29,57),(29,58),(29,59),(29,60),(29,61),(29,62),(29,63),(30,31),(30,32),(30,33),(30,36),(30,37),(30,40),(30,41),(30,42),(30,43),(30,44),(30,45),(30,46),(30,47),(30,48),(30,49),(30,50),(30,51),(30,52),(30,53),(30,55),(30,56),(30,57),(30,58),(30,59),(30,60),(30,61),(30,62),(30,63),(31,34),(31,35),(31,38),(31,39),(31,42),(31,43),(31,44),(31,45),(31,46),(31,47),(31,48),(31,49),(31,50),(31,51),(31,52),(31,53),(31,54),(31,56),(31,57),(31,58),(31,59),(31,60),(31,61),(31,62),(31,63),(32,34),(32,35),(32,38),(32,39),(32,42),(32,43),(32,44),(32,45),(32,46),(32,47),(32,48),(32,49),(32,50),(32,51),(32,52),(32,53),(32,54),(32,56),(32,57),(32,58),(32,59),(32,60),(32,61),(32,62),(32,63),(33,34),(33,35),(33,38),(33,39),(33,42),(33,43),(33,44),(33,45),(33,46),(33,47),(33,48),(33,49),(33,50),(33,51),(33,52),(33,53),(33,54),(33,56),(33,57),(33,58),(33,59),(33,60),(33,61),(33,62),(33,63),(34,35),(34,36),(34,37),(34,39),(34,40),(34,41),(34,42),(34,43),(34,44),(34,45),(34,46),(34,47),(34,48),(34,49),(34,50),(34,51),(34,52),(34,53),(34,55),(34,56),(34,57),(34,58),(34,59),(34,60),(34,62),(34,63),(35,36),(35,37),(35,38),(35,40),(35,41),(35,42),(35,43),(35,44),(35,45),(35,46),(35,47),(35,48),(35,49),(35,50),(35,51),(35,52),(35,53),(35,55),(35,56),(35,57),(35,58),(35,59),(35,61),(35,62),(35,63),(36,37),(36,38),(36,39),(36,41),(36,42),(36,43),(36,44),(36,45),(36,46),(36,47),(36,48),(36,49),(36,50),(36,51),(36,52),(36,53),(36,54),(36,56),(36,57),(36,58),(36,59),(36,60),(36,61),(36,63),(37,38),(37,39),(37,40),(37,42),(37,43),(37,44),(37,45),(37,46),(37,47),(37,48),(37,49),(37,50),(37,51),(37,52),(37,53),(37,54),(37,56),(37,57),(37,58),(37,59),(37,60),(37,61),(37,62),(38,39),(38,40),(38,41),(38,42),(38,43),(38,44),(38,45),(38,46),(38,47),(38,48),(38,49),(38,50),(38,51),(38,52),(38,53),(38,54),(38,55),(38,56),(38,57),(38,58),(38,59),(38,60),(38,62),(38,63),(39,40),(39,41),(39,42),(39,43),(39,44),(39,45),(39,46),(39,47),(39,48),(39,49),(39,50),(39,51),(39,52),(39,53),(39,54),(39,55),(39,56),(39,57),(39,58),(39,59),(39,61),(39,62),(39,63),(40,41),(40,42),(40,43),(40,44),(40,45),(40,46),(40,47),(40,48),(40,49),(40,50),(40,51),(40,52),(40,53),(40,54),(40,55),(40,56),(40,57),(40,58),(40,59),(40,60),(40,61),(40,63),(41,42),(41,43),(41,44),(41,45),(41,46),(41,47),(41,48),(41,49),(41,50),(41,51),(41,52),(41,53),(41,54),(41,55),(41,56),(41,57),(41,58),(41,59),(41,60),(41,61),(41,62),(42,43),(42,45),(42,47),(42,49),(42,51),(42,53),(42,54),(42,55),(42,57),(42,58),(42,59),(42,60),(42,61),(42,62),(42,63),(43,44),(43,46),(43,48),(43,50),(43,52),(43,54),(43,55),(43,56),(43,58),(43,59),(43,60),(43,61),(43,62),(43,63),(44,45),(44,47),(44,49),(44,51),(44,53),(44,54),(44,55),(44,56),(44,57),(44,59),(44,60),(44,61),(44,62),(44,63),(45,46),(45,48),(45,50),(45,52),(45,54),(45,55),(45,56),(45,57),(45,58),(45,60),(45,61),(45,62),(45,63),(46,47),(46,49),(46,51),(46,53),(46,54),(46,55),(46,57),(46,59),(46,60),(46,61),(46,62),(46,63),(47,48),(47,50),(47,52),(47,54),(47,55),(47,56),(47,58),(47,60),(47,61),(47,62),(47,63),(48,49),(48,51),(48,53),(48,54),(48,55),(48,57),(48,59),(48,60),(48,61),(48,62),(48,63),(49,50),(49,52),(49,54),(49,55),(49,56),(49,58),(49,60),(49,61),(49,62),(49,63),(50,51),(50,52),(50,53),(50,54),(50,55),(50,57),(50,58),(50,59),(50,60),(50,61),(50,62),(50,63),(51,52),(51,53),(51,54),(51,55),(51,56),(51,58),(51,59),(51,60),(51,61),(51,62),(51,63),(52,53),(52,54),(52,55),(52,56),(52,57),(52,59),(52,60),(52,61),(52,62),(52,63),(53,54),(53,55),(53,56),(53,57),(53,58),(53,60),(53,61),(53,62),(53,63),(54,55),(54,56),(54,57),(54,58),(54,59),(54,60),(54,61),(54,62),(54,63),(55,56),(55,57),(55,58),(55,59),(55,60),(55,61),(55,62),(55,63),(56,57),(56,58),(56,59),(56,60),(56,61),(56,62),(56,63),(57,58),(57,59),(57,60),(57,61),(57,62),(57,63),(58,59),(58,60),(58,61),(58,62),(58,63),(59,60),(59,61),(59,62),(59,63),(60,61),(60,62),(60,63),(61,62),(61,63),(62,63)],64)
=> ? = 10
[[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,2,2],[3,3,3],[4,4],[5]]
=> ([],1)
=> ([],1)
=> 0
[[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,2],[2,2,2,2],[3,3,3],[4,4],[5]]
=> ([(0,1)],2)
=> ([],2)
=> 1
[[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,2,3],[3,3,3],[4,4],[5]]
=> ([(0,1)],2)
=> ([],2)
=> 1
[[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,2],[2,2,2,3],[3,3,3],[4,4],[5]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 2
[[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,3],[2,2,2,3],[3,3,3],[4,4],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 3
[[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,2,2],[2,2,2,3],[3,3,3],[4,4],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 3
[[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,2,3],[2,2,2,3],[3,3,3],[4,4],[5]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ([(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
=> ? = 4
[[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,2,2],[3,3,4],[4,4],[5]]
=> ([(0,1)],2)
=> ([],2)
=> 1
[[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,2],[2,2,2,2],[3,3,4],[4,4],[5]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 2
[[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,2,3],[3,3,4],[4,4],[5]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 2
[[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,2],[2,2,2,3],[3,3,4],[4,4],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 3
[[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,3],[2,2,2,3],[3,3,4],[4,4],[5]]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(4,5)],6)
=> 4
[[0,1,0,0,0],[0,0,1,0,0],[1,0,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,2,2],[2,2,2,3],[3,3,4],[4,4],[5]]
=> ?
=> ?
=> ? = 4
[[0,0,1,0,0],[0,1,0,0,0],[1,0,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,2,3],[2,2,2,3],[3,3,4],[4,4],[5]]
=> ([(0,3),(0,7),(1,8),(2,8),(3,9),(4,5),(5,1),(6,2),(7,4),(7,9),(9,6)],10)
=> ([(2,9),(3,6),(3,7),(3,8),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,9),(7,9),(8,9)],10)
=> ? = 5
[[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,2,4],[3,3,4],[4,4],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 3
[[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,2],[2,2,2,4],[3,3,4],[4,4],[5]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> 4
[[0,0,1,0,0],[1,0,-1,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,3],[2,2,2,4],[3,3,4],[4,4],[5]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
=> ? = 5
[[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,4],[2,2,2,4],[3,3,4],[4,4],[5]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ([(6,7)],8)
=> ? = 6
[[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,2,2],[2,2,2,4],[3,3,4],[4,4],[5]]
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> ([(3,6),(3,9),(4,5),(4,9),(5,8),(6,8),(7,8),(7,9),(8,9)],10)
=> ? = 5
[[0,0,1,0,0],[0,1,-1,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,2,3],[2,2,2,4],[3,3,4],[4,4],[5]]
=> ?
=> ?
=> ? = 6
[[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,2,4],[2,2,2,4],[3,3,4],[4,4],[5]]
=> ([(0,7),(0,8),(1,12),(2,11),(3,10),(4,10),(4,11),(5,3),(6,1),(6,13),(7,9),(8,5),(9,2),(9,4),(10,14),(11,6),(11,14),(13,12),(14,13)],15)
=> ([(2,11),(3,10),(3,11),(4,12),(4,13),(4,14),(5,12),(5,13),(5,14),(6,8),(6,12),(6,13),(6,14),(7,9),(7,12),(7,13),(7,14),(8,9),(8,10),(8,11),(9,12),(9,13),(9,14),(10,12),(10,13),(10,14),(11,12),(11,13),(11,14)],15)
=> ? = 7
[[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,3,3],[3,3,4],[4,4],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 3
[[0,1,0,0,0],[1,-1,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,2],[2,2,3,3],[3,3,4],[4,4],[5]]
=> ?
=> ?
=> ? = 4
[[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,3],[2,2,3,3],[3,3,4],[4,4],[5]]
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ([(5,6)],7)
=> 5
[[0,1,0,0,0],[0,0,1,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,2,2],[2,2,3,3],[3,3,4],[4,4],[5]]
=> ?
=> ?
=> ? = 5
[[0,0,1,0,0],[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,2,3],[2,2,3,3],[3,3,4],[4,4],[5]]
=> ?
=> ?
=> ? = 6
[[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ([(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
=> ? = 4
[[0,1,0,0,0],[1,-1,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,2],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ([(0,5),(0,6),(1,8),(2,9),(3,8),(3,9),(4,1),(5,4),(6,7),(7,2),(7,3),(8,10),(9,10)],11)
=> ([(2,6),(2,10),(3,7),(3,8),(3,9),(4,7),(4,8),(4,9),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(7,10),(8,10),(9,10)],11)
=> ? = 5
[[0,0,1,0,0],[1,0,-1,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,3],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ?
=> ?
=> ? = 6
[[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,4],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ([(0,5),(0,10),(1,16),(2,15),(3,14),(4,13),(5,12),(6,2),(6,13),(7,4),(7,14),(8,1),(9,6),(10,11),(10,12),(11,3),(11,7),(12,9),(13,15),(14,8),(15,16)],17)
=> ([(2,14),(3,11),(3,12),(3,13),(3,16),(4,10),(4,11),(4,12),(4,13),(4,16),(5,7),(5,8),(5,9),(5,10),(5,14),(6,7),(6,8),(6,9),(6,10),(6,14),(6,16),(7,11),(7,12),(7,13),(7,15),(7,16),(8,11),(8,12),(8,13),(8,15),(8,16),(9,11),(9,12),(9,13),(9,15),(9,16),(10,11),(10,12),(10,13),(10,15),(10,16),(11,14),(11,15),(12,14),(12,15),(13,14),(13,15),(14,15),(14,16),(15,16)],17)
=> ? = 7
[[0,1,0,0,0],[0,0,0,1,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,2,2],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ?
=> ?
=> ? = 6
[[0,0,1,0,0],[0,1,-1,1,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,2,3],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ([(0,9),(0,11),(1,18),(2,17),(3,19),(4,13),(4,19),(5,12),(5,13),(6,16),(7,14),(8,5),(8,18),(9,10),(10,3),(10,4),(11,1),(11,8),(12,17),(13,15),(15,16),(16,14),(17,7),(18,2),(18,12),(19,6),(19,15)],20)
=> ([(2,10),(2,11),(2,13),(2,15),(2,17),(2,19),(3,8),(3,9),(3,12),(3,14),(3,16),(3,18),(4,8),(4,9),(4,12),(4,14),(4,16),(4,18),(4,19),(5,10),(5,11),(5,13),(5,15),(5,17),(5,18),(5,19),(6,8),(6,9),(6,12),(6,13),(6,14),(6,16),(6,17),(6,18),(6,19),(7,10),(7,11),(7,12),(7,13),(7,15),(7,16),(7,17),(7,18),(7,19),(8,10),(8,11),(8,13),(8,15),(8,17),(8,19),(9,10),(9,11),(9,13),(9,15),(9,17),(9,19),(10,12),(10,14),(10,16),(10,18),(11,12),(11,14),(11,16),(11,18),(12,13),(12,15),(12,17),(12,19),(13,14),(13,16),(13,18),(14,15),(14,16),(14,17),(14,19),(15,16),(15,17),(15,18),(16,17),(16,19),(17,18),(18,19)],20)
=> ? = 7
[[0,0,0,1,0],[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,2,4],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ?
=> ?
=> ? = 8
[[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,3,3],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ([(0,8),(2,11),(2,12),(3,10),(4,9),(5,4),(5,14),(6,3),(6,14),(7,1),(8,5),(8,6),(9,11),(9,13),(10,12),(10,13),(11,15),(12,15),(13,15),(14,2),(14,9),(14,10),(15,7)],16)
=> ([(5,14),(5,15),(6,7),(6,15),(7,14),(8,11),(8,12),(8,13),(9,10),(9,12),(9,13),(9,14),(10,11),(10,13),(10,15),(11,12),(11,14),(12,15),(13,14),(13,15),(14,15)],16)
=> ? = 8
[[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,3,4],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ([(0,13),(0,15),(1,17),(2,16),(2,17),(3,19),(4,16),(4,18),(5,22),(6,21),(7,20),(8,23),(8,29),(9,4),(9,28),(10,3),(10,28),(11,6),(12,7),(12,24),(13,14),(14,1),(14,2),(15,9),(15,10),(16,25),(17,12),(17,25),(18,26),(18,29),(19,23),(19,26),(20,27),(22,27),(23,30),(24,20),(24,22),(25,24),(26,30),(27,21),(28,8),(28,18),(28,19),(29,5),(29,30),(30,11)],31)
=> ([(2,11),(2,12),(2,13),(2,21),(2,25),(2,26),(2,30),(3,18),(3,19),(3,22),(3,23),(3,27),(3,28),(3,29),(4,10),(4,11),(4,12),(4,13),(4,21),(4,25),(4,26),(4,29),(4,30),(5,8),(5,18),(5,19),(5,22),(5,23),(5,27),(5,28),(5,29),(5,30),(6,7),(6,9),(6,10),(6,16),(6,18),(6,19),(6,20),(6,22),(6,23),(6,27),(6,28),(6,29),(6,30),(7,8),(7,17),(7,18),(7,19),(7,22),(7,23),(7,25),(7,26),(7,27),(7,28),(7,29),(7,30),(8,9),(8,10),(8,16),(8,18),(8,19),(8,20),(8,22),(8,23),(8,27),(8,28),(8,29),(9,11),(9,12),(9,13),(9,14),(9,15),(9,17),(9,21),(9,24),(9,25),(9,26),(9,29),(9,30),(10,11),(10,12),(10,13),(10,14),(10,15),(10,17),(10,21),(10,24),(10,25),(10,26),(10,30),(11,16),(11,18),(11,19),(11,20),(11,22),(11,23),(11,24),(11,27),(11,28),(11,29),(12,16),(12,18),(12,19),(12,20),(12,22),(12,23),(12,24),(12,27),(12,28),(12,29),(13,16),(13,18),(13,19),(13,20),(13,22),(13,23),(13,24),(13,27),(13,28),(13,29),(14,16),(14,18),(14,19),(14,20),(14,21),(14,22),(14,23),(14,25),(14,26),(14,27),(14,28),(14,29),(14,30),(15,16),(15,17),(15,18),(15,19),(15,20),(15,22),(15,23),(15,25),(15,26),(15,27),(15,28),(15,29),(15,30),(16,17),(16,21),(16,24),(16,25),(16,26),(16,27),(16,29),(16,30),(17,18),(17,19),(17,20),(17,21),(17,22),(17,23),(17,26),(17,27),(17,28),(17,29),(17,30),(18,21),(18,24),(18,25),(18,26),(18,30),(19,21),(19,24),(19,25),(19,26),(19,30),(20,21),(20,23),(20,24),(20,25),(20,26),(20,27),(20,28),(20,29),(20,30),(21,22),(21,23),(21,24),(21,25),(21,27),(21,28),(21,29),(22,24),(22,25),(22,26),(22,28),(22,30),(23,24),(23,25),(23,26),(23,30),(24,25),(24,26),(24,27),(24,28),(24,29),(24,30),(25,27),(25,28),(25,29),(26,27),(26,28),(26,29),(27,30),(28,30),(29,30)],31)
=> ? = 9
[[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,0,0,1]]
=> [[1,1,2,2,2],[2,2,3,3],[3,3,4],[4,4],[5]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ([(6,7)],8)
=> ? = 6
[[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,0,0,1]]
=> [[1,1,2,2,3],[2,2,3,3],[3,3,4],[4,4],[5]]
=> ([(0,5),(0,10),(1,16),(2,15),(3,14),(4,13),(5,12),(6,2),(6,13),(7,4),(7,14),(8,1),(9,6),(10,11),(10,12),(11,3),(11,7),(12,9),(13,15),(14,8),(15,16)],17)
=> ([(2,14),(3,11),(3,12),(3,13),(3,16),(4,10),(4,11),(4,12),(4,13),(4,16),(5,7),(5,8),(5,9),(5,10),(5,14),(6,7),(6,8),(6,9),(6,10),(6,14),(6,16),(7,11),(7,12),(7,13),(7,15),(7,16),(8,11),(8,12),(8,13),(8,15),(8,16),(9,11),(9,12),(9,13),(9,15),(9,16),(10,11),(10,12),(10,13),(10,15),(10,16),(11,14),(11,15),(12,14),(12,15),(13,14),(13,15),(14,15),(14,16),(15,16)],17)
=> ? = 7
[[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1]]
=> [[1,1,2,2,2],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ([(0,7),(0,8),(1,12),(2,11),(3,10),(4,10),(4,11),(5,3),(6,1),(6,13),(7,9),(8,5),(9,2),(9,4),(10,14),(11,6),(11,14),(13,12),(14,13)],15)
=> ([(2,11),(3,10),(3,11),(4,12),(4,13),(4,14),(5,12),(5,13),(5,14),(6,8),(6,12),(6,13),(6,14),(7,9),(7,12),(7,13),(7,14),(8,9),(8,10),(8,11),(9,12),(9,13),(9,14),(10,12),(10,13),(10,14),(11,12),(11,13),(11,14)],15)
=> ? = 7
[[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1]]
=> [[1,1,2,2,3],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ?
=> ?
=> ? = 8
[[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1]]
=> [[1,1,2,2,4],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ([(0,14),(0,15),(1,19),(2,18),(3,29),(4,30),(5,22),(6,23),(7,24),(7,25),(8,9),(9,7),(9,18),(9,19),(10,5),(11,6),(12,2),(12,29),(13,1),(13,30),(14,16),(14,28),(15,17),(15,28),(16,3),(16,12),(17,4),(17,13),(18,24),(18,27),(19,25),(19,27),(20,26),(21,26),(22,20),(23,21),(24,22),(24,31),(25,23),(25,31),(27,31),(28,8),(29,10),(30,11),(31,20),(31,21)],32)
=> ([(2,3),(2,9),(2,27),(2,28),(2,29),(2,31),(3,8),(3,24),(3,25),(3,26),(3,30),(4,5),(4,18),(4,20),(4,23),(4,24),(4,25),(4,26),(4,30),(5,17),(5,19),(5,22),(5,27),(5,28),(5,29),(5,31),(6,8),(6,9),(6,24),(6,25),(6,26),(6,27),(6,28),(6,29),(6,30),(6,31),(7,8),(7,9),(7,13),(7,14),(7,21),(7,24),(7,25),(7,26),(7,27),(7,28),(7,29),(7,30),(7,31),(8,9),(8,13),(8,15),(8,17),(8,19),(8,22),(8,27),(8,28),(8,29),(8,31),(9,14),(9,16),(9,18),(9,20),(9,23),(9,24),(9,25),(9,26),(9,30),(10,17),(10,18),(10,19),(10,20),(10,22),(10,23),(10,24),(10,25),(10,26),(10,27),(10,28),(10,29),(10,30),(10,31),(11,17),(11,18),(11,19),(11,20),(11,22),(11,23),(11,24),(11,25),(11,26),(11,27),(11,28),(11,29),(11,30),(11,31),(12,17),(12,18),(12,19),(12,20),(12,22),(12,23),(12,24),(12,25),(12,26),(12,27),(12,28),(12,29),(12,30),(12,31),(13,14),(13,16),(13,18),(13,20),(13,23),(13,24),(13,25),(13,26),(13,27),(13,28),(13,29),(13,30),(13,31),(14,15),(14,17),(14,19),(14,22),(14,24),(14,25),(14,26),(14,27),(14,28),(14,29),(14,30),(14,31),(15,16),(15,18),(15,20),(15,21),(15,23),(15,24),(15,25),(15,26),(15,27),(15,28),(15,29),(15,30),(15,31),(16,17),(16,19),(16,21),(16,22),(16,24),(16,25),(16,26),(16,27),(16,28),(16,29),(16,30),(16,31),(17,18),(17,20),(17,21),(17,23),(17,24),(17,25),(17,26),(17,30),(18,19),(18,21),(18,22),(18,27),(18,28),(18,29),(18,31),(19,20),(19,21),(19,23),(19,24),(19,25),(19,26),(19,30),(19,31),(20,21),(20,22),(20,27),(20,28),(20,29),(20,30),(20,31),(21,22),(21,23),(21,24),(21,25),(21,26),(21,27),(21,28),(21,29),(21,30),(21,31),(22,23),(22,24),(22,25),(22,26),(22,27),(22,28),(22,29),(22,30),(22,31),(23,24),(23,25),(23,26),(23,27),(23,28),(23,29),(23,30),(23,31),(24,27),(24,28),(24,29),(24,31),(25,27),(25,28),(25,29),(25,31),(26,27),(26,28),(26,29),(26,31),(27,30),(28,30),(29,30),(30,31)],32)
=> ? = 9
[[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,1,1],[2,2,2,2],[3,3,3],[4,5],[5]]
=> ([(0,1)],2)
=> ([],2)
=> 1
[[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,1,2],[2,2,2,2],[3,3,3],[4,5],[5]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 2
[[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,1,1],[2,2,2,3],[3,3,3],[4,5],[5]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 2
[[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,1,2],[2,2,2,3],[3,3,3],[4,5],[5]]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 3
[[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[1,1,1,1,1],[2,2,2,2],[3,3,4],[4,5],[5]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 2
[[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[1,1,1,1,2],[2,2,2,2],[3,3,4],[4,5],[5]]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 3
[[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[1,1,1,1,1],[2,2,2,3],[3,3,4],[4,5],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 3
[[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[1,1,1,1,2],[2,2,2,3],[3,3,4],[4,5],[5]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 4
Description
The cardinality of a minimal non-edge isolating set of a graph.
Let $\mathcal F$ be a set of graphs. A set of vertices $S$ is $\mathcal F$-isolating, if the subgraph induced by the vertices in the complement of the closed neighbourhood of $S$ does not contain any graph in $\mathcal F$.
This statistic returns the cardinality of the smallest isolating set when $\mathcal F$ contains only the graph with two isolated vertices.
Matching statistic: St000273
Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St000273: Graphs ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 9%
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St000273: Graphs ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 9%
Values
[[1]]
=> [[1]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[1,0],[0,1]]
=> [[1,1],[2]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[0,1],[1,0]]
=> [[1,2],[2]]
=> ([(0,1)],2)
=> ([],2)
=> 2 = 1 + 1
[[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[0,1,0],[1,0,0],[0,0,1]]
=> [[1,1,2],[2,2],[3]]
=> ([(0,1)],2)
=> ([],2)
=> 2 = 1 + 1
[[1,0,0],[0,0,1],[0,1,0]]
=> [[1,1,1],[2,3],[3]]
=> ([(0,1)],2)
=> ([],2)
=> 2 = 1 + 1
[[0,1,0],[1,-1,1],[0,1,0]]
=> [[1,1,2],[2,3],[3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 3 = 2 + 1
[[0,0,1],[1,0,0],[0,1,0]]
=> [[1,1,3],[2,3],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 4 = 3 + 1
[[0,1,0],[0,0,1],[1,0,0]]
=> [[1,2,2],[2,3],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 4 = 3 + 1
[[0,0,1],[0,1,0],[1,0,0]]
=> [[1,2,3],[2,3],[3]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ([(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
=> ? = 4 + 1
[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,2],[3,3],[4]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,2],[2,2,2],[3,3],[4]]
=> ([(0,1)],2)
=> ([],2)
=> 2 = 1 + 1
[[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,3],[3,3],[4]]
=> ([(0,1)],2)
=> ([],2)
=> 2 = 1 + 1
[[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
=> [[1,1,1,2],[2,2,3],[3,3],[4]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 3 = 2 + 1
[[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> [[1,1,1,3],[2,2,3],[3,3],[4]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 4 = 3 + 1
[[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]]
=> [[1,1,2,2],[2,2,3],[3,3],[4]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 4 = 3 + 1
[[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> [[1,1,2,3],[2,2,3],[3,3],[4]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ([(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
=> ? = 4 + 1
[[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [[1,1,1,1],[2,2,2],[3,4],[4]]
=> ([(0,1)],2)
=> ([],2)
=> 2 = 1 + 1
[[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> [[1,1,1,2],[2,2,2],[3,4],[4]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 3 = 2 + 1
[[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> [[1,1,1,1],[2,2,3],[3,4],[4]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 3 = 2 + 1
[[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> [[1,1,1,2],[2,2,3],[3,4],[4]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 4 = 3 + 1
[[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]]
=> [[1,1,1,3],[2,2,3],[3,4],[4]]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(4,5)],6)
=> 5 = 4 + 1
[[0,1,0,0],[0,0,1,0],[1,0,-1,1],[0,0,1,0]]
=> [[1,1,2,2],[2,2,3],[3,4],[4]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 5 = 4 + 1
[[0,0,1,0],[0,1,0,0],[1,0,-1,1],[0,0,1,0]]
=> [[1,1,2,3],[2,2,3],[3,4],[4]]
=> ([(0,3),(0,7),(1,8),(2,8),(3,9),(4,5),(5,1),(6,2),(7,4),(7,9),(9,6)],10)
=> ([(2,9),(3,6),(3,7),(3,8),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,9),(7,9),(8,9)],10)
=> ? = 5 + 1
[[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> [[1,1,1,1],[2,2,4],[3,4],[4]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 4 = 3 + 1
[[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
=> [[1,1,1,2],[2,2,4],[3,4],[4]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> 5 = 4 + 1
[[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
=> [[1,1,1,3],[2,2,4],[3,4],[4]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
=> ? = 5 + 1
[[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [[1,1,1,4],[2,2,4],[3,4],[4]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ([(6,7)],8)
=> ? = 6 + 1
[[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
=> [[1,1,2,2],[2,2,4],[3,4],[4]]
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> ([(3,6),(3,9),(4,5),(4,9),(5,8),(6,8),(7,8),(7,9),(8,9)],10)
=> ? = 5 + 1
[[0,0,1,0],[0,1,-1,1],[1,0,0,0],[0,0,1,0]]
=> [[1,1,2,3],[2,2,4],[3,4],[4]]
=> ([(0,6),(1,9),(1,10),(2,8),(3,7),(4,3),(4,12),(5,2),(5,12),(6,4),(6,5),(7,9),(7,11),(8,10),(8,11),(9,13),(10,13),(11,13),(12,1),(12,7),(12,8)],14)
=> ([(3,12),(3,13),(4,5),(4,13),(5,12),(6,9),(6,10),(6,11),(7,8),(7,10),(7,11),(7,12),(8,9),(8,11),(8,13),(9,10),(9,12),(10,13),(11,12),(11,13),(12,13)],14)
=> ? = 6 + 1
[[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
=> [[1,1,2,4],[2,2,4],[3,4],[4]]
=> ([(0,7),(0,8),(1,12),(2,11),(3,10),(4,10),(4,11),(5,3),(6,1),(6,13),(7,9),(8,5),(9,2),(9,4),(10,14),(11,6),(11,14),(13,12),(14,13)],15)
=> ([(2,11),(3,10),(3,11),(4,12),(4,13),(4,14),(5,12),(5,13),(5,14),(6,8),(6,12),(6,13),(6,14),(7,9),(7,12),(7,13),(7,14),(8,9),(8,10),(8,11),(9,12),(9,13),(9,14),(10,12),(10,13),(10,14),(11,12),(11,13),(11,14)],15)
=> ? = 7 + 1
[[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]]
=> [[1,1,1,1],[2,3,3],[3,4],[4]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 4 = 3 + 1
[[0,1,0,0],[1,-1,1,0],[0,0,0,1],[0,1,0,0]]
=> [[1,1,1,2],[2,3,3],[3,4],[4]]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(4,5)],6)
=> 5 = 4 + 1
[[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
=> [[1,1,1,3],[2,3,3],[3,4],[4]]
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ([(5,6)],7)
=> 6 = 5 + 1
[[0,1,0,0],[0,0,1,0],[1,-1,0,1],[0,1,0,0]]
=> [[1,1,2,2],[2,3,3],[3,4],[4]]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ([(5,6)],7)
=> 6 = 5 + 1
[[0,0,1,0],[0,1,0,0],[1,-1,0,1],[0,1,0,0]]
=> [[1,1,2,3],[2,3,3],[3,4],[4]]
=> ([(0,3),(0,8),(1,10),(2,9),(3,11),(4,2),(5,4),(6,7),(7,1),(7,9),(8,5),(8,11),(9,10),(11,6)],12)
=> ([(2,8),(3,7),(4,9),(4,10),(4,11),(5,9),(5,10),(5,11),(6,9),(6,10),(6,11),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11)],12)
=> ? = 6 + 1
[[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> [[1,1,1,1],[2,3,4],[3,4],[4]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ([(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
=> ? = 4 + 1
[[0,1,0,0],[1,-1,0,1],[0,0,1,0],[0,1,0,0]]
=> [[1,1,1,2],[2,3,4],[3,4],[4]]
=> ([(0,5),(0,6),(1,8),(2,9),(3,8),(3,9),(4,1),(5,4),(6,7),(7,2),(7,3),(8,10),(9,10)],11)
=> ([(2,6),(2,10),(3,7),(3,8),(3,9),(4,7),(4,8),(4,9),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(7,10),(8,10),(9,10)],11)
=> ? = 5 + 1
[[0,0,1,0],[1,0,-1,1],[0,0,1,0],[0,1,0,0]]
=> [[1,1,1,3],[2,3,4],[3,4],[4]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ([(4,9),(5,8),(6,7),(7,9),(8,9)],10)
=> ? = 6 + 1
[[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
=> [[1,1,1,4],[2,3,4],[3,4],[4]]
=> ([(0,5),(0,10),(1,16),(2,15),(3,14),(4,13),(5,12),(6,2),(6,13),(7,4),(7,14),(8,1),(9,6),(10,11),(10,12),(11,3),(11,7),(12,9),(13,15),(14,8),(15,16)],17)
=> ([(2,14),(3,11),(3,12),(3,13),(3,16),(4,10),(4,11),(4,12),(4,13),(4,16),(5,7),(5,8),(5,9),(5,10),(5,14),(6,7),(6,8),(6,9),(6,10),(6,14),(6,16),(7,11),(7,12),(7,13),(7,15),(7,16),(8,11),(8,12),(8,13),(8,15),(8,16),(9,11),(9,12),(9,13),(9,15),(9,16),(10,11),(10,12),(10,13),(10,15),(10,16),(11,14),(11,15),(12,14),(12,15),(13,14),(13,15),(14,15),(14,16),(15,16)],17)
=> ? = 7 + 1
[[0,1,0,0],[0,0,0,1],[1,-1,1,0],[0,1,0,0]]
=> [[1,1,2,2],[2,3,4],[3,4],[4]]
=> ([(0,6),(0,7),(1,9),(2,12),(3,9),(3,12),(4,10),(5,1),(6,5),(7,8),(8,2),(8,3),(9,11),(11,10),(12,4),(12,11)],13)
=> ([(2,9),(3,10),(3,11),(3,12),(4,10),(4,11),(4,12),(5,6),(5,8),(5,9),(6,10),(6,11),(6,12),(7,8),(7,10),(7,11),(7,12),(8,10),(8,11),(8,12),(9,10),(9,11),(9,12)],13)
=> ? = 6 + 1
[[0,0,1,0],[0,1,-1,1],[1,-1,1,0],[0,1,0,0]]
=> [[1,1,2,3],[2,3,4],[3,4],[4]]
=> ([(0,9),(0,11),(1,18),(2,17),(3,19),(4,13),(4,19),(5,12),(5,13),(6,16),(7,14),(8,5),(8,18),(9,10),(10,3),(10,4),(11,1),(11,8),(12,17),(13,15),(15,16),(16,14),(17,7),(18,2),(18,12),(19,6),(19,15)],20)
=> ([(2,10),(2,11),(2,13),(2,15),(2,17),(2,19),(3,8),(3,9),(3,12),(3,14),(3,16),(3,18),(4,8),(4,9),(4,12),(4,14),(4,16),(4,18),(4,19),(5,10),(5,11),(5,13),(5,15),(5,17),(5,18),(5,19),(6,8),(6,9),(6,12),(6,13),(6,14),(6,16),(6,17),(6,18),(6,19),(7,10),(7,11),(7,12),(7,13),(7,15),(7,16),(7,17),(7,18),(7,19),(8,10),(8,11),(8,13),(8,15),(8,17),(8,19),(9,10),(9,11),(9,13),(9,15),(9,17),(9,19),(10,12),(10,14),(10,16),(10,18),(11,12),(11,14),(11,16),(11,18),(12,13),(12,15),(12,17),(12,19),(13,14),(13,16),(13,18),(14,15),(14,16),(14,17),(14,19),(15,16),(15,17),(15,18),(16,17),(16,19),(17,18),(18,19)],20)
=> ? = 7 + 1
[[0,0,0,1],[0,1,0,0],[1,-1,1,0],[0,1,0,0]]
=> [[1,1,2,4],[2,3,4],[3,4],[4]]
=> ([(0,12),(0,13),(1,16),(2,15),(3,23),(4,19),(5,17),(5,20),(6,4),(7,5),(7,15),(7,16),(8,10),(9,7),(10,2),(11,1),(11,23),(12,8),(12,22),(13,14),(13,22),(14,3),(14,11),(15,17),(15,21),(16,20),(16,21),(17,24),(19,18),(20,19),(20,24),(21,24),(22,9),(23,6),(24,18)],25)
=> ([(2,6),(2,21),(2,22),(2,23),(3,4),(3,18),(3,19),(3,20),(3,24),(4,5),(4,14),(4,15),(4,21),(4,22),(4,23),(5,12),(5,15),(5,16),(5,18),(5,19),(5,20),(5,24),(6,10),(6,13),(6,17),(6,18),(6,19),(6,20),(6,24),(7,10),(7,13),(7,17),(7,18),(7,19),(7,20),(7,21),(7,22),(7,23),(7,24),(8,10),(8,13),(8,17),(8,18),(8,19),(8,20),(8,21),(8,22),(8,23),(8,24),(9,10),(9,13),(9,17),(9,18),(9,19),(9,20),(9,21),(9,22),(9,23),(9,24),(10,14),(10,15),(10,16),(10,21),(10,22),(10,23),(11,14),(11,15),(11,16),(11,18),(11,19),(11,20),(11,21),(11,22),(11,23),(11,24),(12,14),(12,15),(12,18),(12,19),(12,20),(12,21),(12,22),(12,23),(12,24),(13,14),(13,15),(13,16),(13,21),(13,22),(13,23),(13,24),(14,16),(14,17),(14,18),(14,19),(14,20),(14,24),(15,17),(15,18),(15,19),(15,20),(15,24),(16,17),(16,18),(16,19),(16,20),(16,21),(16,22),(16,23),(16,24),(17,18),(17,19),(17,20),(17,21),(17,22),(17,23),(17,24),(18,21),(18,22),(18,23),(19,21),(19,22),(19,23),(20,21),(20,22),(20,23),(21,24),(22,24),(23,24)],25)
=> ? = 8 + 1
[[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> [[1,1,3,3],[2,3,4],[3,4],[4]]
=> ([(0,8),(2,11),(2,12),(3,10),(4,9),(5,4),(5,14),(6,3),(6,14),(7,1),(8,5),(8,6),(9,11),(9,13),(10,12),(10,13),(11,15),(12,15),(13,15),(14,2),(14,9),(14,10),(15,7)],16)
=> ([(5,14),(5,15),(6,7),(6,15),(7,14),(8,11),(8,12),(8,13),(9,10),(9,12),(9,13),(9,14),(10,11),(10,13),(10,15),(11,12),(11,14),(12,15),(13,14),(13,15),(14,15)],16)
=> ? = 8 + 1
[[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
=> [[1,1,3,4],[2,3,4],[3,4],[4]]
=> ([(0,13),(0,15),(1,17),(2,16),(2,17),(3,19),(4,16),(4,18),(5,22),(6,21),(7,20),(8,23),(8,29),(9,4),(9,28),(10,3),(10,28),(11,6),(12,7),(12,24),(13,14),(14,1),(14,2),(15,9),(15,10),(16,25),(17,12),(17,25),(18,26),(18,29),(19,23),(19,26),(20,27),(22,27),(23,30),(24,20),(24,22),(25,24),(26,30),(27,21),(28,8),(28,18),(28,19),(29,5),(29,30),(30,11)],31)
=> ([(2,11),(2,12),(2,13),(2,21),(2,25),(2,26),(2,30),(3,18),(3,19),(3,22),(3,23),(3,27),(3,28),(3,29),(4,10),(4,11),(4,12),(4,13),(4,21),(4,25),(4,26),(4,29),(4,30),(5,8),(5,18),(5,19),(5,22),(5,23),(5,27),(5,28),(5,29),(5,30),(6,7),(6,9),(6,10),(6,16),(6,18),(6,19),(6,20),(6,22),(6,23),(6,27),(6,28),(6,29),(6,30),(7,8),(7,17),(7,18),(7,19),(7,22),(7,23),(7,25),(7,26),(7,27),(7,28),(7,29),(7,30),(8,9),(8,10),(8,16),(8,18),(8,19),(8,20),(8,22),(8,23),(8,27),(8,28),(8,29),(9,11),(9,12),(9,13),(9,14),(9,15),(9,17),(9,21),(9,24),(9,25),(9,26),(9,29),(9,30),(10,11),(10,12),(10,13),(10,14),(10,15),(10,17),(10,21),(10,24),(10,25),(10,26),(10,30),(11,16),(11,18),(11,19),(11,20),(11,22),(11,23),(11,24),(11,27),(11,28),(11,29),(12,16),(12,18),(12,19),(12,20),(12,22),(12,23),(12,24),(12,27),(12,28),(12,29),(13,16),(13,18),(13,19),(13,20),(13,22),(13,23),(13,24),(13,27),(13,28),(13,29),(14,16),(14,18),(14,19),(14,20),(14,21),(14,22),(14,23),(14,25),(14,26),(14,27),(14,28),(14,29),(14,30),(15,16),(15,17),(15,18),(15,19),(15,20),(15,22),(15,23),(15,25),(15,26),(15,27),(15,28),(15,29),(15,30),(16,17),(16,21),(16,24),(16,25),(16,26),(16,27),(16,29),(16,30),(17,18),(17,19),(17,20),(17,21),(17,22),(17,23),(17,26),(17,27),(17,28),(17,29),(17,30),(18,21),(18,24),(18,25),(18,26),(18,30),(19,21),(19,24),(19,25),(19,26),(19,30),(20,21),(20,23),(20,24),(20,25),(20,26),(20,27),(20,28),(20,29),(20,30),(21,22),(21,23),(21,24),(21,25),(21,27),(21,28),(21,29),(22,24),(22,25),(22,26),(22,28),(22,30),(23,24),(23,25),(23,26),(23,30),(24,25),(24,26),(24,27),(24,28),(24,29),(24,30),(25,27),(25,28),(25,29),(26,27),(26,28),(26,29),(27,30),(28,30),(29,30)],31)
=> ? = 9 + 1
[[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> [[1,2,2,2],[2,3,3],[3,4],[4]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ([(6,7)],8)
=> ? = 6 + 1
[[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]]
=> [[1,2,2,3],[2,3,3],[3,4],[4]]
=> ([(0,5),(0,10),(1,16),(2,15),(3,14),(4,13),(5,12),(6,2),(6,13),(7,4),(7,14),(8,1),(9,6),(10,11),(10,12),(11,3),(11,7),(12,9),(13,15),(14,8),(15,16)],17)
=> ([(2,14),(3,11),(3,12),(3,13),(3,16),(4,10),(4,11),(4,12),(4,13),(4,16),(5,7),(5,8),(5,9),(5,10),(5,14),(6,7),(6,8),(6,9),(6,10),(6,14),(6,16),(7,11),(7,12),(7,13),(7,15),(7,16),(8,11),(8,12),(8,13),(8,15),(8,16),(9,11),(9,12),(9,13),(9,15),(9,16),(10,11),(10,12),(10,13),(10,15),(10,16),(11,14),(11,15),(12,14),(12,15),(13,14),(13,15),(14,15),(14,16),(15,16)],17)
=> ? = 7 + 1
[[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]]
=> [[1,2,2,2],[2,3,4],[3,4],[4]]
=> ([(0,7),(0,8),(1,12),(2,11),(3,10),(4,10),(4,11),(5,3),(6,1),(6,13),(7,9),(8,5),(9,2),(9,4),(10,14),(11,6),(11,14),(13,12),(14,13)],15)
=> ([(2,11),(3,10),(3,11),(4,12),(4,13),(4,14),(5,12),(5,13),(5,14),(6,8),(6,12),(6,13),(6,14),(7,9),(7,12),(7,13),(7,14),(8,9),(8,10),(8,11),(9,12),(9,13),(9,14),(10,12),(10,13),(10,14),(11,12),(11,13),(11,14)],15)
=> ? = 7 + 1
[[0,0,1,0],[0,1,-1,1],[0,0,1,0],[1,0,0,0]]
=> [[1,2,2,3],[2,3,4],[3,4],[4]]
=> ([(0,10),(0,12),(1,16),(2,17),(3,21),(4,22),(5,14),(6,13),(6,14),(7,13),(7,15),(8,7),(8,21),(9,2),(9,18),(10,11),(11,5),(11,6),(12,3),(12,8),(13,19),(14,9),(14,19),(15,22),(16,20),(17,20),(18,16),(18,17),(19,18),(21,4),(21,15),(22,1)],23)
=> ([(2,5),(2,22),(3,15),(3,16),(3,18),(3,19),(3,20),(3,21),(3,22),(4,8),(4,9),(4,12),(4,13),(4,14),(4,17),(4,22),(5,15),(5,16),(5,18),(5,19),(5,20),(5,21),(6,15),(6,16),(6,17),(6,18),(6,19),(6,20),(6,21),(6,22),(7,8),(7,9),(7,12),(7,13),(7,14),(7,17),(7,21),(7,22),(8,11),(8,15),(8,16),(8,18),(8,19),(8,20),(8,21),(9,11),(9,15),(9,16),(9,18),(9,19),(9,20),(9,21),(10,12),(10,14),(10,15),(10,16),(10,17),(10,18),(10,19),(10,20),(10,21),(10,22),(11,12),(11,13),(11,14),(11,17),(11,18),(11,20),(11,21),(11,22),(12,15),(12,16),(12,18),(12,19),(12,20),(12,21),(13,14),(13,15),(13,16),(13,18),(13,19),(13,20),(13,21),(14,15),(14,16),(14,18),(14,19),(14,20),(14,21),(15,17),(15,22),(16,17),(16,22),(17,18),(17,19),(17,20),(17,21),(18,22),(19,20),(19,22),(20,22),(21,22)],23)
=> ? = 8 + 1
[[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
=> [[1,2,2,4],[2,3,4],[3,4],[4]]
=> ([(0,14),(0,15),(1,19),(2,18),(3,29),(4,30),(5,22),(6,23),(7,24),(7,25),(8,9),(9,7),(9,18),(9,19),(10,5),(11,6),(12,2),(12,29),(13,1),(13,30),(14,16),(14,28),(15,17),(15,28),(16,3),(16,12),(17,4),(17,13),(18,24),(18,27),(19,25),(19,27),(20,26),(21,26),(22,20),(23,21),(24,22),(24,31),(25,23),(25,31),(27,31),(28,8),(29,10),(30,11),(31,20),(31,21)],32)
=> ([(2,3),(2,9),(2,27),(2,28),(2,29),(2,31),(3,8),(3,24),(3,25),(3,26),(3,30),(4,5),(4,18),(4,20),(4,23),(4,24),(4,25),(4,26),(4,30),(5,17),(5,19),(5,22),(5,27),(5,28),(5,29),(5,31),(6,8),(6,9),(6,24),(6,25),(6,26),(6,27),(6,28),(6,29),(6,30),(6,31),(7,8),(7,9),(7,13),(7,14),(7,21),(7,24),(7,25),(7,26),(7,27),(7,28),(7,29),(7,30),(7,31),(8,9),(8,13),(8,15),(8,17),(8,19),(8,22),(8,27),(8,28),(8,29),(8,31),(9,14),(9,16),(9,18),(9,20),(9,23),(9,24),(9,25),(9,26),(9,30),(10,17),(10,18),(10,19),(10,20),(10,22),(10,23),(10,24),(10,25),(10,26),(10,27),(10,28),(10,29),(10,30),(10,31),(11,17),(11,18),(11,19),(11,20),(11,22),(11,23),(11,24),(11,25),(11,26),(11,27),(11,28),(11,29),(11,30),(11,31),(12,17),(12,18),(12,19),(12,20),(12,22),(12,23),(12,24),(12,25),(12,26),(12,27),(12,28),(12,29),(12,30),(12,31),(13,14),(13,16),(13,18),(13,20),(13,23),(13,24),(13,25),(13,26),(13,27),(13,28),(13,29),(13,30),(13,31),(14,15),(14,17),(14,19),(14,22),(14,24),(14,25),(14,26),(14,27),(14,28),(14,29),(14,30),(14,31),(15,16),(15,18),(15,20),(15,21),(15,23),(15,24),(15,25),(15,26),(15,27),(15,28),(15,29),(15,30),(15,31),(16,17),(16,19),(16,21),(16,22),(16,24),(16,25),(16,26),(16,27),(16,28),(16,29),(16,30),(16,31),(17,18),(17,20),(17,21),(17,23),(17,24),(17,25),(17,26),(17,30),(18,19),(18,21),(18,22),(18,27),(18,28),(18,29),(18,31),(19,20),(19,21),(19,23),(19,24),(19,25),(19,26),(19,30),(19,31),(20,21),(20,22),(20,27),(20,28),(20,29),(20,30),(20,31),(21,22),(21,23),(21,24),(21,25),(21,26),(21,27),(21,28),(21,29),(21,30),(21,31),(22,23),(22,24),(22,25),(22,26),(22,27),(22,28),(22,29),(22,30),(22,31),(23,24),(23,25),(23,26),(23,27),(23,28),(23,29),(23,30),(23,31),(24,27),(24,28),(24,29),(24,31),(25,27),(25,28),(25,29),(25,31),(26,27),(26,28),(26,29),(26,31),(27,30),(28,30),(29,30),(30,31)],32)
=> ? = 9 + 1
[[0,0,1,0],[0,0,0,1],[0,1,0,0],[1,0,0,0]]
=> [[1,2,3,3],[2,3,4],[3,4],[4]]
=> ([(0,13),(0,15),(1,17),(2,16),(2,17),(3,19),(4,16),(4,18),(5,22),(6,21),(7,20),(8,23),(8,29),(9,4),(9,28),(10,3),(10,28),(11,6),(12,7),(12,24),(13,14),(14,1),(14,2),(15,9),(15,10),(16,25),(17,12),(17,25),(18,26),(18,29),(19,23),(19,26),(20,27),(22,27),(23,30),(24,20),(24,22),(25,24),(26,30),(27,21),(28,8),(28,18),(28,19),(29,5),(29,30),(30,11)],31)
=> ([(2,11),(2,12),(2,13),(2,21),(2,25),(2,26),(2,30),(3,18),(3,19),(3,22),(3,23),(3,27),(3,28),(3,29),(4,10),(4,11),(4,12),(4,13),(4,21),(4,25),(4,26),(4,29),(4,30),(5,8),(5,18),(5,19),(5,22),(5,23),(5,27),(5,28),(5,29),(5,30),(6,7),(6,9),(6,10),(6,16),(6,18),(6,19),(6,20),(6,22),(6,23),(6,27),(6,28),(6,29),(6,30),(7,8),(7,17),(7,18),(7,19),(7,22),(7,23),(7,25),(7,26),(7,27),(7,28),(7,29),(7,30),(8,9),(8,10),(8,16),(8,18),(8,19),(8,20),(8,22),(8,23),(8,27),(8,28),(8,29),(9,11),(9,12),(9,13),(9,14),(9,15),(9,17),(9,21),(9,24),(9,25),(9,26),(9,29),(9,30),(10,11),(10,12),(10,13),(10,14),(10,15),(10,17),(10,21),(10,24),(10,25),(10,26),(10,30),(11,16),(11,18),(11,19),(11,20),(11,22),(11,23),(11,24),(11,27),(11,28),(11,29),(12,16),(12,18),(12,19),(12,20),(12,22),(12,23),(12,24),(12,27),(12,28),(12,29),(13,16),(13,18),(13,19),(13,20),(13,22),(13,23),(13,24),(13,27),(13,28),(13,29),(14,16),(14,18),(14,19),(14,20),(14,21),(14,22),(14,23),(14,25),(14,26),(14,27),(14,28),(14,29),(14,30),(15,16),(15,17),(15,18),(15,19),(15,20),(15,22),(15,23),(15,25),(15,26),(15,27),(15,28),(15,29),(15,30),(16,17),(16,21),(16,24),(16,25),(16,26),(16,27),(16,29),(16,30),(17,18),(17,19),(17,20),(17,21),(17,22),(17,23),(17,26),(17,27),(17,28),(17,29),(17,30),(18,21),(18,24),(18,25),(18,26),(18,30),(19,21),(19,24),(19,25),(19,26),(19,30),(20,21),(20,23),(20,24),(20,25),(20,26),(20,27),(20,28),(20,29),(20,30),(21,22),(21,23),(21,24),(21,25),(21,27),(21,28),(21,29),(22,24),(22,25),(22,26),(22,28),(22,30),(23,24),(23,25),(23,26),(23,30),(24,25),(24,26),(24,27),(24,28),(24,29),(24,30),(25,27),(25,28),(25,29),(26,27),(26,28),(26,29),(27,30),(28,30),(29,30)],31)
=> ? = 9 + 1
[[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> [[1,2,3,4],[2,3,4],[3,4],[4]]
=> ([(0,5),(0,16),(0,17),(1,23),(2,19),(3,11),(3,21),(4,10),(4,20),(5,12),(5,13),(6,50),(7,51),(8,24),(8,58),(9,25),(9,59),(10,14),(10,52),(11,15),(11,53),(12,26),(12,60),(13,27),(13,60),(14,54),(15,55),(16,4),(16,29),(17,3),(17,29),(18,48),(18,49),(19,32),(19,33),(20,46),(20,52),(21,47),(21,53),(22,34),(22,35),(23,18),(23,54),(23,55),(24,40),(24,42),(25,41),(25,43),(26,46),(26,56),(27,47),(27,57),(28,63),(29,1),(30,62),(31,61),(32,61),(33,61),(34,6),(34,62),(35,7),(35,62),(36,58),(37,59),(38,32),(39,33),(40,44),(41,45),(42,38),(43,39),(44,31),(45,31),(46,36),(47,37),(48,40),(48,63),(49,41),(49,63),(50,38),(51,39),(52,8),(52,36),(53,9),(53,37),(54,28),(54,48),(55,28),(55,49),(56,30),(56,34),(57,30),(57,35),(58,42),(58,50),(59,43),(59,51),(60,22),(60,56),(60,57),(62,2),(63,44),(63,45)],64)
=> ([(2,3),(2,6),(2,7),(2,10),(2,11),(2,14),(2,15),(2,18),(2,19),(2,21),(2,23),(2,26),(2,27),(2,31),(2,32),(2,33),(2,36),(2,37),(2,40),(2,41),(2,44),(2,45),(2,48),(2,49),(2,52),(2,53),(2,55),(2,58),(2,59),(2,62),(2,63),(3,4),(3,5),(3,8),(3,9),(3,12),(3,13),(3,16),(3,17),(3,20),(3,22),(3,24),(3,25),(3,28),(3,29),(3,30),(3,34),(3,35),(3,38),(3,39),(3,42),(3,43),(3,46),(3,47),(3,50),(3,51),(3,54),(3,56),(3,57),(3,60),(3,61),(4,5),(4,7),(4,10),(4,11),(4,13),(4,15),(4,17),(4,19),(4,21),(4,23),(4,26),(4,27),(4,31),(4,32),(4,33),(4,36),(4,37),(4,40),(4,41),(4,43),(4,45),(4,47),(4,49),(4,51),(4,53),(4,55),(4,57),(4,59),(4,61),(4,62),(4,63),(5,6),(5,10),(5,11),(5,12),(5,14),(5,16),(5,18),(5,21),(5,23),(5,26),(5,27),(5,31),(5,32),(5,33),(5,36),(5,37),(5,40),(5,41),(5,42),(5,44),(5,46),(5,48),(5,50),(5,52),(5,55),(5,56),(5,58),(5,60),(5,62),(5,63),(6,7),(6,8),(6,9),(6,13),(6,15),(6,17),(6,19),(6,20),(6,22),(6,24),(6,25),(6,28),(6,29),(6,30),(6,34),(6,35),(6,38),(6,39),(6,43),(6,45),(6,47),(6,49),(6,51),(6,53),(6,54),(6,57),(6,59),(6,60),(6,61),(6,62),(7,8),(7,9),(7,12),(7,14),(7,16),(7,18),(7,20),(7,22),(7,24),(7,25),(7,28),(7,29),(7,30),(7,34),(7,35),(7,38),(7,39),(7,42),(7,44),(7,46),(7,48),(7,50),(7,52),(7,54),(7,56),(7,58),(7,60),(7,61),(7,63),(8,9),(8,10),(8,11),(8,14),(8,15),(8,18),(8,19),(8,21),(8,23),(8,25),(8,26),(8,27),(8,31),(8,32),(8,33),(8,36),(8,37),(8,40),(8,41),(8,42),(8,44),(8,45),(8,46),(8,48),(8,49),(8,50),(8,52),(8,53),(8,55),(8,56),(8,58),(8,59),(8,62),(8,63),(9,10),(9,11),(9,14),(9,15),(9,18),(9,19),(9,21),(9,23),(9,24),(9,26),(9,27),(9,31),(9,32),(9,33),(9,36),(9,37),(9,40),(9,41),(9,43),(9,44),(9,45),(9,47),(9,48),(9,49),(9,51),(9,52),(9,53),(9,55),(9,57),(9,58),(9,59),(9,62),(9,63),(10,11),(10,12),(10,13),(10,16),(10,17),(10,20),(10,22),(10,24),(10,25),(10,27),(10,28),(10,29),(10,30),(10,34),(10,35),(10,38),(10,39),(10,42),(10,43),(10,44),(10,46),(10,47),(10,48),(10,50),(10,51),(10,52),(10,54),(10,56),(10,57),(10,58),(10,60),(10,61),(11,12),(11,13),(11,16),(11,17),(11,20),(11,22),(11,24),(11,25),(11,26),(11,28),(11,29),(11,30),(11,34),(11,35),(11,38),(11,39),(11,42),(11,43),(11,45),(11,46),(11,47),(11,49),(11,50),(11,51),(11,53),(11,54),(11,56),(11,57),(11,59),(11,60),(11,61),(12,13),(12,15),(12,17),(12,19),(12,21),(12,23),(12,24),(12,26),(12,27),(12,28),(12,29),(12,30),(12,31),(12,32),(12,33),(12,34),(12,36),(12,37),(12,38),(12,40),(12,41),(12,43),(12,45),(12,47),(12,49),(12,51),(12,53),(12,54),(12,55),(12,57),(12,59),(12,61),(12,62),(12,63),(13,14),(13,16),(13,18),(13,21),(13,23),(13,25),(13,26),(13,27),(13,28),(13,29),(13,30),(13,31),(13,32),(13,33),(13,35),(13,36),(13,37),(13,39),(13,40),(13,41),(13,42),(13,44),(13,46),(13,48),(13,50),(13,52),(13,54),(13,55),(13,56),(13,58),(13,60),(13,62),(13,63),(14,15),(14,17),(14,19),(14,20),(14,22),(14,24),(14,25),(14,26),(14,28),(14,29),(14,30),(14,31),(14,32),(14,33),(14,34),(14,35),(14,36),(14,38),(14,39),(14,40),(14,43),(14,45),(14,47),(14,49),(14,51),(14,53),(14,54),(14,55),(14,57),(14,59),(14,60),(14,61),(14,62),(15,16),(15,18),(15,20),(15,22),(15,24),(15,25),(15,27),(15,28),(15,29),(15,30),(15,31),(15,32),(15,33),(15,34),(15,35),(15,37),(15,38),(15,39),(15,41),(15,42),(15,44),(15,46),(15,48),(15,50),(15,52),(15,54),(15,55),(15,56),(15,58),(15,60),(15,61),(15,63),(16,17),(16,19),(16,21),(16,23),(16,24),(16,26),(16,27),(16,28),(16,29),(16,30),(16,31),(16,32),(16,33),(16,34),(16,36),(16,37),(16,38),(16,40),(16,41),(16,43),(16,45),(16,46),(16,47),(16,49),(16,51),(16,53),(16,54),(16,55),(16,57),(16,58),(16,59),(16,61),(16,62),(16,63),(17,18),(17,21),(17,23),(17,25),(17,26),(17,27),(17,28),(17,29),(17,30),(17,31),(17,32),(17,33),(17,35),(17,36),(17,37),(17,39),(17,40),(17,41),(17,42),(17,44),(17,46),(17,47),(17,48),(17,50),(17,52),(17,54),(17,55),(17,56),(17,58),(17,59),(17,60),(17,62),(17,63),(18,19),(18,20),(18,22),(18,24),(18,25),(18,26),(18,28),(18,29),(18,30),(18,31),(18,32),(18,33),(18,34),(18,35),(18,36),(18,38),(18,39),(18,40),(18,43),(18,45),(18,47),(18,48),(18,49),(18,51),(18,53),(18,54),(18,55),(18,56),(18,57),(18,59),(18,60),(18,61),(18,62),(19,20),(19,22),(19,24),(19,25),(19,27),(19,28),(19,29),(19,30),(19,31),(19,32),(19,33),(19,34),(19,35),(19,37),(19,38),(19,39),(19,41),(19,42),(19,44),(19,46),(19,48),(19,49),(19,50),(19,52),(19,54),(19,55),(19,56),(19,57),(19,58),(19,60),(19,61),(19,63),(20,21),(20,23),(20,24),(20,25),(20,26),(20,27),(20,31),(20,32),(20,33),(20,36),(20,37),(20,40),(20,41),(20,42),(20,43),(20,44),(20,45),(20,46),(20,47),(20,48),(20,49),(20,50),(20,51),(20,52),(20,53),(20,55),(20,56),(20,57),(20,58),(20,59),(20,62),(20,63),(21,22),(21,24),(21,25),(21,26),(21,27),(21,28),(21,29),(21,30),(21,34),(21,35),(21,38),(21,39),(21,42),(21,43),(21,44),(21,45),(21,46),(21,47),(21,48),(21,49),(21,50),(21,51),(21,52),(21,53),(21,54),(21,56),(21,57),(21,58),(21,59),(21,60),(21,61),(22,23),(22,24),(22,25),(22,26),(22,27),(22,31),(22,32),(22,33),(22,34),(22,35),(22,36),(22,37),(22,40),(22,41),(22,42),(22,43),(22,44),(22,45),(22,46),(22,47),(22,48),(22,49),(22,50),(22,51),(22,52),(22,53),(22,54),(22,55),(22,56),(22,57),(22,58),(22,59),(22,62),(22,63),(23,24),(23,25),(23,26),(23,27),(23,28),(23,29),(23,30),(23,34),(23,35),(23,36),(23,37),(23,38),(23,39),(23,42),(23,43),(23,44),(23,45),(23,46),(23,47),(23,48),(23,49),(23,50),(23,51),(23,52),(23,53),(23,54),(23,55),(23,56),(23,57),(23,58),(23,59),(23,60),(23,61),(24,25),(24,26),(24,27),(24,31),(24,32),(24,33),(24,35),(24,36),(24,37),(24,39),(24,40),(24,41),(24,42),(24,44),(24,45),(24,46),(24,48),(24,49),(24,50),(24,52),(24,53),(24,55),(24,56),(24,58),(24,59),(24,60),(24,62),(24,63),(25,26),(25,27),(25,31),(25,32),(25,33),(25,34),(25,36),(25,37),(25,38),(25,40),(25,41),(25,43),(25,44),(25,45),(25,47),(25,48),(25,49),(25,51),(25,52),(25,53),(25,55),(25,57),(25,58),(25,59),(25,61),(25,62),(25,63),(26,27),(26,28),(26,29),(26,30),(26,34),(26,35),(26,37),(26,38),(26,39),(26,41),(26,42),(26,43),(26,44),(26,46),(26,47),(26,48),(26,50),(26,51),(26,52),(26,54),(26,56),(26,57),(26,58),(26,60),(26,61),(26,63),(27,28),(27,29),(27,30),(27,34),(27,35),(27,36),(27,38),(27,39),(27,40),(27,42),(27,43),(27,45),(27,46),(27,47),(27,49),(27,50),(27,51),(27,53),(27,54),(27,56),(27,57),(27,59),(27,60),(27,61),(27,62),(28,31),(28,32),(28,33),(28,36),(28,37),(28,40),(28,41),(28,42),(28,43),(28,44),(28,45),(28,46),(28,47),(28,48),(28,49),(28,50),(28,51),(28,52),(28,53),(28,55),(28,56),(28,57),(28,58),(28,59),(28,60),(28,61),(28,62),(28,63),(29,31),(29,32),(29,33),(29,36),(29,37),(29,40),(29,41),(29,42),(29,43),(29,44),(29,45),(29,46),(29,47),(29,48),(29,49),(29,50),(29,51),(29,52),(29,53),(29,55),(29,56),(29,57),(29,58),(29,59),(29,60),(29,61),(29,62),(29,63),(30,31),(30,32),(30,33),(30,36),(30,37),(30,40),(30,41),(30,42),(30,43),(30,44),(30,45),(30,46),(30,47),(30,48),(30,49),(30,50),(30,51),(30,52),(30,53),(30,55),(30,56),(30,57),(30,58),(30,59),(30,60),(30,61),(30,62),(30,63),(31,34),(31,35),(31,38),(31,39),(31,42),(31,43),(31,44),(31,45),(31,46),(31,47),(31,48),(31,49),(31,50),(31,51),(31,52),(31,53),(31,54),(31,56),(31,57),(31,58),(31,59),(31,60),(31,61),(31,62),(31,63),(32,34),(32,35),(32,38),(32,39),(32,42),(32,43),(32,44),(32,45),(32,46),(32,47),(32,48),(32,49),(32,50),(32,51),(32,52),(32,53),(32,54),(32,56),(32,57),(32,58),(32,59),(32,60),(32,61),(32,62),(32,63),(33,34),(33,35),(33,38),(33,39),(33,42),(33,43),(33,44),(33,45),(33,46),(33,47),(33,48),(33,49),(33,50),(33,51),(33,52),(33,53),(33,54),(33,56),(33,57),(33,58),(33,59),(33,60),(33,61),(33,62),(33,63),(34,35),(34,36),(34,37),(34,39),(34,40),(34,41),(34,42),(34,43),(34,44),(34,45),(34,46),(34,47),(34,48),(34,49),(34,50),(34,51),(34,52),(34,53),(34,55),(34,56),(34,57),(34,58),(34,59),(34,60),(34,62),(34,63),(35,36),(35,37),(35,38),(35,40),(35,41),(35,42),(35,43),(35,44),(35,45),(35,46),(35,47),(35,48),(35,49),(35,50),(35,51),(35,52),(35,53),(35,55),(35,56),(35,57),(35,58),(35,59),(35,61),(35,62),(35,63),(36,37),(36,38),(36,39),(36,41),(36,42),(36,43),(36,44),(36,45),(36,46),(36,47),(36,48),(36,49),(36,50),(36,51),(36,52),(36,53),(36,54),(36,56),(36,57),(36,58),(36,59),(36,60),(36,61),(36,63),(37,38),(37,39),(37,40),(37,42),(37,43),(37,44),(37,45),(37,46),(37,47),(37,48),(37,49),(37,50),(37,51),(37,52),(37,53),(37,54),(37,56),(37,57),(37,58),(37,59),(37,60),(37,61),(37,62),(38,39),(38,40),(38,41),(38,42),(38,43),(38,44),(38,45),(38,46),(38,47),(38,48),(38,49),(38,50),(38,51),(38,52),(38,53),(38,54),(38,55),(38,56),(38,57),(38,58),(38,59),(38,60),(38,62),(38,63),(39,40),(39,41),(39,42),(39,43),(39,44),(39,45),(39,46),(39,47),(39,48),(39,49),(39,50),(39,51),(39,52),(39,53),(39,54),(39,55),(39,56),(39,57),(39,58),(39,59),(39,61),(39,62),(39,63),(40,41),(40,42),(40,43),(40,44),(40,45),(40,46),(40,47),(40,48),(40,49),(40,50),(40,51),(40,52),(40,53),(40,54),(40,55),(40,56),(40,57),(40,58),(40,59),(40,60),(40,61),(40,63),(41,42),(41,43),(41,44),(41,45),(41,46),(41,47),(41,48),(41,49),(41,50),(41,51),(41,52),(41,53),(41,54),(41,55),(41,56),(41,57),(41,58),(41,59),(41,60),(41,61),(41,62),(42,43),(42,45),(42,47),(42,49),(42,51),(42,53),(42,54),(42,55),(42,57),(42,58),(42,59),(42,60),(42,61),(42,62),(42,63),(43,44),(43,46),(43,48),(43,50),(43,52),(43,54),(43,55),(43,56),(43,58),(43,59),(43,60),(43,61),(43,62),(43,63),(44,45),(44,47),(44,49),(44,51),(44,53),(44,54),(44,55),(44,56),(44,57),(44,59),(44,60),(44,61),(44,62),(44,63),(45,46),(45,48),(45,50),(45,52),(45,54),(45,55),(45,56),(45,57),(45,58),(45,60),(45,61),(45,62),(45,63),(46,47),(46,49),(46,51),(46,53),(46,54),(46,55),(46,57),(46,59),(46,60),(46,61),(46,62),(46,63),(47,48),(47,50),(47,52),(47,54),(47,55),(47,56),(47,58),(47,60),(47,61),(47,62),(47,63),(48,49),(48,51),(48,53),(48,54),(48,55),(48,57),(48,59),(48,60),(48,61),(48,62),(48,63),(49,50),(49,52),(49,54),(49,55),(49,56),(49,58),(49,60),(49,61),(49,62),(49,63),(50,51),(50,52),(50,53),(50,54),(50,55),(50,57),(50,58),(50,59),(50,60),(50,61),(50,62),(50,63),(51,52),(51,53),(51,54),(51,55),(51,56),(51,58),(51,59),(51,60),(51,61),(51,62),(51,63),(52,53),(52,54),(52,55),(52,56),(52,57),(52,59),(52,60),(52,61),(52,62),(52,63),(53,54),(53,55),(53,56),(53,57),(53,58),(53,60),(53,61),(53,62),(53,63),(54,55),(54,56),(54,57),(54,58),(54,59),(54,60),(54,61),(54,62),(54,63),(55,56),(55,57),(55,58),(55,59),(55,60),(55,61),(55,62),(55,63),(56,57),(56,58),(56,59),(56,60),(56,61),(56,62),(56,63),(57,58),(57,59),(57,60),(57,61),(57,62),(57,63),(58,59),(58,60),(58,61),(58,62),(58,63),(59,60),(59,61),(59,62),(59,63),(60,61),(60,62),(60,63),(61,62),(61,63),(62,63)],64)
=> ? = 10 + 1
[[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,2,2],[3,3,3],[4,4],[5]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,2],[2,2,2,2],[3,3,3],[4,4],[5]]
=> ([(0,1)],2)
=> ([],2)
=> 2 = 1 + 1
[[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,2,3],[3,3,3],[4,4],[5]]
=> ([(0,1)],2)
=> ([],2)
=> 2 = 1 + 1
[[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,2],[2,2,2,3],[3,3,3],[4,4],[5]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 3 = 2 + 1
[[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,3],[2,2,2,3],[3,3,3],[4,4],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 4 = 3 + 1
[[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,2,2],[2,2,2,3],[3,3,3],[4,4],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 4 = 3 + 1
[[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,2,3],[2,2,2,3],[3,3,3],[4,4],[5]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ([(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
=> ? = 4 + 1
[[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,2,2],[3,3,4],[4,4],[5]]
=> ([(0,1)],2)
=> ([],2)
=> 2 = 1 + 1
[[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,2],[2,2,2,2],[3,3,4],[4,4],[5]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 3 = 2 + 1
[[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,2,3],[3,3,4],[4,4],[5]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 3 = 2 + 1
[[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,2],[2,2,2,3],[3,3,4],[4,4],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 4 = 3 + 1
[[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,3],[2,2,2,3],[3,3,4],[4,4],[5]]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(4,5)],6)
=> 5 = 4 + 1
[[0,1,0,0,0],[0,0,1,0,0],[1,0,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,2,2],[2,2,2,3],[3,3,4],[4,4],[5]]
=> ?
=> ?
=> ? = 4 + 1
[[0,0,1,0,0],[0,1,0,0,0],[1,0,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,2,3],[2,2,2,3],[3,3,4],[4,4],[5]]
=> ([(0,3),(0,7),(1,8),(2,8),(3,9),(4,5),(5,1),(6,2),(7,4),(7,9),(9,6)],10)
=> ([(2,9),(3,6),(3,7),(3,8),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,9),(7,9),(8,9)],10)
=> ? = 5 + 1
[[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,2,4],[3,3,4],[4,4],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 4 = 3 + 1
[[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,2],[2,2,2,4],[3,3,4],[4,4],[5]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> 5 = 4 + 1
[[0,0,1,0,0],[1,0,-1,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,3],[2,2,2,4],[3,3,4],[4,4],[5]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
=> ? = 5 + 1
[[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,4],[2,2,2,4],[3,3,4],[4,4],[5]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ([(6,7)],8)
=> ? = 6 + 1
[[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,2,2],[2,2,2,4],[3,3,4],[4,4],[5]]
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> ([(3,6),(3,9),(4,5),(4,9),(5,8),(6,8),(7,8),(7,9),(8,9)],10)
=> ? = 5 + 1
[[0,0,1,0,0],[0,1,-1,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,2,3],[2,2,2,4],[3,3,4],[4,4],[5]]
=> ?
=> ?
=> ? = 6 + 1
[[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,2,4],[2,2,2,4],[3,3,4],[4,4],[5]]
=> ([(0,7),(0,8),(1,12),(2,11),(3,10),(4,10),(4,11),(5,3),(6,1),(6,13),(7,9),(8,5),(9,2),(9,4),(10,14),(11,6),(11,14),(13,12),(14,13)],15)
=> ([(2,11),(3,10),(3,11),(4,12),(4,13),(4,14),(5,12),(5,13),(5,14),(6,8),(6,12),(6,13),(6,14),(7,9),(7,12),(7,13),(7,14),(8,9),(8,10),(8,11),(9,12),(9,13),(9,14),(10,12),(10,13),(10,14),(11,12),(11,13),(11,14)],15)
=> ? = 7 + 1
[[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,3,3],[3,3,4],[4,4],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 4 = 3 + 1
[[0,1,0,0,0],[1,-1,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,2],[2,2,3,3],[3,3,4],[4,4],[5]]
=> ?
=> ?
=> ? = 4 + 1
[[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,3],[2,2,3,3],[3,3,4],[4,4],[5]]
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ([(5,6)],7)
=> 6 = 5 + 1
[[0,1,0,0,0],[0,0,1,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,2,2],[2,2,3,3],[3,3,4],[4,4],[5]]
=> ?
=> ?
=> ? = 5 + 1
[[0,0,1,0,0],[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,2,3],[2,2,3,3],[3,3,4],[4,4],[5]]
=> ?
=> ?
=> ? = 6 + 1
[[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ([(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
=> ? = 4 + 1
[[0,1,0,0,0],[1,-1,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,2],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ([(0,5),(0,6),(1,8),(2,9),(3,8),(3,9),(4,1),(5,4),(6,7),(7,2),(7,3),(8,10),(9,10)],11)
=> ([(2,6),(2,10),(3,7),(3,8),(3,9),(4,7),(4,8),(4,9),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(7,10),(8,10),(9,10)],11)
=> ? = 5 + 1
[[0,0,1,0,0],[1,0,-1,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,3],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ?
=> ?
=> ? = 6 + 1
[[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,4],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ([(0,5),(0,10),(1,16),(2,15),(3,14),(4,13),(5,12),(6,2),(6,13),(7,4),(7,14),(8,1),(9,6),(10,11),(10,12),(11,3),(11,7),(12,9),(13,15),(14,8),(15,16)],17)
=> ([(2,14),(3,11),(3,12),(3,13),(3,16),(4,10),(4,11),(4,12),(4,13),(4,16),(5,7),(5,8),(5,9),(5,10),(5,14),(6,7),(6,8),(6,9),(6,10),(6,14),(6,16),(7,11),(7,12),(7,13),(7,15),(7,16),(8,11),(8,12),(8,13),(8,15),(8,16),(9,11),(9,12),(9,13),(9,15),(9,16),(10,11),(10,12),(10,13),(10,15),(10,16),(11,14),(11,15),(12,14),(12,15),(13,14),(13,15),(14,15),(14,16),(15,16)],17)
=> ? = 7 + 1
[[0,1,0,0,0],[0,0,0,1,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,2,2],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ?
=> ?
=> ? = 6 + 1
[[0,0,1,0,0],[0,1,-1,1,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,2,3],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ([(0,9),(0,11),(1,18),(2,17),(3,19),(4,13),(4,19),(5,12),(5,13),(6,16),(7,14),(8,5),(8,18),(9,10),(10,3),(10,4),(11,1),(11,8),(12,17),(13,15),(15,16),(16,14),(17,7),(18,2),(18,12),(19,6),(19,15)],20)
=> ([(2,10),(2,11),(2,13),(2,15),(2,17),(2,19),(3,8),(3,9),(3,12),(3,14),(3,16),(3,18),(4,8),(4,9),(4,12),(4,14),(4,16),(4,18),(4,19),(5,10),(5,11),(5,13),(5,15),(5,17),(5,18),(5,19),(6,8),(6,9),(6,12),(6,13),(6,14),(6,16),(6,17),(6,18),(6,19),(7,10),(7,11),(7,12),(7,13),(7,15),(7,16),(7,17),(7,18),(7,19),(8,10),(8,11),(8,13),(8,15),(8,17),(8,19),(9,10),(9,11),(9,13),(9,15),(9,17),(9,19),(10,12),(10,14),(10,16),(10,18),(11,12),(11,14),(11,16),(11,18),(12,13),(12,15),(12,17),(12,19),(13,14),(13,16),(13,18),(14,15),(14,16),(14,17),(14,19),(15,16),(15,17),(15,18),(16,17),(16,19),(17,18),(18,19)],20)
=> ? = 7 + 1
[[0,0,0,1,0],[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,2,4],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ?
=> ?
=> ? = 8 + 1
[[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,3,3],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ([(0,8),(2,11),(2,12),(3,10),(4,9),(5,4),(5,14),(6,3),(6,14),(7,1),(8,5),(8,6),(9,11),(9,13),(10,12),(10,13),(11,15),(12,15),(13,15),(14,2),(14,9),(14,10),(15,7)],16)
=> ([(5,14),(5,15),(6,7),(6,15),(7,14),(8,11),(8,12),(8,13),(9,10),(9,12),(9,13),(9,14),(10,11),(10,13),(10,15),(11,12),(11,14),(12,15),(13,14),(13,15),(14,15)],16)
=> ? = 8 + 1
[[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,3,4],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ([(0,13),(0,15),(1,17),(2,16),(2,17),(3,19),(4,16),(4,18),(5,22),(6,21),(7,20),(8,23),(8,29),(9,4),(9,28),(10,3),(10,28),(11,6),(12,7),(12,24),(13,14),(14,1),(14,2),(15,9),(15,10),(16,25),(17,12),(17,25),(18,26),(18,29),(19,23),(19,26),(20,27),(22,27),(23,30),(24,20),(24,22),(25,24),(26,30),(27,21),(28,8),(28,18),(28,19),(29,5),(29,30),(30,11)],31)
=> ([(2,11),(2,12),(2,13),(2,21),(2,25),(2,26),(2,30),(3,18),(3,19),(3,22),(3,23),(3,27),(3,28),(3,29),(4,10),(4,11),(4,12),(4,13),(4,21),(4,25),(4,26),(4,29),(4,30),(5,8),(5,18),(5,19),(5,22),(5,23),(5,27),(5,28),(5,29),(5,30),(6,7),(6,9),(6,10),(6,16),(6,18),(6,19),(6,20),(6,22),(6,23),(6,27),(6,28),(6,29),(6,30),(7,8),(7,17),(7,18),(7,19),(7,22),(7,23),(7,25),(7,26),(7,27),(7,28),(7,29),(7,30),(8,9),(8,10),(8,16),(8,18),(8,19),(8,20),(8,22),(8,23),(8,27),(8,28),(8,29),(9,11),(9,12),(9,13),(9,14),(9,15),(9,17),(9,21),(9,24),(9,25),(9,26),(9,29),(9,30),(10,11),(10,12),(10,13),(10,14),(10,15),(10,17),(10,21),(10,24),(10,25),(10,26),(10,30),(11,16),(11,18),(11,19),(11,20),(11,22),(11,23),(11,24),(11,27),(11,28),(11,29),(12,16),(12,18),(12,19),(12,20),(12,22),(12,23),(12,24),(12,27),(12,28),(12,29),(13,16),(13,18),(13,19),(13,20),(13,22),(13,23),(13,24),(13,27),(13,28),(13,29),(14,16),(14,18),(14,19),(14,20),(14,21),(14,22),(14,23),(14,25),(14,26),(14,27),(14,28),(14,29),(14,30),(15,16),(15,17),(15,18),(15,19),(15,20),(15,22),(15,23),(15,25),(15,26),(15,27),(15,28),(15,29),(15,30),(16,17),(16,21),(16,24),(16,25),(16,26),(16,27),(16,29),(16,30),(17,18),(17,19),(17,20),(17,21),(17,22),(17,23),(17,26),(17,27),(17,28),(17,29),(17,30),(18,21),(18,24),(18,25),(18,26),(18,30),(19,21),(19,24),(19,25),(19,26),(19,30),(20,21),(20,23),(20,24),(20,25),(20,26),(20,27),(20,28),(20,29),(20,30),(21,22),(21,23),(21,24),(21,25),(21,27),(21,28),(21,29),(22,24),(22,25),(22,26),(22,28),(22,30),(23,24),(23,25),(23,26),(23,30),(24,25),(24,26),(24,27),(24,28),(24,29),(24,30),(25,27),(25,28),(25,29),(26,27),(26,28),(26,29),(27,30),(28,30),(29,30)],31)
=> ? = 9 + 1
[[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,0,0,1]]
=> [[1,1,2,2,2],[2,2,3,3],[3,3,4],[4,4],[5]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ([(6,7)],8)
=> ? = 6 + 1
[[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,0,0,1]]
=> [[1,1,2,2,3],[2,2,3,3],[3,3,4],[4,4],[5]]
=> ([(0,5),(0,10),(1,16),(2,15),(3,14),(4,13),(5,12),(6,2),(6,13),(7,4),(7,14),(8,1),(9,6),(10,11),(10,12),(11,3),(11,7),(12,9),(13,15),(14,8),(15,16)],17)
=> ([(2,14),(3,11),(3,12),(3,13),(3,16),(4,10),(4,11),(4,12),(4,13),(4,16),(5,7),(5,8),(5,9),(5,10),(5,14),(6,7),(6,8),(6,9),(6,10),(6,14),(6,16),(7,11),(7,12),(7,13),(7,15),(7,16),(8,11),(8,12),(8,13),(8,15),(8,16),(9,11),(9,12),(9,13),(9,15),(9,16),(10,11),(10,12),(10,13),(10,15),(10,16),(11,14),(11,15),(12,14),(12,15),(13,14),(13,15),(14,15),(14,16),(15,16)],17)
=> ? = 7 + 1
[[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1]]
=> [[1,1,2,2,2],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ([(0,7),(0,8),(1,12),(2,11),(3,10),(4,10),(4,11),(5,3),(6,1),(6,13),(7,9),(8,5),(9,2),(9,4),(10,14),(11,6),(11,14),(13,12),(14,13)],15)
=> ([(2,11),(3,10),(3,11),(4,12),(4,13),(4,14),(5,12),(5,13),(5,14),(6,8),(6,12),(6,13),(6,14),(7,9),(7,12),(7,13),(7,14),(8,9),(8,10),(8,11),(9,12),(9,13),(9,14),(10,12),(10,13),(10,14),(11,12),(11,13),(11,14)],15)
=> ? = 7 + 1
[[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1]]
=> [[1,1,2,2,3],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ?
=> ?
=> ? = 8 + 1
[[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1]]
=> [[1,1,2,2,4],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ([(0,14),(0,15),(1,19),(2,18),(3,29),(4,30),(5,22),(6,23),(7,24),(7,25),(8,9),(9,7),(9,18),(9,19),(10,5),(11,6),(12,2),(12,29),(13,1),(13,30),(14,16),(14,28),(15,17),(15,28),(16,3),(16,12),(17,4),(17,13),(18,24),(18,27),(19,25),(19,27),(20,26),(21,26),(22,20),(23,21),(24,22),(24,31),(25,23),(25,31),(27,31),(28,8),(29,10),(30,11),(31,20),(31,21)],32)
=> ([(2,3),(2,9),(2,27),(2,28),(2,29),(2,31),(3,8),(3,24),(3,25),(3,26),(3,30),(4,5),(4,18),(4,20),(4,23),(4,24),(4,25),(4,26),(4,30),(5,17),(5,19),(5,22),(5,27),(5,28),(5,29),(5,31),(6,8),(6,9),(6,24),(6,25),(6,26),(6,27),(6,28),(6,29),(6,30),(6,31),(7,8),(7,9),(7,13),(7,14),(7,21),(7,24),(7,25),(7,26),(7,27),(7,28),(7,29),(7,30),(7,31),(8,9),(8,13),(8,15),(8,17),(8,19),(8,22),(8,27),(8,28),(8,29),(8,31),(9,14),(9,16),(9,18),(9,20),(9,23),(9,24),(9,25),(9,26),(9,30),(10,17),(10,18),(10,19),(10,20),(10,22),(10,23),(10,24),(10,25),(10,26),(10,27),(10,28),(10,29),(10,30),(10,31),(11,17),(11,18),(11,19),(11,20),(11,22),(11,23),(11,24),(11,25),(11,26),(11,27),(11,28),(11,29),(11,30),(11,31),(12,17),(12,18),(12,19),(12,20),(12,22),(12,23),(12,24),(12,25),(12,26),(12,27),(12,28),(12,29),(12,30),(12,31),(13,14),(13,16),(13,18),(13,20),(13,23),(13,24),(13,25),(13,26),(13,27),(13,28),(13,29),(13,30),(13,31),(14,15),(14,17),(14,19),(14,22),(14,24),(14,25),(14,26),(14,27),(14,28),(14,29),(14,30),(14,31),(15,16),(15,18),(15,20),(15,21),(15,23),(15,24),(15,25),(15,26),(15,27),(15,28),(15,29),(15,30),(15,31),(16,17),(16,19),(16,21),(16,22),(16,24),(16,25),(16,26),(16,27),(16,28),(16,29),(16,30),(16,31),(17,18),(17,20),(17,21),(17,23),(17,24),(17,25),(17,26),(17,30),(18,19),(18,21),(18,22),(18,27),(18,28),(18,29),(18,31),(19,20),(19,21),(19,23),(19,24),(19,25),(19,26),(19,30),(19,31),(20,21),(20,22),(20,27),(20,28),(20,29),(20,30),(20,31),(21,22),(21,23),(21,24),(21,25),(21,26),(21,27),(21,28),(21,29),(21,30),(21,31),(22,23),(22,24),(22,25),(22,26),(22,27),(22,28),(22,29),(22,30),(22,31),(23,24),(23,25),(23,26),(23,27),(23,28),(23,29),(23,30),(23,31),(24,27),(24,28),(24,29),(24,31),(25,27),(25,28),(25,29),(25,31),(26,27),(26,28),(26,29),(26,31),(27,30),(28,30),(29,30),(30,31)],32)
=> ? = 9 + 1
[[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,1,1],[2,2,2,2],[3,3,3],[4,5],[5]]
=> ([(0,1)],2)
=> ([],2)
=> 2 = 1 + 1
[[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,1,2],[2,2,2,2],[3,3,3],[4,5],[5]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 3 = 2 + 1
[[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,1,1],[2,2,2,3],[3,3,3],[4,5],[5]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 3 = 2 + 1
[[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,1,2],[2,2,2,3],[3,3,3],[4,5],[5]]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 4 = 3 + 1
[[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[1,1,1,1,1],[2,2,2,2],[3,3,4],[4,5],[5]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 3 = 2 + 1
[[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[1,1,1,1,2],[2,2,2,2],[3,3,4],[4,5],[5]]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 4 = 3 + 1
[[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[1,1,1,1,1],[2,2,2,3],[3,3,4],[4,5],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 4 = 3 + 1
[[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[1,1,1,1,2],[2,2,2,3],[3,3,4],[4,5],[5]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 5 = 4 + 1
Description
The domination number of a graph.
The domination number of a graph is given by the minimum size of a dominating set of vertices. A dominating set of vertices is a subset of the vertex set of such that every vertex is either in this subset or adjacent to an element of this subset.
The following 21 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000916The packing number of a graph. St001286The annihilation number of a graph. St001322The size of a minimal independent dominating set in a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001339The irredundance number of a graph. St001829The common independence number of a graph. St000528The height of a poset. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St000080The rank of the poset. St001782The order of rowmotion on the set of order ideals of a poset. St001820The size of the image of the pop stack sorting operator. St001720The minimal length of a chain of small intervals in a lattice. St000906The length of the shortest maximal chain in a poset. St000643The size of the largest orbit of antichains under Panyushev complementation. St001623The number of doubly irreducible elements of a lattice. St001626The number of maximal proper sublattices of a lattice. St001637The number of (upper) dissectors of a poset. St001668The number of points of the poset minus the width of the poset. St000454The largest eigenvalue of a graph if it is integral.
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