- FindStat

Your data matches 82 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000147

Mapping

Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[[1,2]]
 => ([(0,1)],2)
 => [2]
 => 2
[[2,2]]
 => ([(0,2),(2,1)],3)
 => [3]
 => 3
[[1],[2]]
 => ([],1)
 => [1]
 => 1
[[1,3]]
 => ([(0,2),(2,1)],3)
 => [3]
 => 3
[[2,3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => [4,1]
 => 4
[[3,3]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => [5,1]
 => 5
[[1],[3]]
 => ([(0,1)],2)
 => [2]
 => 2
[[2],[3]]
 => ([(0,2),(2,1)],3)
 => [3]
 => 3
[[1,1,2]]
 => ([(0,1)],2)
 => [2]
 => 2
[[1,2,2]]
 => ([(0,2),(2,1)],3)
 => [3]
 => 3
[[2,2,2]]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => 4
[[1,1],[2]]
 => ([],1)
 => [1]
 => 1
[[1,2],[2]]
 => ([(0,1)],2)
 => [2]
 => 2
[[1,4]]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => 4
[[2,4]]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => [5,2]
 => 5
[[3,4]]
 => ([(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
[[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
[[1],[4]]
 => ([(0,2),(2,1)],3)
 => [3]
 => 3
[[2],[4]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => [4,1]
 => 4
[[3],[4]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => [5,1]
 => 5
[[1,1,3]]
 => ([(0,2),(2,1)],3)
 => [3]
 => 3
[[1,2,3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => [4,1]
 => 4
[[1,3,3]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => [5,1]
 => 5
[[2,2,3]]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => [5,2]
 => 5
[[2,3,3]]
 => ([(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
[[3,3,3]]
 => ([(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
[[1,1],[3]]
 => ([(0,1)],2)
 => [2]
 => 2
[[1,2],[3]]
 => ([(0,2),(2,1)],3)
 => [3]
 => 3
[[1,3],[2]]
 => ([(0,2),(2,1)],3)
 => [3]
 => 3
[[1,3],[3]]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => 4
[[2,2],[3]]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => 4
[[2,3],[3]]
 => ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
 => [5,3]
 => 5
[[1],[2],[3]]
 => ([],1)
 => [1]
 => 1
[[1,1,1,2]]
 => ([(0,1)],2)
 => [2]
 => 2
[[1,1,2,2]]
 => ([(0,2),(2,1)],3)
 => [3]
 => 3
[[1,2,2,2]]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => 4
[[2,2,2,2]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [5]
 => 5
[[1,1,1],[2]]
 => ([],1)
 => [1]
 => 1
[[1,1,2],[2]]
 => ([(0,1)],2)
 => [2]
 => 2
[[1,2,2],[2]]
 => ([(0,2),(2,1)],3)
 => [3]
 => 3
[[1,1],[2,2]]
 => ([],1)
 => [1]
 => 1
[[1,5]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [5]
 => 5
[[2,5]]
 => ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
 => [6,3]
 => 6
[[3,5]]
 => ([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
 => [7,4,1]
 => 7
[[4,5]]
 => ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
 => [8,5,1]
 => 8
[[5,5]]
 => ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
 => [9,5,1]
 => 9
[[1],[5]]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => 4
[[2],[5]]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => [5,2]
 => 5
[[3],[5]]
 => ([(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
[[4],[5]]
 => ([(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

Description

The largest part of an integer partition.

Matching statistic: St001392

Mapping

Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St001392: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[[1,2]]
 => ([(0,1)],2)
 => [2]
 => 1 = 2 - 1
[[2,2]]
 => ([(0,2),(2,1)],3)
 => [3]
 => 2 = 3 - 1
[[1],[2]]
 => ([],1)
 => [1]
 => 0 = 1 - 1
[[1,3]]
 => ([(0,2),(2,1)],3)
 => [3]
 => 2 = 3 - 1
[[2,3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => [4,1]
 => 3 = 4 - 1
[[3,3]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => [5,1]
 => 4 = 5 - 1
[[1],[3]]
 => ([(0,1)],2)
 => [2]
 => 1 = 2 - 1
[[2],[3]]
 => ([(0,2),(2,1)],3)
 => [3]
 => 2 = 3 - 1
[[1,1,2]]
 => ([(0,1)],2)
 => [2]
 => 1 = 2 - 1
[[1,2,2]]
 => ([(0,2),(2,1)],3)
 => [3]
 => 2 = 3 - 1
[[2,2,2]]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => 3 = 4 - 1
[[1,1],[2]]
 => ([],1)
 => [1]
 => 0 = 1 - 1
[[1,2],[2]]
 => ([(0,1)],2)
 => [2]
 => 1 = 2 - 1
[[1,4]]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => 3 = 4 - 1
[[2,4]]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => [5,2]
 => 4 = 5 - 1
[[3,4]]
 => ([(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 = 6 - 1
[[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 = 7 - 1
[[1],[4]]
 => ([(0,2),(2,1)],3)
 => [3]
 => 2 = 3 - 1
[[2],[4]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => [4,1]
 => 3 = 4 - 1
[[3],[4]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => [5,1]
 => 4 = 5 - 1
[[1,1,3]]
 => ([(0,2),(2,1)],3)
 => [3]
 => 2 = 3 - 1
[[1,2,3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => [4,1]
 => 3 = 4 - 1
[[1,3,3]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => [5,1]
 => 4 = 5 - 1
[[2,2,3]]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => [5,2]
 => 4 = 5 - 1
[[2,3,3]]
 => ([(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 = 6 - 1
[[3,3,3]]
 => ([(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 = 7 - 1
[[1,1],[3]]
 => ([(0,1)],2)
 => [2]
 => 1 = 2 - 1
[[1,2],[3]]
 => ([(0,2),(2,1)],3)
 => [3]
 => 2 = 3 - 1
[[1,3],[2]]
 => ([(0,2),(2,1)],3)
 => [3]
 => 2 = 3 - 1
[[1,3],[3]]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => 3 = 4 - 1
[[2,2],[3]]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => 3 = 4 - 1
[[2,3],[3]]
 => ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
 => [5,3]
 => 4 = 5 - 1
[[1],[2],[3]]
 => ([],1)
 => [1]
 => 0 = 1 - 1
[[1,1,1,2]]
 => ([(0,1)],2)
 => [2]
 => 1 = 2 - 1
[[1,1,2,2]]
 => ([(0,2),(2,1)],3)
 => [3]
 => 2 = 3 - 1
[[1,2,2,2]]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => 3 = 4 - 1
[[2,2,2,2]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [5]
 => 4 = 5 - 1
[[1,1,1],[2]]
 => ([],1)
 => [1]
 => 0 = 1 - 1
[[1,1,2],[2]]
 => ([(0,1)],2)
 => [2]
 => 1 = 2 - 1
[[1,2,2],[2]]
 => ([(0,2),(2,1)],3)
 => [3]
 => 2 = 3 - 1
[[1,1],[2,2]]
 => ([],1)
 => [1]
 => 0 = 1 - 1
[[1,5]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [5]
 => 4 = 5 - 1
[[2,5]]
 => ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
 => [6,3]
 => 5 = 6 - 1
[[3,5]]
 => ([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
 => [7,4,1]
 => 6 = 7 - 1
[[4,5]]
 => ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
 => [8,5,1]
 => 7 = 8 - 1
[[5,5]]
 => ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
 => [9,5,1]
 => 8 = 9 - 1
[[1],[5]]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => 3 = 4 - 1
[[2],[5]]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => [5,2]
 => 4 = 5 - 1
[[3],[5]]
 => ([(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 = 6 - 1
[[4],[5]]
 => ([(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 = 7 - 1

Description

The largest nonnegative integer which is not a part and is smaller than the largest part of the partition.

Matching statistic: St000010

Mapping

Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[[1,2]]
 => ([(0,1)],2)
 => [2]
 => [1,1]
 => 2
[[2,2]]
 => ([(0,2),(2,1)],3)
 => [3]
 => [1,1,1]
 => 3
[[1],[2]]
 => ([],1)
 => [1]
 => [1]
 => 1
[[1,3]]
 => ([(0,2),(2,1)],3)
 => [3]
 => [1,1,1]
 => 3
[[2,3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => [4,1]
 => [2,1,1,1]
 => 4
[[3,3]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => [5,1]
 => [2,1,1,1,1]
 => 5
[[1],[3]]
 => ([(0,1)],2)
 => [2]
 => [1,1]
 => 2
[[2],[3]]
 => ([(0,2),(2,1)],3)
 => [3]
 => [1,1,1]
 => 3
[[1,1,2]]
 => ([(0,1)],2)
 => [2]
 => [1,1]
 => 2
[[1,2,2]]
 => ([(0,2),(2,1)],3)
 => [3]
 => [1,1,1]
 => 3
[[2,2,2]]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => [1,1,1,1]
 => 4
[[1,1],[2]]
 => ([],1)
 => [1]
 => [1]
 => 1
[[1,2],[2]]
 => ([(0,1)],2)
 => [2]
 => [1,1]
 => 2
[[1,4]]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => [1,1,1,1]
 => 4
[[2,4]]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => [5,2]
 => [2,2,1,1,1]
 => 5
[[3,4]]
 => ([(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]
 => [2,2,2,1,1,1]
 => 6
[[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]
 => [2,2,2,1,1,1,1]
 => 7
[[1],[4]]
 => ([(0,2),(2,1)],3)
 => [3]
 => [1,1,1]
 => 3
[[2],[4]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => [4,1]
 => [2,1,1,1]
 => 4
[[3],[4]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => [5,1]
 => [2,1,1,1,1]
 => 5
[[1,1,3]]
 => ([(0,2),(2,1)],3)
 => [3]
 => [1,1,1]
 => 3
[[1,2,3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => [4,1]
 => [2,1,1,1]
 => 4
[[1,3,3]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => [5,1]
 => [2,1,1,1,1]
 => 5
[[2,2,3]]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => [5,2]
 => [2,2,1,1,1]
 => 5
[[2,3,3]]
 => ([(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]
 => [2,2,2,1,1,1]
 => 6
[[3,3,3]]
 => ([(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]
 => [2,2,2,1,1,1,1]
 => 7
[[1,1],[3]]
 => ([(0,1)],2)
 => [2]
 => [1,1]
 => 2
[[1,2],[3]]
 => ([(0,2),(2,1)],3)
 => [3]
 => [1,1,1]
 => 3
[[1,3],[2]]
 => ([(0,2),(2,1)],3)
 => [3]
 => [1,1,1]
 => 3
[[1,3],[3]]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => [1,1,1,1]
 => 4
[[2,2],[3]]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => [1,1,1,1]
 => 4
[[2,3],[3]]
 => ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
 => [5,3]
 => [2,2,2,1,1]
 => 5
[[1],[2],[3]]
 => ([],1)
 => [1]
 => [1]
 => 1
[[1,1,1,2]]
 => ([(0,1)],2)
 => [2]
 => [1,1]
 => 2
[[1,1,2,2]]
 => ([(0,2),(2,1)],3)
 => [3]
 => [1,1,1]
 => 3
[[1,2,2,2]]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => [1,1,1,1]
 => 4
[[2,2,2,2]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [5]
 => [1,1,1,1,1]
 => 5
[[1,1,1],[2]]
 => ([],1)
 => [1]
 => [1]
 => 1
[[1,1,2],[2]]
 => ([(0,1)],2)
 => [2]
 => [1,1]
 => 2
[[1,2,2],[2]]
 => ([(0,2),(2,1)],3)
 => [3]
 => [1,1,1]
 => 3
[[1,1],[2,2]]
 => ([],1)
 => [1]
 => [1]
 => 1
[[1,5]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [5]
 => [1,1,1,1,1]
 => 5
[[2,5]]
 => ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
 => [6,3]
 => [2,2,2,1,1,1]
 => 6
[[3,5]]
 => ([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
 => [7,4,1]
 => [3,2,2,2,1,1,1]
 => 7
[[4,5]]
 => ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
 => [8,5,1]
 => [3,2,2,2,2,1,1,1]
 => 8
[[5,5]]
 => ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
 => [9,5,1]
 => [3,2,2,2,2,1,1,1,1]
 => 9
[[1],[5]]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => [1,1,1,1]
 => 4
[[2],[5]]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => [5,2]
 => [2,2,1,1,1]
 => 5
[[3],[5]]
 => ([(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]
 => [2,2,2,1,1,1]
 => 6
[[4],[5]]
 => ([(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]
 => [2,2,2,1,1,1,1]
 => 7

Description

The length of the partition.

Matching statistic: St000380
(load all 2 compositions to match this statistic)

Mapping

Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000380: Integer partitions ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%

Values

[[1,2]]
 => ([(0,1)],2)
 => [2]
 => 3 = 2 + 1
[[2,2]]
 => ([(0,2),(2,1)],3)
 => [3]
 => 4 = 3 + 1
[[1],[2]]
 => ([],1)
 => [1]
 => 2 = 1 + 1
[[1,3]]
 => ([(0,2),(2,1)],3)
 => [3]
 => 4 = 3 + 1
[[2,3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => [4,1]
 => 5 = 4 + 1
[[3,3]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => [5,1]
 => 6 = 5 + 1
[[1],[3]]
 => ([(0,1)],2)
 => [2]
 => 3 = 2 + 1
[[2],[3]]
 => ([(0,2),(2,1)],3)
 => [3]
 => 4 = 3 + 1
[[1,1,2]]
 => ([(0,1)],2)
 => [2]
 => 3 = 2 + 1
[[1,2,2]]
 => ([(0,2),(2,1)],3)
 => [3]
 => 4 = 3 + 1
[[2,2,2]]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => 5 = 4 + 1
[[1,1],[2]]
 => ([],1)
 => [1]
 => 2 = 1 + 1
[[1,2],[2]]
 => ([(0,1)],2)
 => [2]
 => 3 = 2 + 1
[[1,4]]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => 5 = 4 + 1
[[2,4]]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => [5,2]
 => 6 = 5 + 1
[[3,4]]
 => ([(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 = 6 + 1
[[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 = 7 + 1
[[1],[4]]
 => ([(0,2),(2,1)],3)
 => [3]
 => 4 = 3 + 1
[[2],[4]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => [4,1]
 => 5 = 4 + 1
[[3],[4]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => [5,1]
 => 6 = 5 + 1
[[1,1,3]]
 => ([(0,2),(2,1)],3)
 => [3]
 => 4 = 3 + 1
[[1,2,3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => [4,1]
 => 5 = 4 + 1
[[1,3,3]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => [5,1]
 => 6 = 5 + 1
[[2,2,3]]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => [5,2]
 => 6 = 5 + 1
[[2,3,3]]
 => ([(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 = 6 + 1
[[3,3,3]]
 => ([(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 = 7 + 1
[[1,1],[3]]
 => ([(0,1)],2)
 => [2]
 => 3 = 2 + 1
[[1,2],[3]]
 => ([(0,2),(2,1)],3)
 => [3]
 => 4 = 3 + 1
[[1,3],[2]]
 => ([(0,2),(2,1)],3)
 => [3]
 => 4 = 3 + 1
[[1,3],[3]]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => 5 = 4 + 1
[[2,2],[3]]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => 5 = 4 + 1
[[2,3],[3]]
 => ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
 => [5,3]
 => 6 = 5 + 1
[[1],[2],[3]]
 => ([],1)
 => [1]
 => 2 = 1 + 1
[[1,1,1,2]]
 => ([(0,1)],2)
 => [2]
 => 3 = 2 + 1
[[1,1,2,2]]
 => ([(0,2),(2,1)],3)
 => [3]
 => 4 = 3 + 1
[[1,2,2,2]]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => 5 = 4 + 1
[[2,2,2,2]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [5]
 => 6 = 5 + 1
[[1,1,1],[2]]
 => ([],1)
 => [1]
 => 2 = 1 + 1
[[1,1,2],[2]]
 => ([(0,1)],2)
 => [2]
 => 3 = 2 + 1
[[1,2,2],[2]]
 => ([(0,2),(2,1)],3)
 => [3]
 => 4 = 3 + 1
[[1,1],[2,2]]
 => ([],1)
 => [1]
 => 2 = 1 + 1
[[1,5]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [5]
 => 6 = 5 + 1
[[2,5]]
 => ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
 => [6,3]
 => 7 = 6 + 1
[[3,5]]
 => ([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
 => [7,4,1]
 => 8 = 7 + 1
[[4,5]]
 => ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
 => [8,5,1]
 => 9 = 8 + 1
[[5,5]]
 => ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
 => [9,5,1]
 => 10 = 9 + 1
[[1],[5]]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => 5 = 4 + 1
[[2],[5]]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => [5,2]
 => 6 = 5 + 1
[[3],[5]]
 => ([(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 = 6 + 1
[[4],[5]]
 => ([(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 = 7 + 1
[[1,2,3],[2],[5]]
 => ([(0,3),(0,4),(1,11),(2,10),(3,2),(3,9),(4,1),(4,9),(5,7),(5,8),(6,12),(7,12),(8,12),(9,5),(9,10),(9,11),(10,6),(10,7),(11,6),(11,8)],13)
 => [6,4,3]
 => ? = 6 + 1
[[1,2,4],[2,4],[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 + 1
[[1,3,3],[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 + 1
[[2,2,2],[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 + 1
[[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 + 1
[[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 + 1
[[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 + 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]
 => ? = 8 + 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]
 => ? = 9 + 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]
 => ? = 8 + 1
[[1,1,1,1,1],[2,2,3,5],[3,3,5],[4,5],[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]
 => ? = 8 + 1
[[1,1,1,1,1],[2,2,4,4],[3,4,5],[4,5],[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]
 => ? = 9 + 1
[[1,1,1,1,1],[2,3,3,3],[3,4,5],[4,5],[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]
 => ? = 8 + 1
[[1,1,1,1,1,2],[2,2,2,2,4],[3,3,3,4],[4,4,4],[5,6],[6]]
 => ([(0,1),(0,2),(1,11),(2,3),(2,4),(2,11),(3,8),(3,10),(4,5),(4,9),(4,10),(5,6),(5,7),(6,13),(7,13),(8,12),(9,7),(9,12),(10,6),(10,12),(11,8),(11,9),(12,13)],14)
 => [6,4,3,1]
 => ? = 6 + 1
[[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,4],[4,4,6],[5,6],[6]]
 => ([(0,1),(0,2),(1,11),(2,3),(2,4),(2,11),(3,8),(3,10),(4,5),(4,9),(4,10),(5,6),(5,7),(6,13),(7,13),(8,12),(9,7),(9,12),(10,6),(10,12),(11,8),(11,9),(12,13)],14)
 => [6,4,3,1]
 => ? = 6 + 1
[[1,1,1,1,2,2,2],[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 + 1

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: St000093
(load all 3 compositions to match this statistic)

Mapping

Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St000093: Graphs ⟶ ℤResult quality: 96% ●values known / values provided: 96%●distinct values known / distinct values provided: 100%

Values

[[1,2]]
 => ([(0,1)],2)
 => ([],2)
 => 2
[[2,2]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 3
[[1],[2]]
 => ([],1)
 => ([],1)
 => 1
[[1,3]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 3
[[2,3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => 4
[[3,3]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => 5
[[1],[3]]
 => ([(0,1)],2)
 => ([],2)
 => 2
[[2],[3]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 3
[[1,1,2]]
 => ([(0,1)],2)
 => ([],2)
 => 2
[[1,2,2]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 3
[[2,2,2]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 4
[[1,1],[2]]
 => ([],1)
 => ([],1)
 => 1
[[1,2],[2]]
 => ([(0,1)],2)
 => ([],2)
 => 2
[[1,4]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 4
[[2,4]]
 => ([(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
[[3,4]]
 => ([(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
[[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
[[1],[4]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 3
[[2],[4]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => 4
[[3],[4]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => 5
[[1,1,3]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 3
[[1,2,3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => 4
[[1,3,3]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => 5
[[2,2,3]]
 => ([(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
[[2,3,3]]
 => ([(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
[[3,3,3]]
 => ([(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
[[1,1],[3]]
 => ([(0,1)],2)
 => ([],2)
 => 2
[[1,2],[3]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 3
[[1,3],[2]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 3
[[1,3],[3]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 4
[[2,2],[3]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 4
[[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
[[1],[2],[3]]
 => ([],1)
 => ([],1)
 => 1
[[1,1,1,2]]
 => ([(0,1)],2)
 => ([],2)
 => 2
[[1,1,2,2]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 3
[[1,2,2,2]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 4
[[2,2,2,2]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 5
[[1,1,1],[2]]
 => ([],1)
 => ([],1)
 => 1
[[1,1,2],[2]]
 => ([(0,1)],2)
 => ([],2)
 => 2
[[1,2,2],[2]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 3
[[1,1],[2,2]]
 => ([],1)
 => ([],1)
 => 1
[[1,5]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 5
[[2,5]]
 => ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
 => ([(3,8),(4,7),(5,6),(5,7),(6,8),(7,8)],9)
 => 6
[[3,5]]
 => ([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
 => ([(3,9),(4,5),(4,11),(5,10),(6,10),(6,11),(7,8),(7,11),(8,9),(8,10),(9,11),(10,11)],12)
 => 7
[[4,5]]
 => ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
 => ([(3,11),(4,10),(5,8),(5,13),(6,9),(6,13),(7,12),(7,13),(8,10),(8,12),(9,11),(9,12),(10,13),(11,13),(12,13)],14)
 => 8
[[5,5]]
 => ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
 => ([(4,12),(5,11),(6,13),(6,14),(7,9),(7,14),(8,10),(8,14),(9,11),(9,13),(10,12),(10,13),(11,14),(12,14),(13,14)],15)
 => 9
[[1],[5]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 4
[[2],[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
[[3],[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)
 => 6
[[4],[5]]
 => ([(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
[[2,2,2,2,2,2,2]]
 => ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
 => ([],8)
 => ? = 8
[[1,8]]
 => ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
 => ([],8)
 => ? = 8
[[1,2,3],[2],[5]]
 => ([(0,3),(0,4),(1,11),(2,10),(3,2),(3,9),(4,1),(4,9),(5,7),(5,8),(6,12),(7,12),(8,12),(9,5),(9,10),(9,11),(10,6),(10,7),(11,6),(11,8)],13)
 => ([(2,11),(2,12),(3,4),(3,12),(4,11),(5,8),(5,9),(5,10),(6,7),(6,9),(6,10),(6,11),(7,8),(7,10),(7,12),(8,9),(8,11),(9,12),(10,11),(10,12),(11,12)],13)
 => ? = 6
[[1,2],[2,5],[4]]
 => ([(0,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,2),(4,3),(4,6),(5,1),(5,4),(6,8),(6,9),(7,10),(8,10),(9,10)],11)
 => ([(3,10),(4,7),(4,8),(4,9),(5,6),(5,8),(5,9),(6,7),(6,9),(7,8),(7,10),(8,10),(9,10)],11)
 => ? = 6
[[1,2],[3,3],[5]]
 => ([(0,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,2),(4,3),(4,6),(5,1),(5,4),(6,8),(6,9),(7,10),(8,10),(9,10)],11)
 => ([(3,10),(4,7),(4,8),(4,9),(5,6),(5,8),(5,9),(6,7),(6,9),(7,8),(7,10),(8,10),(9,10)],11)
 => ? = 6
[[1,1,4],[2,4],[4]]
 => ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
 => ([(6,7)],8)
 => ? = 7
[[1,1,4],[3,3],[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)
 => ? = 7
[[1,1,4],[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)
 => ? = 8
[[1,2,4],[2,4],[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)
 => ? = 8
[[1,2,3],[3,3],[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)
 => ? = 7
[[1,3,3],[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)
 => ? = 9
[[2,2,2],[3,3],[4]]
 => ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
 => ([(6,7)],8)
 => ? = 7
[[2,2,2],[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)
 => ? = 8
[[2,2,3],[3,3],[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)
 => ? = 8
[[2,2,2,2,2,2],[3]]
 => ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
 => ([],8)
 => ? = 8
[[1,2,2,2,2,2,2,2]]
 => ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
 => ([],8)
 => ? = 8
[[2,2,2,2,2,2,2,2]]
 => ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
 => ([],9)
 => ? = 9
[[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)
 => ? = 7
[[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)
 => ? = 8
[[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)
 => ? = 7
[[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)
 => ? = 8
[[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)
 => ? = 9
[[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)
 => ? = 7
[[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)
 => ? = 8
[[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)
 => ? = 8
[[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)
 => ? = 7
[[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)
 => ? = 8
[[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)
 => ? = 8
[[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)
 => ? = 9
[[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)
 => ? = 7
[[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)
 => ? = 8
[[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)
 => ? = 8
[[1,1,1,1,1],[2,2,2,5],[3,3,5],[4,5],[5]]
 => ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
 => ([(6,7)],8)
 => ? = 7
[[1,1,1,1,1],[2,2,3,5],[3,3,5],[4,5],[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)
 => ? = 8
[[1,1,1,1,1],[2,2,2,5],[3,4,5],[4,5],[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)
 => ? = 8
[[1,1,1,1,1],[2,2,4,4],[3,4,5],[4,5],[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)
 => ? = 9
[[1,1,1,1,1],[2,3,3,3],[3,4,4],[4,5],[5]]
 => ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
 => ([(6,7)],8)
 => ? = 7
[[1,1,1,1,1],[2,3,3,4],[3,4,4],[4,5],[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)
 => ? = 8
[[1,1,1,1,1],[2,3,3,3],[3,4,5],[4,5],[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)
 => ? = 8
[[1,1,1,1,1,4],[2,2,2,2,4],[3,3,3,4],[4,4,4],[5,5],[6]]
 => ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
 => ([(6,7)],8)
 => ? = 7
[[1,1,1,1,1,1],[2,2,2,2,5],[3,3,3,5],[4,4,5],[5,5],[6]]
 => ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
 => ([(6,7)],8)
 => ? = 7
[[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,4],[4,4,4],[5,6],[6]]
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ([(2,5),(2,6),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6)],8)
 => ? = 4
[[1,1,1,1,1,3],[2,2,2,2,3],[3,3,3,4],[4,4,4],[5,6],[6]]
 => ([(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(3,7),(3,8),(4,6),(4,8),(5,1),(5,9),(6,11),(7,11),(8,5),(8,11),(9,10),(11,9)],12)
 => ([(2,11),(3,7),(3,11),(4,8),(4,9),(4,10),(5,6),(5,9),(5,10),(6,8),(6,10),(7,8),(7,9),(7,10),(8,9),(8,11),(9,11),(10,11)],12)
 => ? = 6
[[1,1,1,1,1,2],[2,2,2,2,4],[3,3,3,4],[4,4,4],[5,6],[6]]
 => ([(0,1),(0,2),(1,11),(2,3),(2,4),(2,11),(3,8),(3,10),(4,5),(4,9),(4,10),(5,6),(5,7),(6,13),(7,13),(8,12),(9,7),(9,12),(10,6),(10,12),(11,8),(11,9),(12,13)],14)
 => ([(2,12),(3,7),(3,11),(3,13),(4,6),(4,9),(4,10),(4,12),(5,8),(5,9),(5,10),(5,11),(5,13),(6,8),(6,10),(6,11),(6,13),(7,8),(7,9),(7,10),(7,11),(8,9),(8,12),(8,13),(9,11),(9,13),(10,12),(10,13),(11,12),(12,13)],14)
 => ? = 6
[[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,5],[4,4,5],[5,6],[6]]
 => ([(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(3,7),(3,8),(4,6),(4,8),(5,1),(5,9),(6,11),(7,11),(8,5),(8,11),(9,10),(11,9)],12)
 => ([(2,11),(3,7),(3,11),(4,8),(4,9),(4,10),(5,6),(5,9),(5,10),(6,8),(6,10),(7,8),(7,9),(7,10),(8,9),(8,11),(9,11),(10,11)],12)
 => ? = 6
[[1,1,1,1,1,3],[2,2,2,2,3],[3,3,3,3],[4,4,6],[5,6],[6]]
 => ([(0,5),(0,6),(1,4),(1,15),(2,3),(2,14),(3,8),(4,9),(5,2),(5,13),(6,1),(6,13),(8,10),(9,11),(10,7),(11,7),(12,10),(12,11),(13,14),(13,15),(14,8),(14,12),(15,9),(15,12)],16)
 => ([(2,3),(2,11),(2,15),(3,10),(3,14),(4,5),(4,13),(4,14),(5,12),(5,15),(6,12),(6,13),(6,14),(6,15),(7,10),(7,11),(7,14),(7,15),(8,9),(8,10),(8,13),(8,14),(8,15),(9,11),(9,12),(9,14),(9,15),(10,11),(10,12),(10,15),(11,13),(11,14),(12,13),(12,14),(13,15),(14,15)],16)
 => ? = 7
[[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,4],[4,4,6],[5,6],[6]]
 => ([(0,1),(0,2),(1,11),(2,3),(2,4),(2,11),(3,8),(3,10),(4,5),(4,9),(4,10),(5,6),(5,7),(6,13),(7,13),(8,12),(9,7),(9,12),(10,6),(10,12),(11,8),(11,9),(12,13)],14)
 => ([(2,12),(3,7),(3,11),(3,13),(4,6),(4,9),(4,10),(4,12),(5,8),(5,9),(5,10),(5,11),(5,13),(6,8),(6,10),(6,11),(6,13),(7,8),(7,9),(7,10),(7,11),(8,9),(8,12),(8,13),(9,11),(9,13),(10,12),(10,13),(11,12),(12,13)],14)
 => ? = 6
[[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,6],[4,4,6],[5,6],[6]]
 => ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
 => ([(6,7)],8)
 => ? = 7
[[1,1,1,1,2,2,2],[3,3],[4]]
 => ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
 => ([(6,7)],8)
 => ? = 7
[[1,1,1,1,2,2,3],[3,3],[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)
 => ? = 8

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
(load all 2 compositions to match this statistic)

Mapping

Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000384: Integer partitions ⟶ ℤResult quality: 82% ●values known / values provided: 82%●distinct values known / distinct values provided: 100%

Values

[[1,2]]
 => ([(0,1)],2)
 => [2]
 => 2
[[2,2]]
 => ([(0,2),(2,1)],3)
 => [3]
 => 3
[[1],[2]]
 => ([],1)
 => [1]
 => 1
[[1,3]]
 => ([(0,2),(2,1)],3)
 => [3]
 => 3
[[2,3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => [4,1]
 => 4
[[3,3]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => [5,1]
 => 5
[[1],[3]]
 => ([(0,1)],2)
 => [2]
 => 2
[[2],[3]]
 => ([(0,2),(2,1)],3)
 => [3]
 => 3
[[1,1,2]]
 => ([(0,1)],2)
 => [2]
 => 2
[[1,2,2]]
 => ([(0,2),(2,1)],3)
 => [3]
 => 3
[[2,2,2]]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => 4
[[1,1],[2]]
 => ([],1)
 => [1]
 => 1
[[1,2],[2]]
 => ([(0,1)],2)
 => [2]
 => 2
[[1,4]]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => 4
[[2,4]]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => [5,2]
 => 5
[[3,4]]
 => ([(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
[[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
[[1],[4]]
 => ([(0,2),(2,1)],3)
 => [3]
 => 3
[[2],[4]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => [4,1]
 => 4
[[3],[4]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => [5,1]
 => 5
[[1,1,3]]
 => ([(0,2),(2,1)],3)
 => [3]
 => 3
[[1,2,3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => [4,1]
 => 4
[[1,3,3]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => [5,1]
 => 5
[[2,2,3]]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => [5,2]
 => 5
[[2,3,3]]
 => ([(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
[[3,3,3]]
 => ([(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
[[1,1],[3]]
 => ([(0,1)],2)
 => [2]
 => 2
[[1,2],[3]]
 => ([(0,2),(2,1)],3)
 => [3]
 => 3
[[1,3],[2]]
 => ([(0,2),(2,1)],3)
 => [3]
 => 3
[[1,3],[3]]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => 4
[[2,2],[3]]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => 4
[[2,3],[3]]
 => ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
 => [5,3]
 => 5
[[1],[2],[3]]
 => ([],1)
 => [1]
 => 1
[[1,1,1,2]]
 => ([(0,1)],2)
 => [2]
 => 2
[[1,1,2,2]]
 => ([(0,2),(2,1)],3)
 => [3]
 => 3
[[1,2,2,2]]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => 4
[[2,2,2,2]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [5]
 => 5
[[1,1,1],[2]]
 => ([],1)
 => [1]
 => 1
[[1,1,2],[2]]
 => ([(0,1)],2)
 => [2]
 => 2
[[1,2,2],[2]]
 => ([(0,2),(2,1)],3)
 => [3]
 => 3
[[1,1],[2,2]]
 => ([],1)
 => [1]
 => 1
[[1,5]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [5]
 => 5
[[2,5]]
 => ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
 => [6,3]
 => 6
[[3,5]]
 => ([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
 => [7,4,1]
 => 7
[[4,5]]
 => ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
 => [8,5,1]
 => ? = 8
[[5,5]]
 => ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
 => [9,5,1]
 => ? = 9
[[1],[5]]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => 4
[[2],[5]]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => [5,2]
 => 5
[[3],[5]]
 => ([(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
[[4],[5]]
 => ([(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
[[1,1,4]]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => 4
[[1,2,4]]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => [5,2]
 => 5
[[2,3,4]]
 => ([(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
[[2,4,4]]
 => ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
 => [8,5,3]
 => ? = 8
[[3,3,4]]
 => ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
 => [8,5,3]
 => ? = 8
[[2,4],[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
[[2,3,3,3]]
 => ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
 => [8,5,1]
 => ? = 8
[[3,3,3,3]]
 => ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
 => [9,5,1]
 => ? = 9
[[2,3,3],[3]]
 => ([(0,9),(0,10),(1,11),(2,14),(3,12),(4,13),(5,4),(5,11),(6,5),(7,3),(8,1),(8,14),(9,6),(10,2),(10,8),(11,13),(13,12),(14,7)],15)
 => [7,5,3]
 => ? = 7
[[3,6]]
 => ([(0,7),(1,14),(2,9),(3,10),(4,5),(4,14),(5,6),(5,8),(6,2),(6,11),(7,1),(7,4),(8,10),(8,11),(9,13),(10,12),(11,9),(11,12),(12,13),(14,3),(14,8)],15)
 => [8,5,2]
 => ? = 8
[[4],[6]]
 => ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
 => [8,5,1]
 => ? = 8
[[5],[6]]
 => ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
 => [9,5,1]
 => ? = 9
[[1,4,5]]
 => ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
 => [8,5,1]
 => ? = 8
[[1,5,5]]
 => ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
 => [9,5,1]
 => ? = 9
[[2,2,5]]
 => ([(0,6),(1,9),(2,8),(3,5),(3,7),(4,1),(4,7),(5,2),(5,10),(6,3),(6,4),(7,9),(7,10),(8,12),(9,11),(10,8),(10,11),(11,12)],13)
 => [7,4,2]
 => ? = 7
[[2,3],[5]]
 => ([(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
[[2,5],[3]]
 => ([(0,6),(0,7),(1,11),(2,9),(3,9),(3,10),(4,2),(5,1),(5,10),(6,4),(7,8),(8,3),(8,5),(9,12),(10,11),(10,12),(11,13),(12,13)],14)
 => [7,5,2]
 => ? = 7
[[2,4],[5]]
 => ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
 => [8,5,3]
 => ? = 8
[[3,3],[5]]
 => ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
 => [8,5,3]
 => ? = 8
[[1,2,3,4]]
 => ([(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
[[1,2,4,4]]
 => ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
 => [8,5,3]
 => ? = 8
[[1,3,3,4]]
 => ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
 => [8,5,3]
 => ? = 8
[[2,2,2,4]]
 => ([(0,6),(1,9),(2,8),(3,5),(3,7),(4,1),(4,7),(5,2),(5,10),(6,3),(6,4),(7,9),(7,10),(8,12),(9,11),(10,8),(10,11),(11,12)],13)
 => [7,4,2]
 => ? = 7
[[1,2,4],[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
[[1,4,4],[3]]
 => ([(0,7),(1,10),(2,11),(3,8),(4,9),(5,2),(5,9),(6,3),(6,12),(7,4),(7,5),(8,10),(9,6),(9,11),(11,12),(12,1),(12,8)],13)
 => [8,5]
 => ? = 8
[[1,4,4],[4]]
 => ([(0,8),(2,13),(3,11),(4,9),(5,10),(6,3),(6,10),(7,4),(7,12),(8,5),(8,6),(9,13),(10,7),(10,11),(11,12),(12,2),(12,9),(13,1)],14)
 => [9,5]
 => ? = 9
[[2,2,4],[3]]
 => ([(0,6),(0,7),(1,11),(2,9),(3,9),(3,10),(4,5),(5,1),(5,10),(6,4),(7,8),(8,2),(8,3),(9,12),(10,11),(10,12),(11,13),(12,13)],14)
 => [7,5,2]
 => ? = 7
[[2,2,4],[4]]
 => ([(0,7),(0,8),(1,16),(2,10),(2,16),(3,11),(4,12),(5,6),(6,4),(6,10),(7,9),(8,5),(9,1),(9,2),(10,12),(10,13),(11,15),(12,14),(13,11),(13,14),(14,15),(16,3),(16,13)],17)
 => [8,6,3]
 => ? = 8
[[2,3,3],[4]]
 => ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
 => [8,5,1]
 => ? = 8
[[3,3,3],[4]]
 => ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
 => [9,5,1]
 => ? = 9
[[1,3],[4,4]]
 => ([(0,7),(1,8),(1,9),(2,9),(2,13),(3,8),(3,13),(4,11),(5,10),(6,5),(7,1),(7,2),(7,3),(8,6),(9,12),(11,10),(12,11),(13,4),(13,12)],14)
 => [7,4,3]
 => ? = 7
[[2,2],[4,4]]
 => ([(0,6),(1,12),(2,11),(3,11),(3,12),(4,8),(5,9),(6,1),(6,2),(6,3),(7,8),(7,9),(8,10),(9,10),(11,4),(11,7),(12,5),(12,7)],13)
 => [7,4,2]
 => ? = 7
[[2,3],[3,4]]
 => ([(0,7),(1,8),(1,9),(2,9),(2,13),(3,8),(3,13),(4,11),(5,10),(6,5),(7,1),(7,2),(7,3),(8,6),(9,12),(11,10),(12,11),(13,4),(13,12)],14)
 => [7,4,3]
 => ? = 7
[[2,4],[3],[4]]
 => ([(0,9),(0,10),(1,12),(2,11),(3,11),(3,12),(4,7),(5,8),(6,3),(7,2),(8,1),(9,4),(9,14),(10,5),(10,14),(11,13),(12,13),(14,6)],15)
 => [7,5,3]
 => ? = 7
[[1,2,3,3,3]]
 => ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
 => [8,5,1]
 => ? = 8
[[1,3,3,3,3]]
 => ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
 => [9,5,1]
 => ? = 9
[[2,2,2,3,3]]
 => ([(0,7),(1,14),(2,9),(3,10),(4,5),(4,14),(5,6),(5,8),(6,2),(6,11),(7,1),(7,4),(8,10),(8,11),(9,13),(10,12),(11,9),(11,12),(12,13),(14,3),(14,8)],15)
 => [8,5,2]
 => ? = 8
[[1,2,3,3],[3]]
 => ([(0,9),(0,10),(1,11),(2,14),(3,12),(4,13),(5,4),(5,11),(6,5),(7,3),(8,1),(8,14),(9,6),(10,2),(10,8),(11,13),(13,12),(14,7)],15)
 => [7,5,3]
 => ? = 7
[[2,2,3],[3,3]]
 => ([(0,9),(0,10),(1,11),(2,14),(3,12),(4,13),(5,4),(5,11),(6,5),(7,3),(8,1),(8,14),(9,6),(10,2),(10,8),(11,13),(13,12),(14,7)],15)
 => [7,5,3]
 => ? = 7
[[3],[7]]
 => ([(0,7),(1,14),(2,9),(3,10),(4,5),(4,14),(5,6),(5,8),(6,2),(6,11),(7,1),(7,4),(8,10),(8,11),(9,13),(10,12),(11,9),(11,12),(12,13),(14,3),(14,8)],15)
 => [8,5,2]
 => ? = 8
[[1,3,6]]
 => ([(0,7),(1,14),(2,9),(3,10),(4,5),(4,14),(5,6),(5,8),(6,2),(6,11),(7,1),(7,4),(8,10),(8,11),(9,13),(10,12),(11,9),(11,12),(12,13),(14,3),(14,8)],15)
 => [8,5,2]
 => ? = 8
[[1,4],[6]]
 => ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
 => [8,5,1]
 => ? = 8
[[1,5],[6]]
 => ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
 => [9,5,1]
 => ? = 9
[[2,2],[6]]
 => ([(0,6),(1,9),(2,8),(3,5),(3,7),(4,1),(4,7),(5,2),(5,10),(6,3),(6,4),(7,9),(7,10),(8,12),(9,11),(10,8),(10,11),(11,12)],13)
 => [7,4,2]
 => ? = 7
[[2],[4],[6]]
 => ([(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
[[2],[5],[6]]
 => ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
 => [8,5,3]
 => ? = 8
[[3],[4],[6]]
 => ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
 => [8,5,3]
 => ? = 8
[[1,1,4,5]]
 => ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
 => [8,5,1]
 => ? = 8
[[1,1,5,5]]
 => ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
 => [9,5,1]
 => ? = 9
[[1,2,2,5]]
 => ([(0,6),(1,9),(2,8),(3,5),(3,7),(4,1),(4,7),(5,2),(5,10),(6,3),(6,4),(7,9),(7,10),(8,12),(9,11),(10,8),(10,11),(11,12)],13)
 => [7,4,2]
 => ? = 7

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
(load all 3 compositions to match this statistic)

Mapping

Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000784: Integer partitions ⟶ ℤResult quality: 82% ●values known / values provided: 82%●distinct values known / distinct values provided: 100%

Values

[[1,2]]
 => ([(0,1)],2)
 => [2]
 => 2
[[2,2]]
 => ([(0,2),(2,1)],3)
 => [3]
 => 3
[[1],[2]]
 => ([],1)
 => [1]
 => 1
[[1,3]]
 => ([(0,2),(2,1)],3)
 => [3]
 => 3
[[2,3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => [4,1]
 => 4
[[3,3]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => [5,1]
 => 5
[[1],[3]]
 => ([(0,1)],2)
 => [2]
 => 2
[[2],[3]]
 => ([(0,2),(2,1)],3)
 => [3]
 => 3
[[1,1,2]]
 => ([(0,1)],2)
 => [2]
 => 2
[[1,2,2]]
 => ([(0,2),(2,1)],3)
 => [3]
 => 3
[[2,2,2]]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => 4
[[1,1],[2]]
 => ([],1)
 => [1]
 => 1
[[1,2],[2]]
 => ([(0,1)],2)
 => [2]
 => 2
[[1,4]]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => 4
[[2,4]]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => [5,2]
 => 5
[[3,4]]
 => ([(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
[[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
[[1],[4]]
 => ([(0,2),(2,1)],3)
 => [3]
 => 3
[[2],[4]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => [4,1]
 => 4
[[3],[4]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => [5,1]
 => 5
[[1,1,3]]
 => ([(0,2),(2,1)],3)
 => [3]
 => 3
[[1,2,3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => [4,1]
 => 4
[[1,3,3]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => [5,1]
 => 5
[[2,2,3]]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => [5,2]
 => 5
[[2,3,3]]
 => ([(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
[[3,3,3]]
 => ([(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
[[1,1],[3]]
 => ([(0,1)],2)
 => [2]
 => 2
[[1,2],[3]]
 => ([(0,2),(2,1)],3)
 => [3]
 => 3
[[1,3],[2]]
 => ([(0,2),(2,1)],3)
 => [3]
 => 3
[[1,3],[3]]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => 4
[[2,2],[3]]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => 4
[[2,3],[3]]
 => ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
 => [5,3]
 => 5
[[1],[2],[3]]
 => ([],1)
 => [1]
 => 1
[[1,1,1,2]]
 => ([(0,1)],2)
 => [2]
 => 2
[[1,1,2,2]]
 => ([(0,2),(2,1)],3)
 => [3]
 => 3
[[1,2,2,2]]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => 4
[[2,2,2,2]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [5]
 => 5
[[1,1,1],[2]]
 => ([],1)
 => [1]
 => 1
[[1,1,2],[2]]
 => ([(0,1)],2)
 => [2]
 => 2
[[1,2,2],[2]]
 => ([(0,2),(2,1)],3)
 => [3]
 => 3
[[1,1],[2,2]]
 => ([],1)
 => [1]
 => 1
[[1,5]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [5]
 => 5
[[2,5]]
 => ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
 => [6,3]
 => 6
[[3,5]]
 => ([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
 => [7,4,1]
 => 7
[[4,5]]
 => ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
 => [8,5,1]
 => ? = 8
[[5,5]]
 => ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
 => [9,5,1]
 => ? = 9
[[1],[5]]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => 4
[[2],[5]]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => [5,2]
 => 5
[[3],[5]]
 => ([(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
[[4],[5]]
 => ([(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
[[1,1,4]]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => 4
[[1,2,4]]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => [5,2]
 => 5
[[2,3,4]]
 => ([(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
[[2,4,4]]
 => ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
 => [8,5,3]
 => ? = 8
[[3,3,4]]
 => ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
 => [8,5,3]
 => ? = 8
[[2,4],[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
[[2,3,3,3]]
 => ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
 => [8,5,1]
 => ? = 8
[[3,3,3,3]]
 => ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
 => [9,5,1]
 => ? = 9
[[2,3,3],[3]]
 => ([(0,9),(0,10),(1,11),(2,14),(3,12),(4,13),(5,4),(5,11),(6,5),(7,3),(8,1),(8,14),(9,6),(10,2),(10,8),(11,13),(13,12),(14,7)],15)
 => [7,5,3]
 => ? = 7
[[3,6]]
 => ([(0,7),(1,14),(2,9),(3,10),(4,5),(4,14),(5,6),(5,8),(6,2),(6,11),(7,1),(7,4),(8,10),(8,11),(9,13),(10,12),(11,9),(11,12),(12,13),(14,3),(14,8)],15)
 => [8,5,2]
 => ? = 8
[[4],[6]]
 => ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
 => [8,5,1]
 => ? = 8
[[5],[6]]
 => ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
 => [9,5,1]
 => ? = 9
[[1,4,5]]
 => ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
 => [8,5,1]
 => ? = 8
[[1,5,5]]
 => ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
 => [9,5,1]
 => ? = 9
[[2,2,5]]
 => ([(0,6),(1,9),(2,8),(3,5),(3,7),(4,1),(4,7),(5,2),(5,10),(6,3),(6,4),(7,9),(7,10),(8,12),(9,11),(10,8),(10,11),(11,12)],13)
 => [7,4,2]
 => ? = 7
[[2,3],[5]]
 => ([(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
[[2,5],[3]]
 => ([(0,6),(0,7),(1,11),(2,9),(3,9),(3,10),(4,2),(5,1),(5,10),(6,4),(7,8),(8,3),(8,5),(9,12),(10,11),(10,12),(11,13),(12,13)],14)
 => [7,5,2]
 => ? = 7
[[2,4],[5]]
 => ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
 => [8,5,3]
 => ? = 8
[[3,3],[5]]
 => ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
 => [8,5,3]
 => ? = 8
[[1,2,3,4]]
 => ([(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
[[1,2,4,4]]
 => ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
 => [8,5,3]
 => ? = 8
[[1,3,3,4]]
 => ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
 => [8,5,3]
 => ? = 8
[[2,2,2,4]]
 => ([(0,6),(1,9),(2,8),(3,5),(3,7),(4,1),(4,7),(5,2),(5,10),(6,3),(6,4),(7,9),(7,10),(8,12),(9,11),(10,8),(10,11),(11,12)],13)
 => [7,4,2]
 => ? = 7
[[1,2,4],[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
[[1,4,4],[3]]
 => ([(0,7),(1,10),(2,11),(3,8),(4,9),(5,2),(5,9),(6,3),(6,12),(7,4),(7,5),(8,10),(9,6),(9,11),(11,12),(12,1),(12,8)],13)
 => [8,5]
 => ? = 8
[[1,4,4],[4]]
 => ([(0,8),(2,13),(3,11),(4,9),(5,10),(6,3),(6,10),(7,4),(7,12),(8,5),(8,6),(9,13),(10,7),(10,11),(11,12),(12,2),(12,9),(13,1)],14)
 => [9,5]
 => ? = 9
[[2,2,4],[3]]
 => ([(0,6),(0,7),(1,11),(2,9),(3,9),(3,10),(4,5),(5,1),(5,10),(6,4),(7,8),(8,2),(8,3),(9,12),(10,11),(10,12),(11,13),(12,13)],14)
 => [7,5,2]
 => ? = 7
[[2,2,4],[4]]
 => ([(0,7),(0,8),(1,16),(2,10),(2,16),(3,11),(4,12),(5,6),(6,4),(6,10),(7,9),(8,5),(9,1),(9,2),(10,12),(10,13),(11,15),(12,14),(13,11),(13,14),(14,15),(16,3),(16,13)],17)
 => [8,6,3]
 => ? = 8
[[2,3,3],[4]]
 => ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
 => [8,5,1]
 => ? = 8
[[3,3,3],[4]]
 => ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
 => [9,5,1]
 => ? = 9
[[1,3],[4,4]]
 => ([(0,7),(1,8),(1,9),(2,9),(2,13),(3,8),(3,13),(4,11),(5,10),(6,5),(7,1),(7,2),(7,3),(8,6),(9,12),(11,10),(12,11),(13,4),(13,12)],14)
 => [7,4,3]
 => ? = 7
[[2,2],[4,4]]
 => ([(0,6),(1,12),(2,11),(3,11),(3,12),(4,8),(5,9),(6,1),(6,2),(6,3),(7,8),(7,9),(8,10),(9,10),(11,4),(11,7),(12,5),(12,7)],13)
 => [7,4,2]
 => ? = 7
[[2,3],[3,4]]
 => ([(0,7),(1,8),(1,9),(2,9),(2,13),(3,8),(3,13),(4,11),(5,10),(6,5),(7,1),(7,2),(7,3),(8,6),(9,12),(11,10),(12,11),(13,4),(13,12)],14)
 => [7,4,3]
 => ? = 7
[[2,4],[3],[4]]
 => ([(0,9),(0,10),(1,12),(2,11),(3,11),(3,12),(4,7),(5,8),(6,3),(7,2),(8,1),(9,4),(9,14),(10,5),(10,14),(11,13),(12,13),(14,6)],15)
 => [7,5,3]
 => ? = 7
[[1,2,3,3,3]]
 => ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
 => [8,5,1]
 => ? = 8
[[1,3,3,3,3]]
 => ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
 => [9,5,1]
 => ? = 9
[[2,2,2,3,3]]
 => ([(0,7),(1,14),(2,9),(3,10),(4,5),(4,14),(5,6),(5,8),(6,2),(6,11),(7,1),(7,4),(8,10),(8,11),(9,13),(10,12),(11,9),(11,12),(12,13),(14,3),(14,8)],15)
 => [8,5,2]
 => ? = 8
[[1,2,3,3],[3]]
 => ([(0,9),(0,10),(1,11),(2,14),(3,12),(4,13),(5,4),(5,11),(6,5),(7,3),(8,1),(8,14),(9,6),(10,2),(10,8),(11,13),(13,12),(14,7)],15)
 => [7,5,3]
 => ? = 7
[[2,2,3],[3,3]]
 => ([(0,9),(0,10),(1,11),(2,14),(3,12),(4,13),(5,4),(5,11),(6,5),(7,3),(8,1),(8,14),(9,6),(10,2),(10,8),(11,13),(13,12),(14,7)],15)
 => [7,5,3]
 => ? = 7
[[3],[7]]
 => ([(0,7),(1,14),(2,9),(3,10),(4,5),(4,14),(5,6),(5,8),(6,2),(6,11),(7,1),(7,4),(8,10),(8,11),(9,13),(10,12),(11,9),(11,12),(12,13),(14,3),(14,8)],15)
 => [8,5,2]
 => ? = 8
[[1,3,6]]
 => ([(0,7),(1,14),(2,9),(3,10),(4,5),(4,14),(5,6),(5,8),(6,2),(6,11),(7,1),(7,4),(8,10),(8,11),(9,13),(10,12),(11,9),(11,12),(12,13),(14,3),(14,8)],15)
 => [8,5,2]
 => ? = 8
[[1,4],[6]]
 => ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
 => [8,5,1]
 => ? = 8
[[1,5],[6]]
 => ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
 => [9,5,1]
 => ? = 9
[[2,2],[6]]
 => ([(0,6),(1,9),(2,8),(3,5),(3,7),(4,1),(4,7),(5,2),(5,10),(6,3),(6,4),(7,9),(7,10),(8,12),(9,11),(10,8),(10,11),(11,12)],13)
 => [7,4,2]
 => ? = 7
[[2],[4],[6]]
 => ([(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
[[2],[5],[6]]
 => ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
 => [8,5,3]
 => ? = 8
[[3],[4],[6]]
 => ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
 => [8,5,3]
 => ? = 8
[[1,1,4,5]]
 => ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
 => [8,5,1]
 => ? = 8
[[1,1,5,5]]
 => ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
 => [9,5,1]
 => ? = 9
[[1,2,2,5]]
 => ([(0,6),(1,9),(2,8),(3,5),(3,7),(4,1),(4,7),(5,2),(5,10),(6,3),(6,4),(7,9),(7,10),(8,12),(9,11),(10,8),(10,11),(11,12)],13)
 => [7,4,2]
 => ? = 7

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: St000676

Mapping

Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000676: Dyck paths ⟶ ℤResult quality: 79% ●values known / values provided: 79%●distinct values known / distinct values provided: 89%

Values

[[1,2]]
 => ([(0,1)],2)
 => [2]
 => [1,0,1,0]
 => 2
[[2,2]]
 => ([(0,2),(2,1)],3)
 => [3]
 => [1,0,1,0,1,0]
 => 3
[[1],[2]]
 => ([],1)
 => [1]
 => [1,0]
 => 1
[[1,3]]
 => ([(0,2),(2,1)],3)
 => [3]
 => [1,0,1,0,1,0]
 => 3
[[2,3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 4
[[3,3]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 5
[[1],[3]]
 => ([(0,1)],2)
 => [2]
 => [1,0,1,0]
 => 2
[[2],[3]]
 => ([(0,2),(2,1)],3)
 => [3]
 => [1,0,1,0,1,0]
 => 3
[[1,1,2]]
 => ([(0,1)],2)
 => [2]
 => [1,0,1,0]
 => 2
[[1,2,2]]
 => ([(0,2),(2,1)],3)
 => [3]
 => [1,0,1,0,1,0]
 => 3
[[2,2,2]]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 4
[[1,1],[2]]
 => ([],1)
 => [1]
 => [1,0]
 => 1
[[1,2],[2]]
 => ([(0,1)],2)
 => [2]
 => [1,0,1,0]
 => 2
[[1,4]]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 4
[[2,4]]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => [5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 5
[[3,4]]
 => ([(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]
 => [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => 6
[[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]
 => [1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => 7
[[1],[4]]
 => ([(0,2),(2,1)],3)
 => [3]
 => [1,0,1,0,1,0]
 => 3
[[2],[4]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 4
[[3],[4]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 5
[[1,1,3]]
 => ([(0,2),(2,1)],3)
 => [3]
 => [1,0,1,0,1,0]
 => 3
[[1,2,3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 4
[[1,3,3]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 5
[[2,2,3]]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => [5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 5
[[2,3,3]]
 => ([(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]
 => [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => 6
[[3,3,3]]
 => ([(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]
 => [1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => 7
[[1,1],[3]]
 => ([(0,1)],2)
 => [2]
 => [1,0,1,0]
 => 2
[[1,2],[3]]
 => ([(0,2),(2,1)],3)
 => [3]
 => [1,0,1,0,1,0]
 => 3
[[1,3],[2]]
 => ([(0,2),(2,1)],3)
 => [3]
 => [1,0,1,0,1,0]
 => 3
[[1,3],[3]]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 4
[[2,2],[3]]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 4
[[2,3],[3]]
 => ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
 => [5,3]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => 5
[[1],[2],[3]]
 => ([],1)
 => [1]
 => [1,0]
 => 1
[[1,1,1,2]]
 => ([(0,1)],2)
 => [2]
 => [1,0,1,0]
 => 2
[[1,1,2,2]]
 => ([(0,2),(2,1)],3)
 => [3]
 => [1,0,1,0,1,0]
 => 3
[[1,2,2,2]]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 4
[[2,2,2,2]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
[[1,1,1],[2]]
 => ([],1)
 => [1]
 => [1,0]
 => 1
[[1,1,2],[2]]
 => ([(0,1)],2)
 => [2]
 => [1,0,1,0]
 => 2
[[1,2,2],[2]]
 => ([(0,2),(2,1)],3)
 => [3]
 => [1,0,1,0,1,0]
 => 3
[[1,1],[2,2]]
 => ([],1)
 => [1]
 => [1,0]
 => 1
[[1,5]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
[[2,5]]
 => ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
 => [6,3]
 => [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => 6
[[3,5]]
 => ([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
 => [7,4,1]
 => [1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,1,0,0]
 => ? = 7
[[4,5]]
 => ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
 => [8,5,1]
 => [1,0,1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,1,0,0]
 => ? = 8
[[5,5]]
 => ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
 => [9,5,1]
 => [1,0,1,0,1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,1,0,0]
 => ? = 9
[[1],[5]]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 4
[[2],[5]]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => [5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 5
[[3],[5]]
 => ([(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]
 => [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => 6
[[4],[5]]
 => ([(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]
 => [1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => 7
[[1,1,4]]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 4
[[1,2,4]]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => [5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 5
[[1,3,4]]
 => ([(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]
 => [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => 6
[[2,3,4]]
 => ([(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]
 => [1,0,1,0,1,0,1,1,1,0,1,1,1,0,0,0,0,0]
 => ? = 7
[[2,4,4]]
 => ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
 => [8,5,3]
 => [1,0,1,0,1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
 => ? = 8
[[3,3,4]]
 => ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
 => [8,5,3]
 => [1,0,1,0,1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
 => ? = 8
[[2,4],[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]
 => [1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,1,0,0]
 => ? = 7
[[2,2,3,3]]
 => ([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
 => [7,4,1]
 => [1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,1,0,0]
 => ? = 7
[[2,3,3,3]]
 => ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
 => [8,5,1]
 => [1,0,1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,1,0,0]
 => ? = 8
[[3,3,3,3]]
 => ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
 => [9,5,1]
 => [1,0,1,0,1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,1,0,0]
 => ? = 9
[[2,3,3],[3]]
 => ([(0,9),(0,10),(1,11),(2,14),(3,12),(4,13),(5,4),(5,11),(6,5),(7,3),(8,1),(8,14),(9,6),(10,2),(10,8),(11,13),(13,12),(14,7)],15)
 => [7,5,3]
 => [1,0,1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
 => ? = 7
[[3,6]]
 => ([(0,7),(1,14),(2,9),(3,10),(4,5),(4,14),(5,6),(5,8),(6,2),(6,11),(7,1),(7,4),(8,10),(8,11),(9,13),(10,12),(11,9),(11,12),(12,13),(14,3),(14,8)],15)
 => [8,5,2]
 => [1,0,1,0,1,0,1,1,1,0,1,0,1,0,1,1,0,0,0,0]
 => ? = 8
[[3],[6]]
 => ([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
 => [7,4,1]
 => [1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,1,0,0]
 => ? = 7
[[4],[6]]
 => ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
 => [8,5,1]
 => [1,0,1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,1,0,0]
 => ? = 8
[[5],[6]]
 => ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
 => [9,5,1]
 => [1,0,1,0,1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,1,0,0]
 => ? = 9
[[1,3,5]]
 => ([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
 => [7,4,1]
 => [1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,1,0,0]
 => ? = 7
[[1,4,5]]
 => ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
 => [8,5,1]
 => [1,0,1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,1,0,0]
 => ? = 8
[[1,5,5]]
 => ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
 => [9,5,1]
 => [1,0,1,0,1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,1,0,0]
 => ? = 9
[[2,2,5]]
 => ([(0,6),(1,9),(2,8),(3,5),(3,7),(4,1),(4,7),(5,2),(5,10),(6,3),(6,4),(7,9),(7,10),(8,12),(9,11),(10,8),(10,11),(11,12)],13)
 => [7,4,2]
 => [1,0,1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,0]
 => ? = 7
[[1,5],[5]]
 => ([(0,6),(1,8),(2,10),(4,9),(5,1),(5,10),(6,7),(7,2),(7,5),(8,9),(9,3),(10,4),(10,8)],11)
 => [8,3]
 => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => ? = 8
[[2,3],[5]]
 => ([(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]
 => [1,0,1,0,1,0,1,1,1,0,1,1,1,0,0,0,0,0]
 => ? = 7
[[2,5],[3]]
 => ([(0,6),(0,7),(1,11),(2,9),(3,9),(3,10),(4,2),(5,1),(5,10),(6,4),(7,8),(8,3),(8,5),(9,12),(10,11),(10,12),(11,13),(12,13)],14)
 => [7,5,2]
 => [1,0,1,0,1,1,1,0,1,0,1,0,1,1,0,0,0,0]
 => ? = 7
[[2,4],[5]]
 => ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
 => [8,5,3]
 => [1,0,1,0,1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
 => ? = 8
[[3,3],[5]]
 => ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
 => [8,5,3]
 => [1,0,1,0,1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
 => ? = 8
[[1,2,3,4]]
 => ([(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]
 => [1,0,1,0,1,0,1,1,1,0,1,1,1,0,0,0,0,0]
 => ? = 7
[[1,2,4,4]]
 => ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
 => [8,5,3]
 => [1,0,1,0,1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
 => ? = 8
[[1,3,3,4]]
 => ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
 => [8,5,3]
 => [1,0,1,0,1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
 => ? = 8
[[2,2,2,4]]
 => ([(0,6),(1,9),(2,8),(3,5),(3,7),(4,1),(4,7),(5,2),(5,10),(6,3),(6,4),(7,9),(7,10),(8,12),(9,11),(10,8),(10,11),(11,12)],13)
 => [7,4,2]
 => [1,0,1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,0]
 => ? = 7
[[1,2,4],[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]
 => [1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,1,0,0]
 => ? = 7
[[1,4,4],[3]]
 => ([(0,7),(1,10),(2,11),(3,8),(4,9),(5,2),(5,9),(6,3),(6,12),(7,4),(7,5),(8,10),(9,6),(9,11),(11,12),(12,1),(12,8)],13)
 => [8,5]
 => [1,0,1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,0]
 => ? = 8
[[1,4,4],[4]]
 => ([(0,8),(2,13),(3,11),(4,9),(5,10),(6,3),(6,10),(7,4),(7,12),(8,5),(8,6),(9,13),(10,7),(10,11),(11,12),(12,2),(12,9),(13,1)],14)
 => [9,5]
 => [1,0,1,0,1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,0]
 => ? = 9
[[2,2,3],[4]]
 => ([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
 => [7,4,1]
 => [1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,1,0,0]
 => ? = 7
[[2,2,4],[3]]
 => ([(0,6),(0,7),(1,11),(2,9),(3,9),(3,10),(4,5),(5,1),(5,10),(6,4),(7,8),(8,2),(8,3),(9,12),(10,11),(10,12),(11,13),(12,13)],14)
 => [7,5,2]
 => [1,0,1,0,1,1,1,0,1,0,1,0,1,1,0,0,0,0]
 => ? = 7
[[2,2,4],[4]]
 => ([(0,7),(0,8),(1,16),(2,10),(2,16),(3,11),(4,12),(5,6),(6,4),(6,10),(7,9),(8,5),(9,1),(9,2),(10,12),(10,13),(11,15),(12,14),(13,11),(13,14),(14,15),(16,3),(16,13)],17)
 => [8,6,3]
 => [1,0,1,0,1,1,1,0,1,0,1,0,1,1,1,0,0,0,0,0]
 => ? = 8
[[2,3,3],[4]]
 => ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
 => [8,5,1]
 => [1,0,1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,1,0,0]
 => ? = 8
[[3,3,3],[4]]
 => ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
 => [9,5,1]
 => [1,0,1,0,1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,1,0,0]
 => ? = 9
[[1,3],[4,4]]
 => ([(0,7),(1,8),(1,9),(2,9),(2,13),(3,8),(3,13),(4,11),(5,10),(6,5),(7,1),(7,2),(7,3),(8,6),(9,12),(11,10),(12,11),(13,4),(13,12)],14)
 => [7,4,3]
 => [1,0,1,0,1,0,1,1,1,0,1,1,1,0,0,0,0,0]
 => ? = 7
[[2,2],[4,4]]
 => ([(0,6),(1,12),(2,11),(3,11),(3,12),(4,8),(5,9),(6,1),(6,2),(6,3),(7,8),(7,9),(8,10),(9,10),(11,4),(11,7),(12,5),(12,7)],13)
 => [7,4,2]
 => [1,0,1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,0]
 => ? = 7
[[2,3],[3,4]]
 => ([(0,7),(1,8),(1,9),(2,9),(2,13),(3,8),(3,13),(4,11),(5,10),(6,5),(7,1),(7,2),(7,3),(8,6),(9,12),(11,10),(12,11),(13,4),(13,12)],14)
 => [7,4,3]
 => [1,0,1,0,1,0,1,1,1,0,1,1,1,0,0,0,0,0]
 => ? = 7
[[2,4],[3],[4]]
 => ([(0,9),(0,10),(1,12),(2,11),(3,11),(3,12),(4,7),(5,8),(6,3),(7,2),(8,1),(9,4),(9,14),(10,5),(10,14),(11,13),(12,13),(14,6)],15)
 => [7,5,3]
 => [1,0,1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
 => ? = 7
[[1,2,2,3,3]]
 => ([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
 => [7,4,1]
 => [1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,1,0,0]
 => ? = 7
[[1,2,3,3,3]]
 => ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
 => [8,5,1]
 => [1,0,1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,1,0,0]
 => ? = 8
[[1,3,3,3,3]]
 => ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
 => [9,5,1]
 => [1,0,1,0,1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,1,0,0]
 => ? = 9
[[2,2,2,3,3]]
 => ([(0,7),(1,14),(2,9),(3,10),(4,5),(4,14),(5,6),(5,8),(6,2),(6,11),(7,1),(7,4),(8,10),(8,11),(9,13),(10,12),(11,9),(11,12),(12,13),(14,3),(14,8)],15)
 => [8,5,2]
 => [1,0,1,0,1,0,1,1,1,0,1,0,1,0,1,1,0,0,0,0]
 => ? = 8
[[1,2,3,3],[3]]
 => ([(0,9),(0,10),(1,11),(2,14),(3,12),(4,13),(5,4),(5,11),(6,5),(7,3),(8,1),(8,14),(9,6),(10,2),(10,8),(11,13),(13,12),(14,7)],15)
 => [7,5,3]
 => [1,0,1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
 => ? = 7
[[1,3,3,3],[3]]
 => ([(0,7),(2,9),(3,10),(4,8),(5,4),(5,10),(6,1),(7,3),(7,5),(8,9),(9,6),(10,2),(10,8)],11)
 => [8,3]
 => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => ? = 8
[[2,2,3],[3,3]]
 => ([(0,9),(0,10),(1,11),(2,14),(3,12),(4,13),(5,4),(5,11),(6,5),(7,3),(8,1),(8,14),(9,6),(10,2),(10,8),(11,13),(13,12),(14,7)],15)
 => [7,5,3]
 => [1,0,1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
 => ? = 7
[[3],[7]]
 => ([(0,7),(1,14),(2,9),(3,10),(4,5),(4,14),(5,6),(5,8),(6,2),(6,11),(7,1),(7,4),(8,10),(8,11),(9,13),(10,12),(11,9),(11,12),(12,13),(14,3),(14,8)],15)
 => [8,5,2]
 => [1,0,1,0,1,0,1,1,1,0,1,0,1,0,1,1,0,0,0,0]
 => ? = 8
[[1,3,6]]
 => ([(0,7),(1,14),(2,9),(3,10),(4,5),(4,14),(5,6),(5,8),(6,2),(6,11),(7,1),(7,4),(8,10),(8,11),(9,13),(10,12),(11,9),(11,12),(12,13),(14,3),(14,8)],15)
 => [8,5,2]
 => [1,0,1,0,1,0,1,1,1,0,1,0,1,0,1,1,0,0,0,0]
 => ? = 8
[[1,3],[6]]
 => ([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
 => [7,4,1]
 => [1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,1,0,0]
 => ? = 7

Description

The number of odd rises of a Dyck path. This is the number of ones at an odd position, with the initial position equal to 1. The number of Dyck paths of semilength

$n$ with

$k$ up steps in odd positions and

$k$ returns to the main diagonal are counted by the binomial coefficient

$\binom{n-1}{k-1}$ [3,4].

Matching statistic: St001039

Mapping

Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001039: Dyck paths ⟶ ℤResult quality: 75% ●values known / values provided: 75%●distinct values known / distinct values provided: 78%

Values

[[1,2]]
 => ([(0,1)],2)
 => [2]
 => [1,0,1,0]
 => 2
[[2,2]]
 => ([(0,2),(2,1)],3)
 => [3]
 => [1,0,1,0,1,0]
 => 3
[[1],[2]]
 => ([],1)
 => [1]
 => [1,0]
 => ? = 1
[[1,3]]
 => ([(0,2),(2,1)],3)
 => [3]
 => [1,0,1,0,1,0]
 => 3
[[2,3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 4
[[3,3]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 5
[[1],[3]]
 => ([(0,1)],2)
 => [2]
 => [1,0,1,0]
 => 2
[[2],[3]]
 => ([(0,2),(2,1)],3)
 => [3]
 => [1,0,1,0,1,0]
 => 3
[[1,1,2]]
 => ([(0,1)],2)
 => [2]
 => [1,0,1,0]
 => 2
[[1,2,2]]
 => ([(0,2),(2,1)],3)
 => [3]
 => [1,0,1,0,1,0]
 => 3
[[2,2,2]]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 4
[[1,1],[2]]
 => ([],1)
 => [1]
 => [1,0]
 => ? = 1
[[1,2],[2]]
 => ([(0,1)],2)
 => [2]
 => [1,0,1,0]
 => 2
[[1,4]]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 4
[[2,4]]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => [5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 5
[[3,4]]
 => ([(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]
 => [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => 6
[[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]
 => [1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => 7
[[1],[4]]
 => ([(0,2),(2,1)],3)
 => [3]
 => [1,0,1,0,1,0]
 => 3
[[2],[4]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 4
[[3],[4]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 5
[[1,1,3]]
 => ([(0,2),(2,1)],3)
 => [3]
 => [1,0,1,0,1,0]
 => 3
[[1,2,3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 4
[[1,3,3]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 5
[[2,2,3]]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => [5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 5
[[2,3,3]]
 => ([(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]
 => [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => 6
[[3,3,3]]
 => ([(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]
 => [1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => 7
[[1,1],[3]]
 => ([(0,1)],2)
 => [2]
 => [1,0,1,0]
 => 2
[[1,2],[3]]
 => ([(0,2),(2,1)],3)
 => [3]
 => [1,0,1,0,1,0]
 => 3
[[1,3],[2]]
 => ([(0,2),(2,1)],3)
 => [3]
 => [1,0,1,0,1,0]
 => 3
[[1,3],[3]]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 4
[[2,2],[3]]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 4
[[2,3],[3]]
 => ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
 => [5,3]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => 5
[[1],[2],[3]]
 => ([],1)
 => [1]
 => [1,0]
 => ? = 1
[[1,1,1,2]]
 => ([(0,1)],2)
 => [2]
 => [1,0,1,0]
 => 2
[[1,1,2,2]]
 => ([(0,2),(2,1)],3)
 => [3]
 => [1,0,1,0,1,0]
 => 3
[[1,2,2,2]]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 4
[[2,2,2,2]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
[[1,1,1],[2]]
 => ([],1)
 => [1]
 => [1,0]
 => ? = 1
[[1,1,2],[2]]
 => ([(0,1)],2)
 => [2]
 => [1,0,1,0]
 => 2
[[1,2,2],[2]]
 => ([(0,2),(2,1)],3)
 => [3]
 => [1,0,1,0,1,0]
 => 3
[[1,1],[2,2]]
 => ([],1)
 => [1]
 => [1,0]
 => ? = 1
[[1,5]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
[[2,5]]
 => ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
 => [6,3]
 => [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => 6
[[3,5]]
 => ([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
 => [7,4,1]
 => [1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,1,0,0]
 => ? = 7
[[4,5]]
 => ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
 => [8,5,1]
 => [1,0,1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,1,0,0]
 => ? = 8
[[5,5]]
 => ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
 => [9,5,1]
 => [1,0,1,0,1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,1,0,0]
 => ? = 9
[[1],[5]]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 4
[[2],[5]]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => [5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 5
[[3],[5]]
 => ([(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]
 => [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => 6
[[4],[5]]
 => ([(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]
 => [1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => 7
[[1,1,4]]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 4
[[1,2,4]]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => [5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 5
[[1,3,4]]
 => ([(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]
 => [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => 6
[[1,4,4]]
 => ([(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]
 => [1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => 7
[[2,2,4]]
 => ([(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]
 => [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => 6
[[2,3,4]]
 => ([(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]
 => [1,0,1,0,1,0,1,1,1,0,1,1,1,0,0,0,0,0]
 => ? = 7
[[2,4,4]]
 => ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
 => [8,5,3]
 => [1,0,1,0,1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
 => ? = 8
[[3,3,4]]
 => ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
 => [8,5,3]
 => [1,0,1,0,1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
 => ? = 8
[[1,1],[4]]
 => ([(0,2),(2,1)],3)
 => [3]
 => [1,0,1,0,1,0]
 => 3
[[1,2],[4]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 4
[[1,4],[2]]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 4
[[2,4],[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]
 => [1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,1,0,0]
 => ? = 7
[[2,2,3,3]]
 => ([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
 => [7,4,1]
 => [1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,1,0,0]
 => ? = 7
[[2,3,3,3]]
 => ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
 => [8,5,1]
 => [1,0,1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,1,0,0]
 => ? = 8
[[3,3,3,3]]
 => ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
 => [9,5,1]
 => [1,0,1,0,1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,1,0,0]
 => ? = 9
[[2,3,3],[3]]
 => ([(0,9),(0,10),(1,11),(2,14),(3,12),(4,13),(5,4),(5,11),(6,5),(7,3),(8,1),(8,14),(9,6),(10,2),(10,8),(11,13),(13,12),(14,7)],15)
 => [7,5,3]
 => [1,0,1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
 => ? = 7
[[1,1],[2],[3]]
 => ([],1)
 => [1]
 => [1,0]
 => ? = 1
[[1,1,1,1],[2]]
 => ([],1)
 => [1]
 => [1,0]
 => ? = 1
[[1,1,1],[2,2]]
 => ([],1)
 => [1]
 => [1,0]
 => ? = 1
[[3,6]]
 => ([(0,7),(1,14),(2,9),(3,10),(4,5),(4,14),(5,6),(5,8),(6,2),(6,11),(7,1),(7,4),(8,10),(8,11),(9,13),(10,12),(11,9),(11,12),(12,13),(14,3),(14,8)],15)
 => [8,5,2]
 => [1,0,1,0,1,0,1,1,1,0,1,0,1,0,1,1,0,0,0,0]
 => ? = 8
[[3],[6]]
 => ([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
 => [7,4,1]
 => [1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,1,0,0]
 => ? = 7
[[4],[6]]
 => ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
 => [8,5,1]
 => [1,0,1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,1,0,0]
 => ? = 8
[[5],[6]]
 => ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
 => [9,5,1]
 => [1,0,1,0,1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,1,0,0]
 => ? = 9
[[1,3,5]]
 => ([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
 => [7,4,1]
 => [1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,1,0,0]
 => ? = 7
[[1,4,5]]
 => ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
 => [8,5,1]
 => [1,0,1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,1,0,0]
 => ? = 8
[[1,5,5]]
 => ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
 => [9,5,1]
 => [1,0,1,0,1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,1,0,0]
 => ? = 9
[[2,2,5]]
 => ([(0,6),(1,9),(2,8),(3,5),(3,7),(4,1),(4,7),(5,2),(5,10),(6,3),(6,4),(7,9),(7,10),(8,12),(9,11),(10,8),(10,11),(11,12)],13)
 => [7,4,2]
 => [1,0,1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,0]
 => ? = 7
[[1,5],[5]]
 => ([(0,6),(1,8),(2,10),(4,9),(5,1),(5,10),(6,7),(7,2),(7,5),(8,9),(9,3),(10,4),(10,8)],11)
 => [8,3]
 => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => ? = 8
[[2,3],[5]]
 => ([(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]
 => [1,0,1,0,1,0,1,1,1,0,1,1,1,0,0,0,0,0]
 => ? = 7
[[2,5],[3]]
 => ([(0,6),(0,7),(1,11),(2,9),(3,9),(3,10),(4,2),(5,1),(5,10),(6,4),(7,8),(8,3),(8,5),(9,12),(10,11),(10,12),(11,13),(12,13)],14)
 => [7,5,2]
 => [1,0,1,0,1,1,1,0,1,0,1,0,1,1,0,0,0,0]
 => ? = 7
[[2,4],[5]]
 => ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
 => [8,5,3]
 => [1,0,1,0,1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
 => ? = 8
[[3,3],[5]]
 => ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
 => [8,5,3]
 => [1,0,1,0,1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
 => ? = 8
[[1,2,3,4]]
 => ([(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]
 => [1,0,1,0,1,0,1,1,1,0,1,1,1,0,0,0,0,0]
 => ? = 7
[[1,2,4,4]]
 => ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
 => [8,5,3]
 => [1,0,1,0,1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
 => ? = 8
[[1,3,3,4]]
 => ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
 => [8,5,3]
 => [1,0,1,0,1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
 => ? = 8
[[2,2,2,4]]
 => ([(0,6),(1,9),(2,8),(3,5),(3,7),(4,1),(4,7),(5,2),(5,10),(6,3),(6,4),(7,9),(7,10),(8,12),(9,11),(10,8),(10,11),(11,12)],13)
 => [7,4,2]
 => [1,0,1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,0]
 => ? = 7
[[1,2,4],[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]
 => [1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,1,0,0]
 => ? = 7
[[1,4,4],[3]]
 => ([(0,7),(1,10),(2,11),(3,8),(4,9),(5,2),(5,9),(6,3),(6,12),(7,4),(7,5),(8,10),(9,6),(9,11),(11,12),(12,1),(12,8)],13)
 => [8,5]
 => [1,0,1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,0]
 => ? = 8
[[1,4,4],[4]]
 => ([(0,8),(2,13),(3,11),(4,9),(5,10),(6,3),(6,10),(7,4),(7,12),(8,5),(8,6),(9,13),(10,7),(10,11),(11,12),(12,2),(12,9),(13,1)],14)
 => [9,5]
 => [1,0,1,0,1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,0]
 => ? = 9
[[2,2,3],[4]]
 => ([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
 => [7,4,1]
 => [1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,1,0,0]
 => ? = 7
[[2,2,4],[3]]
 => ([(0,6),(0,7),(1,11),(2,9),(3,9),(3,10),(4,5),(5,1),(5,10),(6,4),(7,8),(8,2),(8,3),(9,12),(10,11),(10,12),(11,13),(12,13)],14)
 => [7,5,2]
 => [1,0,1,0,1,1,1,0,1,0,1,0,1,1,0,0,0,0]
 => ? = 7
[[2,2,4],[4]]
 => ([(0,7),(0,8),(1,16),(2,10),(2,16),(3,11),(4,12),(5,6),(6,4),(6,10),(7,9),(8,5),(9,1),(9,2),(10,12),(10,13),(11,15),(12,14),(13,11),(13,14),(14,15),(16,3),(16,13)],17)
 => [8,6,3]
 => [1,0,1,0,1,1,1,0,1,0,1,0,1,1,1,0,0,0,0,0]
 => ? = 8
[[2,3,3],[4]]
 => ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
 => [8,5,1]
 => [1,0,1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,1,0,0]
 => ? = 8
[[3,3,3],[4]]
 => ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
 => [9,5,1]
 => [1,0,1,0,1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,1,0,0]
 => ? = 9
[[1,3],[4,4]]
 => ([(0,7),(1,8),(1,9),(2,9),(2,13),(3,8),(3,13),(4,11),(5,10),(6,5),(7,1),(7,2),(7,3),(8,6),(9,12),(11,10),(12,11),(13,4),(13,12)],14)
 => [7,4,3]
 => [1,0,1,0,1,0,1,1,1,0,1,1,1,0,0,0,0,0]
 => ? = 7
[[2,2],[4,4]]
 => ([(0,6),(1,12),(2,11),(3,11),(3,12),(4,8),(5,9),(6,1),(6,2),(6,3),(7,8),(7,9),(8,10),(9,10),(11,4),(11,7),(12,5),(12,7)],13)
 => [7,4,2]
 => [1,0,1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,0]
 => ? = 7
[[2,3],[3,4]]
 => ([(0,7),(1,8),(1,9),(2,9),(2,13),(3,8),(3,13),(4,11),(5,10),(6,5),(7,1),(7,2),(7,3),(8,6),(9,12),(11,10),(12,11),(13,4),(13,12)],14)
 => [7,4,3]
 => [1,0,1,0,1,0,1,1,1,0,1,1,1,0,0,0,0,0]
 => ? = 7
[[2,4],[3],[4]]
 => ([(0,9),(0,10),(1,12),(2,11),(3,11),(3,12),(4,7),(5,8),(6,3),(7,2),(8,1),(9,4),(9,14),(10,5),(10,14),(11,13),(12,13),(14,6)],15)
 => [7,5,3]
 => [1,0,1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
 => ? = 7
[[1],[2],[3],[4]]
 => ([],1)
 => [1]
 => [1,0]
 => ? = 1
[[1,2,2,3,3]]
 => ([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
 => [7,4,1]
 => [1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,1,0,0]
 => ? = 7

Description

The maximal height of a column in the parallelogram polyomino associated with a Dyck path.

Matching statistic: St000734

Mapping

Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000734: Standard tableaux ⟶ ℤResult quality: 75% ●values known / values provided: 75%●distinct values known / distinct values provided: 100%

Values

[[1,2]]
 => ([(0,1)],2)
 => [2]
 => [[1,2]]
 => 2
[[2,2]]
 => ([(0,2),(2,1)],3)
 => [3]
 => [[1,2,3]]
 => 3
[[1],[2]]
 => ([],1)
 => [1]
 => [[1]]
 => 1
[[1,3]]
 => ([(0,2),(2,1)],3)
 => [3]
 => [[1,2,3]]
 => 3
[[2,3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => [4,1]
 => [[1,2,3,4],[5]]
 => 4
[[3,3]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => [5,1]
 => [[1,2,3,4,5],[6]]
 => 5
[[1],[3]]
 => ([(0,1)],2)
 => [2]
 => [[1,2]]
 => 2
[[2],[3]]
 => ([(0,2),(2,1)],3)
 => [3]
 => [[1,2,3]]
 => 3
[[1,1,2]]
 => ([(0,1)],2)
 => [2]
 => [[1,2]]
 => 2
[[1,2,2]]
 => ([(0,2),(2,1)],3)
 => [3]
 => [[1,2,3]]
 => 3
[[2,2,2]]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => [[1,2,3,4]]
 => 4
[[1,1],[2]]
 => ([],1)
 => [1]
 => [[1]]
 => 1
[[1,2],[2]]
 => ([(0,1)],2)
 => [2]
 => [[1,2]]
 => 2
[[1,4]]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => [[1,2,3,4]]
 => 4
[[2,4]]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => [5,2]
 => [[1,2,3,4,5],[6,7]]
 => 5
[[3,4]]
 => ([(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]
 => [[1,2,3,4,5,6],[7,8,9]]
 => 6
[[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]
 => [[1,2,3,4,5,6,7],[8,9,10]]
 => 7
[[1],[4]]
 => ([(0,2),(2,1)],3)
 => [3]
 => [[1,2,3]]
 => 3
[[2],[4]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => [4,1]
 => [[1,2,3,4],[5]]
 => 4
[[3],[4]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => [5,1]
 => [[1,2,3,4,5],[6]]
 => 5
[[1,1,3]]
 => ([(0,2),(2,1)],3)
 => [3]
 => [[1,2,3]]
 => 3
[[1,2,3]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => [4,1]
 => [[1,2,3,4],[5]]
 => 4
[[1,3,3]]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => [5,1]
 => [[1,2,3,4,5],[6]]
 => 5
[[2,2,3]]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => [5,2]
 => [[1,2,3,4,5],[6,7]]
 => 5
[[2,3,3]]
 => ([(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]
 => [[1,2,3,4,5,6],[7,8,9]]
 => 6
[[3,3,3]]
 => ([(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]
 => [[1,2,3,4,5,6,7],[8,9,10]]
 => 7
[[1,1],[3]]
 => ([(0,1)],2)
 => [2]
 => [[1,2]]
 => 2
[[1,2],[3]]
 => ([(0,2),(2,1)],3)
 => [3]
 => [[1,2,3]]
 => 3
[[1,3],[2]]
 => ([(0,2),(2,1)],3)
 => [3]
 => [[1,2,3]]
 => 3
[[1,3],[3]]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => [[1,2,3,4]]
 => 4
[[2,2],[3]]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => [[1,2,3,4]]
 => 4
[[2,3],[3]]
 => ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
 => [5,3]
 => [[1,2,3,4,5],[6,7,8]]
 => 5
[[1],[2],[3]]
 => ([],1)
 => [1]
 => [[1]]
 => 1
[[1,1,1,2]]
 => ([(0,1)],2)
 => [2]
 => [[1,2]]
 => 2
[[1,1,2,2]]
 => ([(0,2),(2,1)],3)
 => [3]
 => [[1,2,3]]
 => 3
[[1,2,2,2]]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => [[1,2,3,4]]
 => 4
[[2,2,2,2]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [5]
 => [[1,2,3,4,5]]
 => 5
[[1,1,1],[2]]
 => ([],1)
 => [1]
 => [[1]]
 => 1
[[1,1,2],[2]]
 => ([(0,1)],2)
 => [2]
 => [[1,2]]
 => 2
[[1,2,2],[2]]
 => ([(0,2),(2,1)],3)
 => [3]
 => [[1,2,3]]
 => 3
[[1,1],[2,2]]
 => ([],1)
 => [1]
 => [[1]]
 => 1
[[1,5]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [5]
 => [[1,2,3,4,5]]
 => 5
[[2,5]]
 => ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
 => [6,3]
 => [[1,2,3,4,5,6],[7,8,9]]
 => 6
[[3,5]]
 => ([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
 => [7,4,1]
 => [[1,2,3,4,5,6,7],[8,9,10,11],[12]]
 => ? = 7
[[4,5]]
 => ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
 => [8,5,1]
 => [[1,2,3,4,5,6,7,8],[9,10,11,12,13],[14]]
 => ? = 8
[[5,5]]
 => ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
 => [9,5,1]
 => [[1,2,3,4,5,6,7,8,9],[10,11,12,13,14],[15]]
 => ? = 9
[[1],[5]]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => [[1,2,3,4]]
 => 4
[[2],[5]]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => [5,2]
 => [[1,2,3,4,5],[6,7]]
 => 5
[[3],[5]]
 => ([(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]
 => [[1,2,3,4,5,6],[7,8,9]]
 => 6
[[4],[5]]
 => ([(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]
 => [[1,2,3,4,5,6,7],[8,9,10]]
 => 7
[[1,1,4]]
 => ([(0,3),(2,1),(3,2)],4)
 => [4]
 => [[1,2,3,4]]
 => 4
[[1,2,4]]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => [5,2]
 => [[1,2,3,4,5],[6,7]]
 => 5
[[1,3,4]]
 => ([(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]
 => [[1,2,3,4,5,6],[7,8,9]]
 => 6
[[2,3,4]]
 => ([(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]
 => [[1,2,3,4,5,6,7],[8,9,10,11],[12,13,14]]
 => ? = 7
[[2,4,4]]
 => ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
 => [8,5,3]
 => [[1,2,3,4,5,6,7,8],[9,10,11,12,13],[14,15,16]]
 => ? = 8
[[3,3,4]]
 => ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
 => [8,5,3]
 => [[1,2,3,4,5,6,7,8],[9,10,11,12,13],[14,15,16]]
 => ? = 8
[[2,4],[3]]
 => ([(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]
 => [[1,2,3,4,5,6],[7,8,9,10],[11]]
 => ? = 6
[[2,4],[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]
 => [[1,2,3,4,5,6,7],[8,9,10,11,12],[13]]
 => ? = 7
[[2,2,3,3]]
 => ([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
 => [7,4,1]
 => [[1,2,3,4,5,6,7],[8,9,10,11],[12]]
 => ? = 7
[[2,3,3,3]]
 => ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
 => [8,5,1]
 => [[1,2,3,4,5,6,7,8],[9,10,11,12,13],[14]]
 => ? = 8
[[3,3,3,3]]
 => ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
 => [9,5,1]
 => [[1,2,3,4,5,6,7,8,9],[10,11,12,13,14],[15]]
 => ? = 9
[[2,3,3],[3]]
 => ([(0,9),(0,10),(1,11),(2,14),(3,12),(4,13),(5,4),(5,11),(6,5),(7,3),(8,1),(8,14),(9,6),(10,2),(10,8),(11,13),(13,12),(14,7)],15)
 => [7,5,3]
 => [[1,2,3,4,5,6,7],[8,9,10,11,12],[13,14,15]]
 => ? = 7
[[2,6]]
 => ([(0,6),(1,7),(2,8),(3,4),(3,7),(4,5),(4,10),(5,2),(5,9),(6,1),(6,3),(7,10),(9,8),(10,9)],11)
 => [7,4]
 => [[1,2,3,4,5,6,7],[8,9,10,11]]
 => ? = 7
[[3,6]]
 => ([(0,7),(1,14),(2,9),(3,10),(4,5),(4,14),(5,6),(5,8),(6,2),(6,11),(7,1),(7,4),(8,10),(8,11),(9,13),(10,12),(11,9),(11,12),(12,13),(14,3),(14,8)],15)
 => [8,5,2]
 => [[1,2,3,4,5,6,7,8],[9,10,11,12,13],[14,15]]
 => ? = 8
[[3],[6]]
 => ([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
 => [7,4,1]
 => [[1,2,3,4,5,6,7],[8,9,10,11],[12]]
 => ? = 7
[[4],[6]]
 => ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
 => [8,5,1]
 => [[1,2,3,4,5,6,7,8],[9,10,11,12,13],[14]]
 => ? = 8
[[5],[6]]
 => ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
 => [9,5,1]
 => [[1,2,3,4,5,6,7,8,9],[10,11,12,13,14],[15]]
 => ? = 9
[[1,3,5]]
 => ([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
 => [7,4,1]
 => [[1,2,3,4,5,6,7],[8,9,10,11],[12]]
 => ? = 7
[[1,4,5]]
 => ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
 => [8,5,1]
 => [[1,2,3,4,5,6,7,8],[9,10,11,12,13],[14]]
 => ? = 8
[[1,5,5]]
 => ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
 => [9,5,1]
 => [[1,2,3,4,5,6,7,8,9],[10,11,12,13,14],[15]]
 => ? = 9
[[2,2,5]]
 => ([(0,6),(1,9),(2,8),(3,5),(3,7),(4,1),(4,7),(5,2),(5,10),(6,3),(6,4),(7,9),(7,10),(8,12),(9,11),(10,8),(10,11),(11,12)],13)
 => [7,4,2]
 => [[1,2,3,4,5,6,7],[8,9,10,11],[12,13]]
 => ? = 7
[[1,5],[5]]
 => ([(0,6),(1,8),(2,10),(4,9),(5,1),(5,10),(6,7),(7,2),(7,5),(8,9),(9,3),(10,4),(10,8)],11)
 => [8,3]
 => [[1,2,3,4,5,6,7,8],[9,10,11]]
 => ? = 8
[[2,3],[5]]
 => ([(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]
 => [[1,2,3,4,5,6,7],[8,9,10,11],[12,13,14]]
 => ? = 7
[[2,5],[3]]
 => ([(0,6),(0,7),(1,11),(2,9),(3,9),(3,10),(4,2),(5,1),(5,10),(6,4),(7,8),(8,3),(8,5),(9,12),(10,11),(10,12),(11,13),(12,13)],14)
 => [7,5,2]
 => [[1,2,3,4,5,6,7],[8,9,10,11,12],[13,14]]
 => ? = 7
[[2,4],[5]]
 => ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
 => [8,5,3]
 => [[1,2,3,4,5,6,7,8],[9,10,11,12,13],[14,15,16]]
 => ? = 8
[[3,3],[5]]
 => ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
 => [8,5,3]
 => [[1,2,3,4,5,6,7,8],[9,10,11,12,13],[14,15,16]]
 => ? = 8
[[1,2,3,4]]
 => ([(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]
 => [[1,2,3,4,5,6,7],[8,9,10,11],[12,13,14]]
 => ? = 7
[[1,2,4,4]]
 => ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
 => [8,5,3]
 => [[1,2,3,4,5,6,7,8],[9,10,11,12,13],[14,15,16]]
 => ? = 8
[[1,3,3,4]]
 => ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
 => [8,5,3]
 => [[1,2,3,4,5,6,7,8],[9,10,11,12,13],[14,15,16]]
 => ? = 8
[[2,2,2,4]]
 => ([(0,6),(1,9),(2,8),(3,5),(3,7),(4,1),(4,7),(5,2),(5,10),(6,3),(6,4),(7,9),(7,10),(8,12),(9,11),(10,8),(10,11),(11,12)],13)
 => [7,4,2]
 => [[1,2,3,4,5,6,7],[8,9,10,11],[12,13]]
 => ? = 7
[[1,2,4],[3]]
 => ([(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]
 => [[1,2,3,4,5,6],[7,8,9,10],[11]]
 => ? = 6
[[1,2,4],[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]
 => [[1,2,3,4,5,6,7],[8,9,10,11,12],[13]]
 => ? = 7
[[1,3,4],[3]]
 => ([(0,6),(1,7),(2,8),(3,9),(4,3),(4,7),(5,2),(5,10),(6,1),(6,4),(7,5),(7,9),(9,10),(10,8)],11)
 => [7,4]
 => [[1,2,3,4,5,6,7],[8,9,10,11]]
 => ? = 7
[[1,4,4],[3]]
 => ([(0,7),(1,10),(2,11),(3,8),(4,9),(5,2),(5,9),(6,3),(6,12),(7,4),(7,5),(8,10),(9,6),(9,11),(11,12),(12,1),(12,8)],13)
 => [8,5]
 => [[1,2,3,4,5,6,7,8],[9,10,11,12,13]]
 => ? = 8
[[1,4,4],[4]]
 => ([(0,8),(2,13),(3,11),(4,9),(5,10),(6,3),(6,10),(7,4),(7,12),(8,5),(8,6),(9,13),(10,7),(10,11),(11,12),(12,2),(12,9),(13,1)],14)
 => [9,5]
 => [[1,2,3,4,5,6,7,8,9],[10,11,12,13,14]]
 => ? = 9
[[2,2,3],[4]]
 => ([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
 => [7,4,1]
 => [[1,2,3,4,5,6,7],[8,9,10,11],[12]]
 => ? = 7
[[2,2,4],[3]]
 => ([(0,6),(0,7),(1,11),(2,9),(3,9),(3,10),(4,5),(5,1),(5,10),(6,4),(7,8),(8,2),(8,3),(9,12),(10,11),(10,12),(11,13),(12,13)],14)
 => [7,5,2]
 => [[1,2,3,4,5,6,7],[8,9,10,11,12],[13,14]]
 => ? = 7
[[2,2,4],[4]]
 => ([(0,7),(0,8),(1,16),(2,10),(2,16),(3,11),(4,12),(5,6),(6,4),(6,10),(7,9),(8,5),(9,1),(9,2),(10,12),(10,13),(11,15),(12,14),(13,11),(13,14),(14,15),(16,3),(16,13)],17)
 => [8,6,3]
 => [[1,2,3,4,5,6,7,8],[9,10,11,12,13,14],[15,16,17]]
 => ? = 8
[[2,3,3],[4]]
 => ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
 => [8,5,1]
 => [[1,2,3,4,5,6,7,8],[9,10,11,12,13],[14]]
 => ? = 8
[[3,3,3],[4]]
 => ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
 => [9,5,1]
 => [[1,2,3,4,5,6,7,8,9],[10,11,12,13,14],[15]]
 => ? = 9
[[1,3],[4,4]]
 => ([(0,7),(1,8),(1,9),(2,9),(2,13),(3,8),(3,13),(4,11),(5,10),(6,5),(7,1),(7,2),(7,3),(8,6),(9,12),(11,10),(12,11),(13,4),(13,12)],14)
 => [7,4,3]
 => [[1,2,3,4,5,6,7],[8,9,10,11],[12,13,14]]
 => ? = 7
[[2,2],[4,4]]
 => ([(0,6),(1,12),(2,11),(3,11),(3,12),(4,8),(5,9),(6,1),(6,2),(6,3),(7,8),(7,9),(8,10),(9,10),(11,4),(11,7),(12,5),(12,7)],13)
 => [7,4,2]
 => [[1,2,3,4,5,6,7],[8,9,10,11],[12,13]]
 => ? = 7
[[2,3],[3,4]]
 => ([(0,7),(1,8),(1,9),(2,9),(2,13),(3,8),(3,13),(4,11),(5,10),(6,5),(7,1),(7,2),(7,3),(8,6),(9,12),(11,10),(12,11),(13,4),(13,12)],14)
 => [7,4,3]
 => [[1,2,3,4,5,6,7],[8,9,10,11],[12,13,14]]
 => ? = 7
[[2,4],[3],[4]]
 => ([(0,9),(0,10),(1,12),(2,11),(3,11),(3,12),(4,7),(5,8),(6,3),(7,2),(8,1),(9,4),(9,14),(10,5),(10,14),(11,13),(12,13),(14,6)],15)
 => [7,5,3]
 => [[1,2,3,4,5,6,7],[8,9,10,11,12],[13,14,15]]
 => ? = 7
[[1,2,2,3,3]]
 => ([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
 => [7,4,1]
 => [[1,2,3,4,5,6,7],[8,9,10,11],[12]]
 => ? = 7
[[1,2,3,3,3]]
 => ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
 => [8,5,1]
 => [[1,2,3,4,5,6,7,8],[9,10,11,12,13],[14]]
 => ? = 8
[[1,3,3,3,3]]
 => ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
 => [9,5,1]
 => [[1,2,3,4,5,6,7,8,9],[10,11,12,13,14],[15]]
 => ? = 9
[[2,2,2,2,3]]
 => ([(0,6),(1,7),(2,8),(3,4),(3,7),(4,5),(4,10),(5,2),(5,9),(6,1),(6,3),(7,10),(9,8),(10,9)],11)
 => [7,4]
 => [[1,2,3,4,5,6,7],[8,9,10,11]]
 => ? = 7
[[2,2,2,3,3]]
 => ([(0,7),(1,14),(2,9),(3,10),(4,5),(4,14),(5,6),(5,8),(6,2),(6,11),(7,1),(7,4),(8,10),(8,11),(9,13),(10,12),(11,9),(11,12),(12,13),(14,3),(14,8)],15)
 => [8,5,2]
 => [[1,2,3,4,5,6,7,8],[9,10,11,12,13],[14,15]]
 => ? = 8
[[1,2,3,3],[3]]
 => ([(0,9),(0,10),(1,11),(2,14),(3,12),(4,13),(5,4),(5,11),(6,5),(7,3),(8,1),(8,14),(9,6),(10,2),(10,8),(11,13),(13,12),(14,7)],15)
 => [7,5,3]
 => [[1,2,3,4,5,6,7],[8,9,10,11,12],[13,14,15]]
 => ? = 7

Description

The last entry in the first row of a standard tableau.

The following 72 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St001622The number of join-irreducible elements of a lattice. St000744The length of the path to the largest entry in a standard Young tableau. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000474Dyson's crank of a partition. St000273The domination number of a graph. 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. St000259The diameter of a connected graph. St001340The cardinality of a minimal non-edge isolating set of a graph. St001029The size of the core of a graph. St001108The 2-dynamic chromatic number of a graph. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St000272The treewidth of a graph. St000536The pathwidth of a graph. St000778The metric dimension of a graph. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001190Number of simple modules with projective dimension at most 4 in the corresponding Nakayama algebra. St001644The dimension 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. St001820The size of the image of the pop stack sorting operator. St001720The minimal length of a chain of small intervals in a lattice. St000080The rank of the poset. St000822The Hadwiger number of the graph. St001316The domatic number of a graph. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000310The minimal degree of a vertex of a graph. St000741The Colin de Verdière graph invariant. St001270The bandwidth of a graph. St001277The degeneracy of a graph. St001358The largest degree of a regular subgraph of a graph. St001962The proper pathwidth of a graph. St001875The number of simple modules with projective dimension at most 1. 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. St000907The number of maximal antichains of minimal length in a poset. St001812The biclique partition number of a graph. St001330The hat guessing number of a graph. St000454The largest eigenvalue of a graph if it is integral. St001637The number of (upper) dissectors of a poset. St001668The number of points of the poset minus the width of the poset. St001613The binary logarithm of the size of the center of a lattice. St001615The number of join prime elements of a lattice. St001617The dimension of the space of valuations of a lattice. St000327The number of cover relations in a poset. St000112The sum of the entries reduced by the index of their row in a semistandard tableau. St001621The number of atoms of a lattice. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001877Number of indecomposable injective modules with projective dimension 2. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001645The pebbling number of a connected graph. St000260The radius of a connected graph. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph.