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Matching statistic: St000001
St000001: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 1
[1,2] => 1
[2,1] => 1
[1,2,3] => 1
[1,3,2] => 1
[2,1,3] => 1
[2,3,1] => 1
[3,1,2] => 1
[3,2,1] => 2
[1,2,3,4] => 1
[1,2,4,3] => 1
[1,3,2,4] => 1
[1,3,4,2] => 1
[1,4,2,3] => 1
[1,4,3,2] => 2
[2,1,3,4] => 1
[2,1,4,3] => 2
[2,3,1,4] => 1
[2,3,4,1] => 1
[2,4,1,3] => 2
[2,4,3,1] => 3
[3,1,2,4] => 1
[3,1,4,2] => 2
[3,2,1,4] => 2
[3,2,4,1] => 3
[3,4,1,2] => 2
[3,4,2,1] => 5
[4,1,2,3] => 1
[4,1,3,2] => 3
[4,2,1,3] => 3
[4,2,3,1] => 6
[4,3,1,2] => 5
[4,3,2,1] => 16
[1,2,3,4,5] => 1
[1,2,3,5,4] => 1
[1,2,4,3,5] => 1
[1,2,4,5,3] => 1
[1,2,5,3,4] => 1
[1,2,5,4,3] => 2
[1,3,2,4,5] => 1
[1,3,2,5,4] => 2
[1,3,4,2,5] => 1
[1,3,4,5,2] => 1
[1,3,5,2,4] => 2
[1,3,5,4,2] => 3
[1,4,2,3,5] => 1
[1,4,2,5,3] => 2
[1,4,3,2,5] => 2
[1,4,3,5,2] => 3
[1,4,5,2,3] => 2
Description
The number of reduced words for a permutation.
This is the number of ways to write a permutation as a minimal length product of simple transpositions. E.g., there are two reduced words for the permutation $[3,2,1]$, which are $(1,2)(2,3)(1,2) = (2,3)(1,2)(2,3)$.
Matching statistic: St000909
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
St000909: Posets ⟶ ℤResult quality: 4% ●values known / values provided: 9%●distinct values known / distinct values provided: 4%
Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
St000909: Posets ⟶ ℤResult quality: 4% ●values known / values provided: 9%●distinct values known / distinct values provided: 4%
Values
[1] => [[1]]
=> [[1]]
=> ([],1)
=> 1
[1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> ([],1)
=> 1
[2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> ([(0,1)],2)
=> 1
[1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ([],1)
=> 1
[1,3,2] => [[1,0,0],[0,0,1],[0,1,0]]
=> [[1,1,1],[2,3],[3]]
=> ([(0,1)],2)
=> 1
[2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
=> [[1,1,2],[2,2],[3]]
=> ([(0,1)],2)
=> 1
[2,3,1] => [[0,0,1],[1,0,0],[0,1,0]]
=> [[1,1,3],[2,3],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[3,1,2] => [[0,1,0],[0,0,1],[1,0,0]]
=> [[1,2,2],[2,3],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[3,2,1] => [[0,0,1],[0,1,0],[1,0,0]]
=> [[1,2,3],[2,3],[3]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> 2
[1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,2],[3,3],[4]]
=> ([],1)
=> 1
[1,2,4,3] => [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [[1,1,1,1],[2,2,2],[3,4],[4]]
=> ([(0,1)],2)
=> 1
[1,3,2,4] => [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,3],[3,3],[4]]
=> ([(0,1)],2)
=> 1
[1,3,4,2] => [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> [[1,1,1,1],[2,2,4],[3,4],[4]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,4,2,3] => [[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]]
=> [[1,1,1,1],[2,3,3],[3,4],[4]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,4,3,2] => [[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> [[1,1,1,1],[2,3,4],[3,4],[4]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> 2
[2,1,3,4] => [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,2],[2,2,2],[3,3],[4]]
=> ([(0,1)],2)
=> 1
[2,1,4,3] => [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> [[1,1,1,2],[2,2,2],[3,4],[4]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,3,1,4] => [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> [[1,1,1,3],[2,2,3],[3,3],[4]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[2,3,4,1] => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [[1,1,1,4],[2,2,4],[3,4],[4]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> 2
[2,4,1,3] => [[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
=> [[1,1,1,3],[2,3,3],[3,4],[4]]
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> 2
[2,4,3,1] => [[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
=> [[1,1,1,4],[2,3,4],[3,4],[4]]
=> ([(0,5),(0,10),(1,16),(2,15),(3,14),(4,13),(5,12),(6,2),(6,13),(7,4),(7,14),(8,1),(9,6),(10,11),(10,12),(11,3),(11,7),(12,9),(13,15),(14,8),(15,16)],17)
=> ? ∊ {1,1,3,3,3,3,5,5,6,16}
[3,1,2,4] => [[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]]
=> [[1,1,2,2],[2,2,3],[3,3],[4]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[3,1,4,2] => [[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
=> [[1,1,2,2],[2,2,4],[3,4],[4]]
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> ? ∊ {1,1,3,3,3,3,5,5,6,16}
[3,2,1,4] => [[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> [[1,1,2,3],[2,2,3],[3,3],[4]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> 2
[3,2,4,1] => [[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
=> [[1,1,2,4],[2,2,4],[3,4],[4]]
=> ([(0,7),(0,8),(1,12),(2,11),(3,10),(4,10),(4,11),(5,3),(6,1),(6,13),(7,9),(8,5),(9,2),(9,4),(10,14),(11,6),(11,14),(13,12),(14,13)],15)
=> ? ∊ {1,1,3,3,3,3,5,5,6,16}
[3,4,1,2] => [[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> [[1,1,3,3],[2,3,4],[3,4],[4]]
=> ([(0,8),(2,11),(2,12),(3,10),(4,9),(5,4),(5,14),(6,3),(6,14),(7,1),(8,5),(8,6),(9,11),(9,13),(10,12),(10,13),(11,15),(12,15),(13,15),(14,2),(14,9),(14,10),(15,7)],16)
=> ? ∊ {1,1,3,3,3,3,5,5,6,16}
[3,4,2,1] => [[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
=> [[1,1,3,4],[2,3,4],[3,4],[4]]
=> ([(0,13),(0,15),(1,17),(2,16),(2,17),(3,19),(4,16),(4,18),(5,22),(6,21),(7,20),(8,23),(8,29),(9,4),(9,28),(10,3),(10,28),(11,6),(12,7),(12,24),(13,14),(14,1),(14,2),(15,9),(15,10),(16,25),(17,12),(17,25),(18,26),(18,29),(19,23),(19,26),(20,27),(22,27),(23,30),(24,20),(24,22),(25,24),(26,30),(27,21),(28,8),(28,18),(28,19),(29,5),(29,30),(30,11)],31)
=> ? ∊ {1,1,3,3,3,3,5,5,6,16}
[4,1,2,3] => [[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> [[1,2,2,2],[2,3,3],[3,4],[4]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> 2
[4,1,3,2] => [[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]]
=> [[1,2,2,2],[2,3,4],[3,4],[4]]
=> ([(0,7),(0,8),(1,12),(2,11),(3,10),(4,10),(4,11),(5,3),(6,1),(6,13),(7,9),(8,5),(9,2),(9,4),(10,14),(11,6),(11,14),(13,12),(14,13)],15)
=> ? ∊ {1,1,3,3,3,3,5,5,6,16}
[4,2,1,3] => [[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]]
=> [[1,2,2,3],[2,3,3],[3,4],[4]]
=> ([(0,5),(0,10),(1,16),(2,15),(3,14),(4,13),(5,12),(6,2),(6,13),(7,4),(7,14),(8,1),(9,6),(10,11),(10,12),(11,3),(11,7),(12,9),(13,15),(14,8),(15,16)],17)
=> ? ∊ {1,1,3,3,3,3,5,5,6,16}
[4,2,3,1] => [[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
=> [[1,2,2,4],[2,3,4],[3,4],[4]]
=> ([(0,14),(0,15),(1,19),(2,18),(3,29),(4,30),(5,22),(6,23),(7,24),(7,25),(8,9),(9,7),(9,18),(9,19),(10,5),(11,6),(12,2),(12,29),(13,1),(13,30),(14,16),(14,28),(15,17),(15,28),(16,3),(16,12),(17,4),(17,13),(18,24),(18,27),(19,25),(19,27),(20,26),(21,26),(22,20),(23,21),(24,22),(24,31),(25,23),(25,31),(27,31),(28,8),(29,10),(30,11),(31,20),(31,21)],32)
=> ? ∊ {1,1,3,3,3,3,5,5,6,16}
[4,3,1,2] => [[0,0,1,0],[0,0,0,1],[0,1,0,0],[1,0,0,0]]
=> [[1,2,3,3],[2,3,4],[3,4],[4]]
=> ([(0,13),(0,15),(1,17),(2,16),(2,17),(3,19),(4,16),(4,18),(5,22),(6,21),(7,20),(8,23),(8,29),(9,4),(9,28),(10,3),(10,28),(11,6),(12,7),(12,24),(13,14),(14,1),(14,2),(15,9),(15,10),(16,25),(17,12),(17,25),(18,26),(18,29),(19,23),(19,26),(20,27),(22,27),(23,30),(24,20),(24,22),(25,24),(26,30),(27,21),(28,8),(28,18),(28,19),(29,5),(29,30),(30,11)],31)
=> ? ∊ {1,1,3,3,3,3,5,5,6,16}
[4,3,2,1] => [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> [[1,2,3,4],[2,3,4],[3,4],[4]]
=> ([(0,5),(0,16),(0,17),(1,23),(2,19),(3,11),(3,21),(4,10),(4,20),(5,12),(5,13),(6,50),(7,51),(8,24),(8,58),(9,25),(9,59),(10,14),(10,52),(11,15),(11,53),(12,26),(12,60),(13,27),(13,60),(14,54),(15,55),(16,4),(16,29),(17,3),(17,29),(18,48),(18,49),(19,32),(19,33),(20,46),(20,52),(21,47),(21,53),(22,34),(22,35),(23,18),(23,54),(23,55),(24,40),(24,42),(25,41),(25,43),(26,46),(26,56),(27,47),(27,57),(28,63),(29,1),(30,62),(31,61),(32,61),(33,61),(34,6),(34,62),(35,7),(35,62),(36,58),(37,59),(38,32),(39,33),(40,44),(41,45),(42,38),(43,39),(44,31),(45,31),(46,36),(47,37),(48,40),(48,63),(49,41),(49,63),(50,38),(51,39),(52,8),(52,36),(53,9),(53,37),(54,28),(54,48),(55,28),(55,49),(56,30),(56,34),(57,30),(57,35),(58,42),(58,50),(59,43),(59,51),(60,22),(60,56),(60,57),(62,2),(63,44),(63,45)],64)
=> ? ∊ {1,1,3,3,3,3,5,5,6,16}
[1,2,3,4,5] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,2,2],[3,3,3],[4,4],[5]]
=> ([],1)
=> 1
[1,2,3,5,4] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,1,1],[2,2,2,2],[3,3,3],[4,5],[5]]
=> ([(0,1)],2)
=> 1
[1,2,4,3,5] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,2,2],[3,3,4],[4,4],[5]]
=> ([(0,1)],2)
=> 1
[1,2,4,5,3] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [[1,1,1,1,1],[2,2,2,2],[3,3,5],[4,5],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,2,5,3,4] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,0,1,0,0]]
=> [[1,1,1,1,1],[2,2,2,2],[3,4,4],[4,5],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,2,5,4,3] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [[1,1,1,1,1],[2,2,2,2],[3,4,5],[4,5],[5]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> 2
[1,3,2,4,5] => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,2,3],[3,3,3],[4,4],[5]]
=> ([(0,1)],2)
=> 1
[1,3,2,5,4] => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,1,1],[2,2,2,3],[3,3,3],[4,5],[5]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,3,4,2,5] => [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,2,4],[3,3,4],[4,4],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,3,4,5,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [[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)
=> 2
[1,3,5,2,4] => [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0]]
=> [[1,1,1,1,1],[2,2,2,4],[3,4,4],[4,5],[5]]
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> 2
[1,3,5,4,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]
=> [[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)
=> ? ∊ {1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[1,4,2,3,5] => [[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,3,3],[3,3,4],[4,4],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,4,2,5,3] => [[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [[1,1,1,1,1],[2,2,3,3],[3,3,5],[4,5],[5]]
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> ? ∊ {1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[1,4,3,2,5] => [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> 2
[1,4,3,5,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [[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)
=> ? ∊ {1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[1,4,5,2,3] => [[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0]]
=> [[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)
=> ? ∊ {1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[1,4,5,3,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0]]
=> [[1,1,1,1,1],[2,2,4,5],[3,4,5],[4,5],[5]]
=> ([(0,13),(0,15),(1,17),(2,16),(2,17),(3,19),(4,16),(4,18),(5,22),(6,21),(7,20),(8,23),(8,29),(9,4),(9,28),(10,3),(10,28),(11,6),(12,7),(12,24),(13,14),(14,1),(14,2),(15,9),(15,10),(16,25),(17,12),(17,25),(18,26),(18,29),(19,23),(19,26),(20,27),(22,27),(23,30),(24,20),(24,22),(25,24),(26,30),(27,21),(28,8),(28,18),(28,19),(29,5),(29,30),(30,11)],31)
=> ? ∊ {1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[1,5,2,3,4] => [[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0]]
=> [[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)
=> 2
[1,5,2,4,3] => [[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,1,0,0,0]]
=> [[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)
=> ? ∊ {1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[1,5,3,2,4] => [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1],[0,1,0,0,0]]
=> [[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)
=> ? ∊ {1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[1,5,3,4,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0]]
=> [[1,1,1,1,1],[2,3,3,5],[3,4,5],[4,5],[5]]
=> ([(0,14),(0,15),(1,19),(2,18),(3,29),(4,30),(5,22),(6,23),(7,24),(7,25),(8,9),(9,7),(9,18),(9,19),(10,5),(11,6),(12,2),(12,29),(13,1),(13,30),(14,16),(14,28),(15,17),(15,28),(16,3),(16,12),(17,4),(17,13),(18,24),(18,27),(19,25),(19,27),(20,26),(21,26),(22,20),(23,21),(24,22),(24,31),(25,23),(25,31),(27,31),(28,8),(29,10),(30,11),(31,20),(31,21)],32)
=> ? ∊ {1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[1,5,4,2,3] => [[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0]]
=> [[1,1,1,1,1],[2,3,4,4],[3,4,5],[4,5],[5]]
=> ([(0,13),(0,15),(1,17),(2,16),(2,17),(3,19),(4,16),(4,18),(5,22),(6,21),(7,20),(8,23),(8,29),(9,4),(9,28),(10,3),(10,28),(11,6),(12,7),(12,24),(13,14),(14,1),(14,2),(15,9),(15,10),(16,25),(17,12),(17,25),(18,26),(18,29),(19,23),(19,26),(20,27),(22,27),(23,30),(24,20),(24,22),(25,24),(26,30),(27,21),(28,8),(28,18),(28,19),(29,5),(29,30),(30,11)],31)
=> ? ∊ {1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[1,5,4,3,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0]]
=> [[1,1,1,1,1],[2,3,4,5],[3,4,5],[4,5],[5]]
=> ([(0,5),(0,16),(0,17),(1,23),(2,19),(3,11),(3,21),(4,10),(4,20),(5,12),(5,13),(6,50),(7,51),(8,24),(8,58),(9,25),(9,59),(10,14),(10,52),(11,15),(11,53),(12,26),(12,60),(13,27),(13,60),(14,54),(15,55),(16,4),(16,29),(17,3),(17,29),(18,48),(18,49),(19,32),(19,33),(20,46),(20,52),(21,47),(21,53),(22,34),(22,35),(23,18),(23,54),(23,55),(24,40),(24,42),(25,41),(25,43),(26,46),(26,56),(27,47),(27,57),(28,63),(29,1),(30,62),(31,61),(32,61),(33,61),(34,6),(34,62),(35,7),(35,62),(36,58),(37,59),(38,32),(39,33),(40,44),(41,45),(42,38),(43,39),(44,31),(45,31),(46,36),(47,37),(48,40),(48,63),(49,41),(49,63),(50,38),(51,39),(52,8),(52,36),(53,9),(53,37),(54,28),(54,48),(55,28),(55,49),(56,30),(56,34),(57,30),(57,35),(58,42),(58,50),(59,43),(59,51),(60,22),(60,56),(60,57),(62,2),(63,44),(63,45)],64)
=> ? ∊ {1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[2,1,3,4,5] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,2],[2,2,2,2],[3,3,3],[4,4],[5]]
=> ([(0,1)],2)
=> 1
[2,1,3,5,4] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,1,2],[2,2,2,2],[3,3,3],[4,5],[5]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,1,4,3,5] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,2],[2,2,2,2],[3,3,4],[4,4],[5]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,1,4,5,3] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [[1,1,1,1,2],[2,2,2,2],[3,3,5],[4,5],[5]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> 4
[2,1,5,3,4] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,0,1,0,0]]
=> [[1,1,1,1,2],[2,2,2,2],[3,4,4],[4,5],[5]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> 4
[2,1,5,4,3] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [[1,1,1,1,2],[2,2,2,2],[3,4,5],[4,5],[5]]
=> ([(0,3),(0,6),(0,7),(1,8),(1,12),(2,8),(2,11),(3,9),(3,10),(4,2),(4,13),(5,1),(5,14),(6,4),(6,9),(7,5),(7,10),(8,15),(9,13),(10,14),(11,15),(12,15),(13,11),(14,12)],16)
=> ? ∊ {1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[2,3,1,4,5] => [[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,3],[2,2,2,3],[3,3,3],[4,4],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[2,3,1,5,4] => [[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,1,3],[2,2,2,3],[3,3,3],[4,5],[5]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> 4
[2,3,4,1,5] => [[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,4],[2,2,2,4],[3,3,4],[4,4],[5]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> 2
[2,3,4,5,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [[1,1,1,1,5],[2,2,2,5],[3,3,5],[4,5],[5]]
=> ([(0,9),(1,13),(2,14),(4,12),(5,11),(6,1),(6,12),(7,3),(8,5),(8,15),(9,10),(10,4),(10,6),(11,14),(12,8),(12,13),(13,15),(14,7),(15,2),(15,11)],16)
=> ? ∊ {1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[2,3,5,1,4] => [[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0]]
=> [[1,1,1,1,4],[2,2,2,4],[3,4,4],[4,5],[5]]
=> ([(0,4),(0,8),(1,11),(3,10),(4,9),(5,2),(6,3),(6,12),(7,5),(8,6),(8,9),(9,12),(10,11),(11,7),(12,1),(12,10)],13)
=> ? ∊ {1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[2,3,5,4,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]
=> [[1,1,1,1,5],[2,2,2,5],[3,4,5],[4,5],[5]]
=> ([(0,8),(0,16),(1,19),(1,20),(2,21),(3,23),(4,26),(5,24),(6,22),(7,27),(8,18),(9,10),(9,20),(10,4),(10,31),(11,2),(11,30),(12,3),(13,7),(13,29),(14,11),(14,28),(15,6),(15,25),(16,17),(16,18),(17,1),(17,9),(17,33),(18,33),(19,24),(20,13),(20,31),(21,32),(22,32),(24,14),(25,12),(26,28),(27,25),(28,30),(29,15),(29,27),(30,21),(30,22),(31,26),(31,29),(32,23),(33,5),(33,19)],34)
=> ? ∊ {1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[2,4,1,3,5] => [[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,3],[2,2,3,3],[3,3,4],[4,4],[5]]
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> 2
[2,4,1,5,3] => [[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [[1,1,1,1,3],[2,2,3,3],[3,3,5],[4,5],[5]]
=> ([(0,6),(0,7),(1,5),(1,15),(2,4),(2,14),(3,13),(4,12),(5,3),(5,16),(6,9),(7,2),(7,9),(9,1),(9,14),(10,11),(11,8),(12,10),(13,8),(14,12),(14,15),(15,10),(15,16),(16,11),(16,13)],17)
=> ? ∊ {1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[2,4,3,1,5] => [[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,4],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ([(0,5),(0,10),(1,16),(2,15),(3,14),(4,13),(5,12),(6,2),(6,13),(7,4),(7,14),(8,1),(9,6),(10,11),(10,12),(11,3),(11,7),(12,9),(13,15),(14,8),(15,16)],17)
=> ? ∊ {1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[2,4,3,5,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [[1,1,1,1,5],[2,2,3,5],[3,3,5],[4,5],[5]]
=> ([(0,8),(0,13),(1,19),(2,16),(2,18),(3,21),(4,26),(5,24),(6,16),(6,22),(7,20),(7,23),(8,17),(9,4),(9,18),(10,9),(11,7),(11,30),(12,3),(12,25),(13,14),(13,17),(14,1),(14,15),(15,2),(15,6),(15,19),(16,28),(17,10),(18,26),(18,28),(19,11),(19,22),(20,24),(20,31),(22,30),(23,31),(24,12),(24,27),(25,21),(26,23),(26,29),(27,25),(28,29),(29,31),(30,5),(30,20),(31,27)],32)
=> ? ∊ {1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[2,4,5,1,3] => [[0,0,0,1,0],[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0]]
=> [[1,1,1,1,4],[2,2,4,4],[3,4,5],[4,5],[5]]
=> ([(0,9),(0,10),(1,2),(3,7),(3,23),(4,6),(4,22),(5,15),(6,16),(7,8),(7,24),(8,20),(9,19),(10,4),(10,19),(11,14),(11,18),(12,26),(13,26),(14,25),(15,1),(16,21),(17,13),(17,25),(18,12),(18,25),(19,3),(19,22),(20,12),(20,13),(21,14),(21,17),(22,16),(22,23),(23,11),(23,21),(23,24),(24,17),(24,18),(24,20),(25,5),(25,26),(26,15)],27)
=> ? ∊ {1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[2,4,5,3,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0]]
=> [[1,1,1,1,5],[2,2,4,5],[3,4,5],[4,5],[5]]
=> ([(0,17),(0,18),(1,11),(1,12),(2,53),(3,50),(4,33),(5,16),(5,65),(6,13),(6,61),(7,14),(7,62),(8,15),(8,51),(9,25),(9,63),(10,24),(10,52),(11,54),(12,22),(12,23),(12,54),(13,36),(14,35),(15,40),(16,59),(17,1),(17,48),(18,7),(18,48),(19,32),(19,45),(20,29),(20,38),(21,55),(21,58),(22,49),(22,60),(23,42),(23,49),(24,31),(24,46),(25,21),(25,60),(25,64),(27,70),(28,70),(29,67),(30,66),(31,68),(32,3),(32,69),(33,8),(34,39),(35,57),(36,39),(37,32),(37,66),(38,19),(38,37),(38,67),(39,26),(40,26),(41,30),(41,67),(42,52),(43,44),(43,68),(44,28),(44,69),(45,27),(45,69),(46,53),(46,68),(47,61),(48,9),(48,62),(49,5),(49,56),(50,34),(51,40),(52,2),(52,46),(53,6),(53,47),(54,10),(54,42),(55,31),(55,43),(56,43),(56,65),(57,29),(57,41),(58,30),(58,37),(59,27),(59,28),(60,55),(60,56),(61,34),(61,36),(62,35),(62,63),(63,20),(63,57),(63,64),(64,38),(64,41),(64,58),(65,44),(65,45),(65,59),(66,33),(67,4),(67,66),(68,47),(69,50),(69,70),(70,51)],71)
=> ? ∊ {1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[2,5,1,3,4] => [[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0]]
=> [[1,1,1,1,3],[2,3,3,3],[3,4,4],[4,5],[5]]
=> ([(0,4),(0,8),(1,11),(3,10),(4,9),(5,2),(6,3),(6,12),(7,5),(8,6),(8,9),(9,12),(10,11),(11,7),(12,1),(12,10)],13)
=> ? ∊ {1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[2,5,1,4,3] => [[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,1,0,0,0]]
=> [[1,1,1,1,3],[2,3,3,3],[3,4,5],[4,5],[5]]
=> ([(0,7),(0,8),(0,11),(1,23),(2,5),(2,20),(3,10),(3,21),(4,9),(4,22),(5,6),(5,15),(6,12),(7,4),(7,17),(8,3),(8,16),(9,14),(9,18),(10,13),(10,14),(11,16),(11,17),(13,24),(14,24),(15,12),(16,21),(17,22),(18,23),(18,24),(19,20),(20,15),(21,13),(22,1),(22,18),(23,2),(23,19),(24,19)],25)
=> ? ∊ {1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[2,5,3,1,4] => [[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,1,0,0,0]]
=> [[1,1,1,1,4],[2,3,3,4],[3,4,4],[4,5],[5]]
=> ([(0,17),(0,18),(1,21),(2,20),(3,28),(4,27),(5,23),(6,24),(7,2),(7,30),(8,1),(8,31),(9,15),(10,16),(11,13),(11,32),(12,14),(12,33),(13,3),(13,22),(14,4),(14,22),(15,5),(15,25),(16,6),(16,26),(17,7),(17,19),(18,8),(18,19),(19,30),(19,31),(20,32),(21,33),(22,27),(22,28),(23,29),(24,29),(25,23),(26,24),(27,25),(28,26),(30,11),(30,20),(31,12),(31,21),(32,9),(33,10)],34)
=> ? ∊ {1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[2,5,3,4,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0]]
=> [[1,1,1,1,5],[2,3,3,5],[3,4,5],[4,5],[5]]
=> ([(0,20),(0,21),(1,11),(2,10),(3,27),(3,36),(4,16),(4,34),(5,17),(5,35),(6,26),(6,52),(7,12),(7,53),(8,13),(8,55),(9,15),(9,25),(9,54),(10,51),(11,14),(11,63),(12,38),(13,39),(14,60),(15,19),(15,50),(16,18),(16,64),(17,56),(18,59),(19,22),(19,57),(20,9),(20,58),(21,8),(21,58),(22,47),(22,61),(23,31),(23,46),(24,45),(24,48),(25,37),(25,50),(26,24),(26,49),(26,62),(27,23),(27,61),(27,64),(29,68),(30,68),(31,67),(32,65),(33,69),(34,1),(35,2),(36,6),(37,36),(38,35),(39,34),(40,32),(40,67),(41,51),(41,69),(42,28),(43,28),(44,29),(45,41),(45,66),(46,63),(46,67),(47,52),(48,30),(48,66),(49,48),(49,65),(50,7),(50,57),(51,43),(52,62),(53,5),(53,38),(54,3),(54,37),(55,4),(55,39),(56,33),(56,41),(57,47),(57,53),(58,54),(58,55),(59,32),(59,49),(60,29),(60,30),(61,31),(61,40),(62,45),(62,56),(62,65),(63,44),(63,60),(64,40),(64,46),(64,59),(65,33),(65,66),(66,68),(66,69),(67,44),(68,42),(69,42),(69,43)],70)
=> ? ∊ {1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[2,5,4,1,3] => [[0,0,0,1,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0]]
=> [[1,1,1,1,4],[2,3,4,4],[3,4,5],[4,5],[5]]
=> ([(0,15),(0,16),(0,20),(1,9),(2,13),(2,47),(3,38),(4,14),(4,46),(5,40),(6,10),(6,26),(7,18),(7,45),(8,19),(8,39),(9,25),(10,32),(11,31),(12,44),(13,12),(13,48),(14,11),(14,42),(15,8),(15,28),(16,7),(16,37),(17,24),(17,33),(18,21),(18,36),(19,21),(19,35),(20,28),(20,37),(21,51),(22,49),(23,49),(24,3),(24,50),(26,1),(27,25),(28,39),(29,46),(30,27),(31,30),(32,27),(33,23),(33,50),(34,22),(34,50),(35,40),(35,51),(36,41),(36,51),(37,2),(37,45),(38,43),(39,5),(39,35),(40,4),(40,29),(41,24),(41,34),(42,31),(42,43),(43,30),(43,32),(44,22),(44,23),(45,36),(45,47),(46,38),(46,42),(47,17),(47,41),(47,48),(48,33),(48,34),(48,44),(49,26),(50,6),(50,49),(51,29)],52)
=> ? ∊ {1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[2,5,4,3,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0]]
=> [[1,1,1,1,5],[2,3,4,5],[3,4,5],[4,5],[5]]
=> ([(0,21),(0,22),(0,44),(1,103),(2,102),(3,19),(3,105),(4,20),(4,104),(5,40),(5,60),(6,37),(6,101),(7,42),(7,129),(8,36),(8,132),(9,34),(9,126),(10,41),(10,127),(11,38),(11,108),(12,32),(12,107),(13,43),(13,59),(14,35),(14,100),(15,18),(15,33),(15,106),(16,39),(16,128),(17,119),(18,26),(18,27),(18,99),(19,17),(19,130),(20,113),(21,16),(21,97),(22,11),(22,116),(23,78),(23,120),(24,58),(24,74),(25,57),(25,70),(26,98),(26,124),(27,23),(27,98),(27,115),(28,63),(28,117),(29,64),(29,73),(30,93),(30,95),(31,111),(31,118),(32,50),(32,89),(33,88),(33,99),(34,85),(34,87),(35,30),(35,86),(35,122),(36,84),(36,125),(37,55),(37,96),(38,51),(38,90),(39,51),(39,91),(40,49),(40,92),(41,28),(41,121),(41,123),(42,31),(42,124),(42,131),(43,29),(43,120),(43,130),(44,15),(44,97),(44,116),(45,145),(46,135),(47,133),(48,146),(49,134),(50,141),(51,142),(52,139),(53,136),(54,136),(55,138),(56,147),(57,2),(57,143),(58,1),(58,140),(59,14),(60,6),(61,77),(62,49),(62,145),(63,75),(64,84),(64,144),(65,54),(66,57),(66,133),(67,65),(68,53),(69,53),(70,62),(70,143),(71,126),(72,127),(73,121),(73,144),(74,25),(74,66),(74,140),(75,94),(76,52),(76,146),(77,52),(77,134),(78,100),(79,85),(79,147),(80,81),(80,141),(81,45),(81,143),(82,46),(82,144),(83,47),(83,140),(84,61),(85,68),(86,95),(86,135),(87,65),(88,59),(89,104),(89,141),(90,105),(90,142),(91,110),(91,142),(92,94),(92,134),(93,79),(93,137),(94,55),(94,139),(95,48),(95,137),(96,54),(96,138),(97,7),(97,128),(98,8),(98,114),(99,12),(99,115),(100,122),(101,96),(102,67),(103,109),(104,9),(104,71),(105,10),(105,72),(106,13),(106,88),(107,4),(107,89),(108,3),(108,90),(109,75),(109,92),(110,58),(110,83),(111,50),(111,80),(112,76),(112,77),(113,56),(113,79),(114,80),(114,132),(115,78),(115,107),(116,106),(116,108),(117,48),(117,76),(118,47),(118,66),(119,46),(119,86),(120,64),(120,82),(121,112),(121,117),(122,93),(122,113),(122,135),(123,63),(123,109),(124,111),(124,114),(125,45),(125,62),(126,67),(126,87),(127,103),(127,123),(128,91),(128,129),(129,24),(129,110),(129,131),(130,73),(130,82),(130,119),(131,74),(131,83),(131,118),(132,70),(132,81),(132,125),(133,60),(134,139),(135,56),(135,137),(137,146),(137,147),(138,136),(139,138),(140,5),(140,133),(141,71),(142,72),(143,102),(143,145),(144,61),(144,112),(145,101),(146,69),(147,68),(147,69)],148)
=> ? ∊ {1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[3,1,2,4,5] => [[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,2,2],[2,2,2,3],[3,3,3],[4,4],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[3,1,2,5,4] => [[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,2,2],[2,2,2,3],[3,3,3],[4,5],[5]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> 4
[3,1,4,2,5] => [[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,2,2],[2,2,2,4],[3,3,4],[4,4],[5]]
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> ? ∊ {1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[3,1,4,5,2] => [[0,1,0,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [[1,1,1,2,2],[2,2,2,5],[3,3,5],[4,5],[5]]
=> ([(0,1),(1,3),(1,4),(2,5),(2,18),(3,6),(3,19),(4,7),(4,8),(4,19),(5,13),(6,17),(7,12),(7,15),(8,12),(8,14),(10,21),(11,21),(12,2),(12,20),(13,9),(14,10),(14,20),(15,11),(15,20),(16,9),(17,10),(17,11),(18,13),(18,16),(19,14),(19,15),(19,17),(20,18),(20,21),(21,16)],22)
=> ? ∊ {1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[3,1,5,2,4] => [[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0]]
=> [[1,1,1,2,2],[2,2,2,4],[3,4,4],[4,5],[5]]
=> ([(0,6),(0,7),(1,5),(1,15),(2,4),(2,14),(3,13),(4,12),(5,3),(5,16),(6,9),(7,2),(7,9),(9,1),(9,14),(10,11),(11,8),(12,10),(13,8),(14,12),(14,15),(15,10),(15,16),(16,11),(16,13)],17)
=> ? ∊ {1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[3,1,5,4,2] => [[0,1,0,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]
=> [[1,1,1,2,2],[2,2,2,5],[3,4,5],[4,5],[5]]
=> ([(0,4),(0,5),(1,10),(1,33),(2,9),(2,34),(3,6),(3,7),(3,38),(4,25),(5,3),(5,8),(5,25),(6,19),(6,30),(7,12),(7,19),(7,37),(8,13),(8,35),(8,38),(9,18),(9,31),(10,11),(10,29),(10,36),(11,26),(11,28),(12,22),(12,29),(13,21),(13,32),(14,45),(15,44),(16,46),(17,41),(18,42),(19,2),(19,39),(20,27),(21,20),(22,17),(22,40),(23,17),(23,46),(24,14),(25,1),(25,35),(26,18),(26,43),(27,15),(27,41),(28,15),(28,43),(29,26),(29,40),(30,16),(30,39),(31,14),(31,42),(32,16),(32,23),(33,20),(33,36),(34,24),(34,31),(35,21),(35,33),(36,27),(36,28),(36,40),(37,22),(37,23),(37,39),(38,30),(38,32),(38,37),(39,34),(39,46),(40,41),(40,43),(41,44),(42,45),(43,42),(43,44),(44,45),(46,24)],47)
=> ? ∊ {1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[3,2,1,4,5] => [[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,2,3],[2,2,2,3],[3,3,3],[4,4],[5]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> 2
[3,2,1,5,4] => [[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,2,3],[2,2,2,3],[3,3,3],[4,5],[5]]
=> ([(0,3),(0,6),(0,7),(1,8),(1,12),(2,8),(2,11),(3,9),(3,10),(4,2),(4,13),(5,1),(5,14),(6,4),(6,9),(7,5),(7,10),(8,15),(9,13),(10,14),(11,15),(12,15),(13,11),(14,12)],16)
=> ? ∊ {1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[3,2,4,1,5] => [[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,2,4],[2,2,2,4],[3,3,4],[4,4],[5]]
=> ([(0,7),(0,8),(1,12),(2,11),(3,10),(4,10),(4,11),(5,3),(6,1),(6,13),(7,9),(8,5),(9,2),(9,4),(10,14),(11,6),(11,14),(13,12),(14,13)],15)
=> ? ∊ {1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[3,2,4,5,1] => [[0,0,0,0,1],[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [[1,1,1,2,5],[2,2,2,5],[3,3,5],[4,5],[5]]
=> ([(0,11),(0,12),(1,22),(2,19),(3,18),(3,23),(4,14),(4,20),(5,15),(6,15),(6,17),(7,5),(8,1),(8,17),(9,3),(9,21),(9,26),(10,2),(10,16),(11,13),(12,7),(13,6),(13,8),(14,29),(15,24),(16,19),(17,9),(17,22),(17,24),(18,20),(18,28),(20,10),(20,29),(21,23),(21,27),(22,25),(22,26),(23,28),(24,21),(24,25),(25,27),(26,4),(26,18),(26,27),(27,14),(27,28),(28,29),(29,16)],30)
=> ? ∊ {1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[3,2,5,1,4] => [[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0]]
=> [[1,1,1,2,4],[2,2,2,4],[3,4,4],[4,5],[5]]
=> ([(0,7),(0,8),(0,11),(1,23),(2,5),(2,20),(3,10),(3,21),(4,9),(4,22),(5,6),(5,15),(6,12),(7,4),(7,17),(8,3),(8,16),(9,14),(9,18),(10,13),(10,14),(11,16),(11,17),(13,24),(14,24),(15,12),(16,21),(17,22),(18,23),(18,24),(19,20),(20,15),(21,13),(22,1),(22,18),(23,2),(23,19),(24,19)],25)
=> ? ∊ {1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[3,2,5,4,1] => [[0,0,0,0,1],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]
=> [[1,1,1,2,5],[2,2,2,5],[3,4,5],[4,5],[5]]
=> ([(0,8),(0,9),(0,19),(1,42),(2,4),(2,20),(2,29),(3,50),(4,5),(4,52),(5,3),(5,53),(6,11),(6,35),(7,12),(7,31),(8,7),(8,34),(9,2),(9,33),(10,18),(10,46),(10,48),(11,27),(11,28),(12,25),(12,47),(13,15),(13,43),(13,44),(14,16),(14,26),(14,45),(15,17),(15,41),(15,51),(16,24),(16,40),(17,23),(17,39),(18,22),(18,32),(19,33),(19,34),(20,38),(20,47),(20,52),(21,55),(22,54),(23,57),(24,56),(25,62),(26,6),(26,63),(27,58),(28,58),(29,1),(29,38),(30,27),(30,60),(31,25),(32,26),(32,54),(33,29),(34,31),(35,28),(36,43),(37,21),(37,61),(38,42),(38,62),(39,30),(39,57),(40,30),(40,56),(41,23),(41,59),(42,13),(42,36),(43,41),(43,55),(44,51),(44,55),(45,24),(45,63),(46,22),(46,61),(47,49),(47,62),(48,14),(48,32),(48,61),(49,37),(49,46),(50,21),(50,44),(51,39),(51,40),(51,59),(52,10),(52,49),(52,53),(53,37),(53,48),(53,50),(54,63),(55,59),(56,60),(57,60),(59,56),(59,57),(60,58),(61,45),(61,54),(62,36),(63,35)],64)
=> ? ∊ {1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[3,4,1,2,5] => [[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,3,3],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ([(0,8),(2,11),(2,12),(3,10),(4,9),(5,4),(5,14),(6,3),(6,14),(7,1),(8,5),(8,6),(9,11),(9,13),(10,12),(10,13),(11,15),(12,15),(13,15),(14,2),(14,9),(14,10),(15,7)],16)
=> ? ∊ {1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[3,4,1,5,2] => [[0,0,1,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [[1,1,1,3,3],[2,2,3,5],[3,3,5],[4,5],[5]]
=> ([(0,1),(1,4),(1,5),(2,8),(2,32),(3,6),(3,30),(4,7),(4,33),(5,10),(5,11),(5,33),(6,20),(7,31),(8,27),(9,15),(9,19),(10,18),(10,28),(11,18),(11,25),(13,39),(14,36),(14,39),(15,36),(16,38),(17,38),(18,2),(18,37),(19,26),(19,36),(20,21),(21,12),(22,12),(23,17),(23,35),(24,16),(24,35),(25,13),(25,37),(26,24),(26,34),(27,16),(27,17),(28,14),(28,19),(28,37),(29,21),(29,22),(30,20),(30,29),(31,13),(31,14),(31,15),(32,23),(32,24),(32,27),(33,9),(33,25),(33,28),(33,31),(34,30),(34,35),(35,29),(35,38),(36,3),(36,34),(37,26),(37,32),(37,39),(38,22),(39,23),(39,34)],40)
=> ? ∊ {1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[3,4,2,1,5] => [[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,3,4],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ([(0,13),(0,15),(1,17),(2,16),(2,17),(3,19),(4,16),(4,18),(5,22),(6,21),(7,20),(8,23),(8,29),(9,4),(9,28),(10,3),(10,28),(11,6),(12,7),(12,24),(13,14),(14,1),(14,2),(15,9),(15,10),(16,25),(17,12),(17,25),(18,26),(18,29),(19,23),(19,26),(20,27),(22,27),(23,30),(24,20),(24,22),(25,24),(26,30),(27,21),(28,8),(28,18),(28,19),(29,5),(29,30),(30,11)],31)
=> ? ∊ {1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[3,4,2,5,1] => [[0,0,0,0,1],[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [[1,1,1,3,5],[2,2,3,5],[3,3,5],[4,5],[5]]
=> ([(0,3),(0,4),(1,10),(1,18),(2,7),(3,1),(4,8),(4,9),(5,11),(5,48),(6,46),(7,38),(8,19),(8,49),(9,12),(9,49),(10,13),(10,47),(11,31),(12,43),(13,42),(14,17),(14,37),(14,44),(15,25),(15,26),(16,23),(16,27),(17,28),(17,33),(18,36),(18,47),(19,36),(19,41),(21,51),(22,50),(23,51),(24,54),(25,52),(26,5),(26,52),(27,6),(27,51),(28,34),(28,53),(29,20),(30,20),(31,29),(32,30),(33,26),(33,53),(34,38),(34,54),(35,32),(36,40),(37,15),(37,33),(37,50),(38,35),(39,29),(39,30),(40,22),(40,44),(41,21),(41,27),(42,22),(42,37),(43,21),(43,23),(44,28),(44,46),(44,50),(45,32),(45,39),(46,24),(46,34),(47,14),(47,40),(47,42),(48,31),(48,39),(49,16),(49,41),(49,43),(50,24),(50,25),(50,53),(51,2),(52,45),(52,48),(53,52),(53,54),(54,35),(54,45)],55)
=> ? ∊ {1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[3,4,5,1,2] => [[0,0,0,1,0],[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0]]
=> [[1,1,1,4,4],[2,2,4,5],[3,4,5],[4,5],[5]]
=> ([(0,3),(2,1),(3,5),(3,8),(4,11),(4,46),(5,10),(5,47),(6,9),(6,45),(7,13),(7,37),(8,14),(8,15),(8,47),(9,39),(10,44),(11,38),(12,24),(12,25),(13,32),(13,33),(14,27),(14,41),(15,27),(15,40),(16,20),(16,26),(16,29),(17,49),(18,58),(19,60),(20,48),(20,59),(21,52),(22,48),(22,54),(23,54),(23,59),(24,51),(25,51),(26,7),(26,48),(27,6),(27,50),(28,17),(28,60),(29,35),(29,59),(30,19),(30,53),(31,18),(31,52),(32,18),(32,55),(33,30),(33,55),(34,17),(34,61),(35,42),(35,56),(36,21),(36,57),(37,33),(38,19),(38,28),(39,21),(39,31),(40,23),(40,29),(40,50),(41,22),(41,26),(41,50),(42,31),(42,32),(42,57),(43,28),(43,34),(43,53),(44,20),(44,22),(44,23),(45,36),(45,39),(45,42),(46,30),(46,38),(46,43),(47,16),(47,40),(47,41),(47,44),(48,37),(49,51),(50,35),(50,45),(50,54),(51,2),(52,34),(52,58),(53,12),(53,60),(53,61),(54,36),(54,56),(55,53),(55,58),(56,46),(56,57),(57,43),(57,52),(57,55),(58,61),(59,4),(59,56),(60,24),(60,49),(61,25),(61,49)],62)
=> ? ∊ {1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[3,4,5,2,1] => [[0,0,0,0,1],[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0]]
=> [[1,1,1,4,5],[2,2,4,5],[3,4,5],[4,5],[5]]
=> ([(0,3),(0,4),(1,14),(1,22),(2,9),(3,1),(4,7),(4,11),(5,15),(5,91),(6,76),(7,27),(7,28),(7,96),(8,29),(8,75),(9,47),(10,25),(10,95),(11,24),(11,96),(12,26),(12,89),(13,16),(13,94),(14,17),(14,90),(15,51),(16,82),(17,81),(18,66),(18,84),(19,21),(19,67),(19,85),(20,45),(20,74),(21,48),(21,49),(22,60),(22,90),(23,41),(23,46),(24,60),(24,93),(25,56),(25,77),(26,69),(26,70),(27,42),(27,79),(28,42),(28,80),(29,34),(29,54),(30,39),(30,44),(30,65),(31,115),(32,115),(33,108),(34,97),(35,113),(35,114),(36,116),(37,109),(37,114),(38,117),(39,18),(39,109),(39,113),(40,100),(41,103),(42,13),(42,112),(43,8),(44,12),(44,113),(45,5),(45,101),(46,6),(46,103),(48,57),(48,99),(49,45),(49,99),(50,86),(50,98),(51,83),(52,63),(52,97),(53,75),(54,64),(54,97),(55,47),(56,59),(57,56),(57,111),(58,53),(58,111),(59,52),(60,78),(61,40),(61,116),(62,33),(62,107),(63,31),(63,110),(64,32),(64,110),(65,50),(65,109),(66,43),(67,20),(67,49),(67,117),(68,71),(68,105),(69,40),(69,106),(70,68),(70,106),(71,34),(71,102),(72,33),(72,102),(73,36),(73,104),(74,53),(74,101),(75,54),(76,55),(77,71),(77,72),(78,38),(78,85),(79,37),(79,65),(79,112),(80,35),(80,44),(80,112),(81,38),(81,67),(82,36),(82,61),(83,31),(83,32),(84,57),(84,58),(85,48),(85,84),(85,117),(86,61),(86,69),(86,104),(87,52),(87,92),(88,62),(88,72),(88,105),(89,43),(89,70),(90,19),(90,78),(90,81),(91,51),(91,92),(92,63),(92,64),(92,83),(93,35),(93,37),(93,39),(94,73),(94,82),(94,86),(95,68),(95,77),(95,88),(96,30),(96,79),(96,80),(96,93),(97,110),(98,95),(98,104),(99,101),(99,111),(100,107),(101,87),(101,91),(102,46),(102,108),(103,2),(104,88),(104,106),(104,116),(105,23),(105,102),(105,107),(106,100),(106,105),(107,41),(107,108),(108,103),(109,10),(109,98),(110,76),(110,115),(111,59),(111,87),(112,50),(112,94),(112,114),(113,66),(113,89),(114,73),(114,98),(115,55),(116,62),(116,100),(117,58),(117,74),(117,99)],118)
=> ? ∊ {1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[4,1,2,3,5] => [[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,0,0,1]]
=> [[1,1,2,2,2],[2,2,3,3],[3,3,4],[4,4],[5]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> 2
Description
The number of maximal chains of maximal size in a poset.
Matching statistic: St000527
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
St000527: Posets ⟶ ℤResult quality: 3% ●values known / values provided: 7%●distinct values known / distinct values provided: 3%
Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
St000527: Posets ⟶ ℤResult quality: 3% ●values known / values provided: 7%●distinct values known / distinct values provided: 3%
Values
[1] => [[1]]
=> [[1]]
=> ([],1)
=> 1
[1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> ([],1)
=> 1
[2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> ([(0,1)],2)
=> 1
[1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ([],1)
=> 1
[1,3,2] => [[1,0,0],[0,0,1],[0,1,0]]
=> [[1,1,1],[2,3],[3]]
=> ([(0,1)],2)
=> 1
[2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
=> [[1,1,2],[2,2],[3]]
=> ([(0,1)],2)
=> 1
[2,3,1] => [[0,0,1],[1,0,0],[0,1,0]]
=> [[1,1,3],[2,3],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[3,1,2] => [[0,1,0],[0,0,1],[1,0,0]]
=> [[1,2,2],[2,3],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[3,2,1] => [[0,0,1],[0,1,0],[1,0,0]]
=> [[1,2,3],[2,3],[3]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ? = 2
[1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,2],[3,3],[4]]
=> ([],1)
=> 1
[1,2,4,3] => [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [[1,1,1,1],[2,2,2],[3,4],[4]]
=> ([(0,1)],2)
=> 1
[1,3,2,4] => [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,3],[3,3],[4]]
=> ([(0,1)],2)
=> 1
[1,3,4,2] => [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> [[1,1,1,1],[2,2,4],[3,4],[4]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,4,2,3] => [[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]]
=> [[1,1,1,1],[2,3,3],[3,4],[4]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,4,3,2] => [[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> [[1,1,1,1],[2,3,4],[3,4],[4]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ? ∊ {1,1,2,2,2,2,3,3,3,5,5,6,16}
[2,1,3,4] => [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,2],[2,2,2],[3,3],[4]]
=> ([(0,1)],2)
=> 1
[2,1,4,3] => [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> [[1,1,1,2],[2,2,2],[3,4],[4]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,3,1,4] => [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> [[1,1,1,3],[2,2,3],[3,3],[4]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[2,3,4,1] => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [[1,1,1,4],[2,2,4],[3,4],[4]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ? ∊ {1,1,2,2,2,2,3,3,3,5,5,6,16}
[2,4,1,3] => [[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
=> [[1,1,1,3],[2,3,3],[3,4],[4]]
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> 2
[2,4,3,1] => [[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
=> [[1,1,1,4],[2,3,4],[3,4],[4]]
=> ([(0,5),(0,10),(1,16),(2,15),(3,14),(4,13),(5,12),(6,2),(6,13),(7,4),(7,14),(8,1),(9,6),(10,11),(10,12),(11,3),(11,7),(12,9),(13,15),(14,8),(15,16)],17)
=> ? ∊ {1,1,2,2,2,2,3,3,3,5,5,6,16}
[3,1,2,4] => [[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]]
=> [[1,1,2,2],[2,2,3],[3,3],[4]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[3,1,4,2] => [[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
=> [[1,1,2,2],[2,2,4],[3,4],[4]]
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> 3
[3,2,1,4] => [[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> [[1,1,2,3],[2,2,3],[3,3],[4]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ? ∊ {1,1,2,2,2,2,3,3,3,5,5,6,16}
[3,2,4,1] => [[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
=> [[1,1,2,4],[2,2,4],[3,4],[4]]
=> ([(0,7),(0,8),(1,12),(2,11),(3,10),(4,10),(4,11),(5,3),(6,1),(6,13),(7,9),(8,5),(9,2),(9,4),(10,14),(11,6),(11,14),(13,12),(14,13)],15)
=> ? ∊ {1,1,2,2,2,2,3,3,3,5,5,6,16}
[3,4,1,2] => [[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> [[1,1,3,3],[2,3,4],[3,4],[4]]
=> ([(0,8),(2,11),(2,12),(3,10),(4,9),(5,4),(5,14),(6,3),(6,14),(7,1),(8,5),(8,6),(9,11),(9,13),(10,12),(10,13),(11,15),(12,15),(13,15),(14,2),(14,9),(14,10),(15,7)],16)
=> ? ∊ {1,1,2,2,2,2,3,3,3,5,5,6,16}
[3,4,2,1] => [[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
=> [[1,1,3,4],[2,3,4],[3,4],[4]]
=> ([(0,13),(0,15),(1,17),(2,16),(2,17),(3,19),(4,16),(4,18),(5,22),(6,21),(7,20),(8,23),(8,29),(9,4),(9,28),(10,3),(10,28),(11,6),(12,7),(12,24),(13,14),(14,1),(14,2),(15,9),(15,10),(16,25),(17,12),(17,25),(18,26),(18,29),(19,23),(19,26),(20,27),(22,27),(23,30),(24,20),(24,22),(25,24),(26,30),(27,21),(28,8),(28,18),(28,19),(29,5),(29,30),(30,11)],31)
=> ? ∊ {1,1,2,2,2,2,3,3,3,5,5,6,16}
[4,1,2,3] => [[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> [[1,2,2,2],[2,3,3],[3,4],[4]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ? ∊ {1,1,2,2,2,2,3,3,3,5,5,6,16}
[4,1,3,2] => [[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]]
=> [[1,2,2,2],[2,3,4],[3,4],[4]]
=> ([(0,7),(0,8),(1,12),(2,11),(3,10),(4,10),(4,11),(5,3),(6,1),(6,13),(7,9),(8,5),(9,2),(9,4),(10,14),(11,6),(11,14),(13,12),(14,13)],15)
=> ? ∊ {1,1,2,2,2,2,3,3,3,5,5,6,16}
[4,2,1,3] => [[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]]
=> [[1,2,2,3],[2,3,3],[3,4],[4]]
=> ([(0,5),(0,10),(1,16),(2,15),(3,14),(4,13),(5,12),(6,2),(6,13),(7,4),(7,14),(8,1),(9,6),(10,11),(10,12),(11,3),(11,7),(12,9),(13,15),(14,8),(15,16)],17)
=> ? ∊ {1,1,2,2,2,2,3,3,3,5,5,6,16}
[4,2,3,1] => [[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
=> [[1,2,2,4],[2,3,4],[3,4],[4]]
=> ([(0,14),(0,15),(1,19),(2,18),(3,29),(4,30),(5,22),(6,23),(7,24),(7,25),(8,9),(9,7),(9,18),(9,19),(10,5),(11,6),(12,2),(12,29),(13,1),(13,30),(14,16),(14,28),(15,17),(15,28),(16,3),(16,12),(17,4),(17,13),(18,24),(18,27),(19,25),(19,27),(20,26),(21,26),(22,20),(23,21),(24,22),(24,31),(25,23),(25,31),(27,31),(28,8),(29,10),(30,11),(31,20),(31,21)],32)
=> ? ∊ {1,1,2,2,2,2,3,3,3,5,5,6,16}
[4,3,1,2] => [[0,0,1,0],[0,0,0,1],[0,1,0,0],[1,0,0,0]]
=> [[1,2,3,3],[2,3,4],[3,4],[4]]
=> ([(0,13),(0,15),(1,17),(2,16),(2,17),(3,19),(4,16),(4,18),(5,22),(6,21),(7,20),(8,23),(8,29),(9,4),(9,28),(10,3),(10,28),(11,6),(12,7),(12,24),(13,14),(14,1),(14,2),(15,9),(15,10),(16,25),(17,12),(17,25),(18,26),(18,29),(19,23),(19,26),(20,27),(22,27),(23,30),(24,20),(24,22),(25,24),(26,30),(27,21),(28,8),(28,18),(28,19),(29,5),(29,30),(30,11)],31)
=> ? ∊ {1,1,2,2,2,2,3,3,3,5,5,6,16}
[4,3,2,1] => [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> [[1,2,3,4],[2,3,4],[3,4],[4]]
=> ([(0,5),(0,16),(0,17),(1,23),(2,19),(3,11),(3,21),(4,10),(4,20),(5,12),(5,13),(6,50),(7,51),(8,24),(8,58),(9,25),(9,59),(10,14),(10,52),(11,15),(11,53),(12,26),(12,60),(13,27),(13,60),(14,54),(15,55),(16,4),(16,29),(17,3),(17,29),(18,48),(18,49),(19,32),(19,33),(20,46),(20,52),(21,47),(21,53),(22,34),(22,35),(23,18),(23,54),(23,55),(24,40),(24,42),(25,41),(25,43),(26,46),(26,56),(27,47),(27,57),(28,63),(29,1),(30,62),(31,61),(32,61),(33,61),(34,6),(34,62),(35,7),(35,62),(36,58),(37,59),(38,32),(39,33),(40,44),(41,45),(42,38),(43,39),(44,31),(45,31),(46,36),(47,37),(48,40),(48,63),(49,41),(49,63),(50,38),(51,39),(52,8),(52,36),(53,9),(53,37),(54,28),(54,48),(55,28),(55,49),(56,30),(56,34),(57,30),(57,35),(58,42),(58,50),(59,43),(59,51),(60,22),(60,56),(60,57),(62,2),(63,44),(63,45)],64)
=> ? ∊ {1,1,2,2,2,2,3,3,3,5,5,6,16}
[1,2,3,4,5] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,2,2],[3,3,3],[4,4],[5]]
=> ([],1)
=> 1
[1,2,3,5,4] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,1,1],[2,2,2,2],[3,3,3],[4,5],[5]]
=> ([(0,1)],2)
=> 1
[1,2,4,3,5] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,2,2],[3,3,4],[4,4],[5]]
=> ([(0,1)],2)
=> 1
[1,2,4,5,3] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [[1,1,1,1,1],[2,2,2,2],[3,3,5],[4,5],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,2,5,3,4] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,0,1,0,0]]
=> [[1,1,1,1,1],[2,2,2,2],[3,4,4],[4,5],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,2,5,4,3] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [[1,1,1,1,1],[2,2,2,2],[3,4,5],[4,5],[5]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ? ∊ {1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[1,3,2,4,5] => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,2,3],[3,3,3],[4,4],[5]]
=> ([(0,1)],2)
=> 1
[1,3,2,5,4] => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,1,1],[2,2,2,3],[3,3,3],[4,5],[5]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,3,4,2,5] => [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,2,4],[3,3,4],[4,4],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,3,4,5,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [[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)
=> ? ∊ {1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[1,3,5,2,4] => [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0]]
=> [[1,1,1,1,1],[2,2,2,4],[3,4,4],[4,5],[5]]
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> 2
[1,3,5,4,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]
=> [[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)
=> ? ∊ {1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[1,4,2,3,5] => [[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,3,3],[3,3,4],[4,4],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,4,2,5,3] => [[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [[1,1,1,1,1],[2,2,3,3],[3,3,5],[4,5],[5]]
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> 3
[1,4,3,2,5] => [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ? ∊ {1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[1,4,3,5,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [[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)
=> ? ∊ {1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[1,4,5,2,3] => [[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0]]
=> [[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)
=> ? ∊ {1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[1,4,5,3,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0]]
=> [[1,1,1,1,1],[2,2,4,5],[3,4,5],[4,5],[5]]
=> ([(0,13),(0,15),(1,17),(2,16),(2,17),(3,19),(4,16),(4,18),(5,22),(6,21),(7,20),(8,23),(8,29),(9,4),(9,28),(10,3),(10,28),(11,6),(12,7),(12,24),(13,14),(14,1),(14,2),(15,9),(15,10),(16,25),(17,12),(17,25),(18,26),(18,29),(19,23),(19,26),(20,27),(22,27),(23,30),(24,20),(24,22),(25,24),(26,30),(27,21),(28,8),(28,18),(28,19),(29,5),(29,30),(30,11)],31)
=> ? ∊ {1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[1,5,2,3,4] => [[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0]]
=> [[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)
=> ? ∊ {1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[1,5,2,4,3] => [[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,1,0,0,0]]
=> [[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)
=> ? ∊ {1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[1,5,3,2,4] => [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1],[0,1,0,0,0]]
=> [[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)
=> ? ∊ {1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[1,5,3,4,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0]]
=> [[1,1,1,1,1],[2,3,3,5],[3,4,5],[4,5],[5]]
=> ([(0,14),(0,15),(1,19),(2,18),(3,29),(4,30),(5,22),(6,23),(7,24),(7,25),(8,9),(9,7),(9,18),(9,19),(10,5),(11,6),(12,2),(12,29),(13,1),(13,30),(14,16),(14,28),(15,17),(15,28),(16,3),(16,12),(17,4),(17,13),(18,24),(18,27),(19,25),(19,27),(20,26),(21,26),(22,20),(23,21),(24,22),(24,31),(25,23),(25,31),(27,31),(28,8),(29,10),(30,11),(31,20),(31,21)],32)
=> ? ∊ {1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[1,5,4,2,3] => [[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0]]
=> [[1,1,1,1,1],[2,3,4,4],[3,4,5],[4,5],[5]]
=> ([(0,13),(0,15),(1,17),(2,16),(2,17),(3,19),(4,16),(4,18),(5,22),(6,21),(7,20),(8,23),(8,29),(9,4),(9,28),(10,3),(10,28),(11,6),(12,7),(12,24),(13,14),(14,1),(14,2),(15,9),(15,10),(16,25),(17,12),(17,25),(18,26),(18,29),(19,23),(19,26),(20,27),(22,27),(23,30),(24,20),(24,22),(25,24),(26,30),(27,21),(28,8),(28,18),(28,19),(29,5),(29,30),(30,11)],31)
=> ? ∊ {1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[1,5,4,3,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0]]
=> [[1,1,1,1,1],[2,3,4,5],[3,4,5],[4,5],[5]]
=> ([(0,5),(0,16),(0,17),(1,23),(2,19),(3,11),(3,21),(4,10),(4,20),(5,12),(5,13),(6,50),(7,51),(8,24),(8,58),(9,25),(9,59),(10,14),(10,52),(11,15),(11,53),(12,26),(12,60),(13,27),(13,60),(14,54),(15,55),(16,4),(16,29),(17,3),(17,29),(18,48),(18,49),(19,32),(19,33),(20,46),(20,52),(21,47),(21,53),(22,34),(22,35),(23,18),(23,54),(23,55),(24,40),(24,42),(25,41),(25,43),(26,46),(26,56),(27,47),(27,57),(28,63),(29,1),(30,62),(31,61),(32,61),(33,61),(34,6),(34,62),(35,7),(35,62),(36,58),(37,59),(38,32),(39,33),(40,44),(41,45),(42,38),(43,39),(44,31),(45,31),(46,36),(47,37),(48,40),(48,63),(49,41),(49,63),(50,38),(51,39),(52,8),(52,36),(53,9),(53,37),(54,28),(54,48),(55,28),(55,49),(56,30),(56,34),(57,30),(57,35),(58,42),(58,50),(59,43),(59,51),(60,22),(60,56),(60,57),(62,2),(63,44),(63,45)],64)
=> ? ∊ {1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[2,1,3,4,5] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,2],[2,2,2,2],[3,3,3],[4,4],[5]]
=> ([(0,1)],2)
=> 1
[2,1,3,5,4] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,1,2],[2,2,2,2],[3,3,3],[4,5],[5]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,1,4,3,5] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,2],[2,2,2,2],[3,3,4],[4,4],[5]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,1,4,5,3] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [[1,1,1,1,2],[2,2,2,2],[3,3,5],[4,5],[5]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> 2
[2,1,5,3,4] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,0,1,0,0]]
=> [[1,1,1,1,2],[2,2,2,2],[3,4,4],[4,5],[5]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> 2
[2,1,5,4,3] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [[1,1,1,1,2],[2,2,2,2],[3,4,5],[4,5],[5]]
=> ([(0,3),(0,6),(0,7),(1,8),(1,12),(2,8),(2,11),(3,9),(3,10),(4,2),(4,13),(5,1),(5,14),(6,4),(6,9),(7,5),(7,10),(8,15),(9,13),(10,14),(11,15),(12,15),(13,11),(14,12)],16)
=> ? ∊ {1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[2,3,1,4,5] => [[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,3],[2,2,2,3],[3,3,3],[4,4],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[2,3,1,5,4] => [[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,1,3],[2,2,2,3],[3,3,3],[4,5],[5]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> 2
[2,3,4,1,5] => [[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,4],[2,2,2,4],[3,3,4],[4,4],[5]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ? ∊ {1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[2,3,4,5,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [[1,1,1,1,5],[2,2,2,5],[3,3,5],[4,5],[5]]
=> ([(0,9),(1,13),(2,14),(4,12),(5,11),(6,1),(6,12),(7,3),(8,5),(8,15),(9,10),(10,4),(10,6),(11,14),(12,8),(12,13),(13,15),(14,7),(15,2),(15,11)],16)
=> ? ∊ {1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[2,3,5,1,4] => [[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0]]
=> [[1,1,1,1,4],[2,2,2,4],[3,4,4],[4,5],[5]]
=> ([(0,4),(0,8),(1,11),(3,10),(4,9),(5,2),(6,3),(6,12),(7,5),(8,6),(8,9),(9,12),(10,11),(11,7),(12,1),(12,10)],13)
=> ? ∊ {1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[2,3,5,4,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]
=> [[1,1,1,1,5],[2,2,2,5],[3,4,5],[4,5],[5]]
=> ([(0,8),(0,16),(1,19),(1,20),(2,21),(3,23),(4,26),(5,24),(6,22),(7,27),(8,18),(9,10),(9,20),(10,4),(10,31),(11,2),(11,30),(12,3),(13,7),(13,29),(14,11),(14,28),(15,6),(15,25),(16,17),(16,18),(17,1),(17,9),(17,33),(18,33),(19,24),(20,13),(20,31),(21,32),(22,32),(24,14),(25,12),(26,28),(27,25),(28,30),(29,15),(29,27),(30,21),(30,22),(31,26),(31,29),(32,23),(33,5),(33,19)],34)
=> ? ∊ {1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[2,4,1,3,5] => [[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,3],[2,2,3,3],[3,3,4],[4,4],[5]]
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> 2
[2,4,1,5,3] => [[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [[1,1,1,1,3],[2,2,3,3],[3,3,5],[4,5],[5]]
=> ([(0,6),(0,7),(1,5),(1,15),(2,4),(2,14),(3,13),(4,12),(5,3),(5,16),(6,9),(7,2),(7,9),(9,1),(9,14),(10,11),(11,8),(12,10),(13,8),(14,12),(14,15),(15,10),(15,16),(16,11),(16,13)],17)
=> ? ∊ {1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[2,4,3,1,5] => [[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,4],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ([(0,5),(0,10),(1,16),(2,15),(3,14),(4,13),(5,12),(6,2),(6,13),(7,4),(7,14),(8,1),(9,6),(10,11),(10,12),(11,3),(11,7),(12,9),(13,15),(14,8),(15,16)],17)
=> ? ∊ {1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[2,4,3,5,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [[1,1,1,1,5],[2,2,3,5],[3,3,5],[4,5],[5]]
=> ([(0,8),(0,13),(1,19),(2,16),(2,18),(3,21),(4,26),(5,24),(6,16),(6,22),(7,20),(7,23),(8,17),(9,4),(9,18),(10,9),(11,7),(11,30),(12,3),(12,25),(13,14),(13,17),(14,1),(14,15),(15,2),(15,6),(15,19),(16,28),(17,10),(18,26),(18,28),(19,11),(19,22),(20,24),(20,31),(22,30),(23,31),(24,12),(24,27),(25,21),(26,23),(26,29),(27,25),(28,29),(29,31),(30,5),(30,20),(31,27)],32)
=> ? ∊ {1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[2,4,5,1,3] => [[0,0,0,1,0],[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0]]
=> [[1,1,1,1,4],[2,2,4,4],[3,4,5],[4,5],[5]]
=> ([(0,9),(0,10),(1,2),(3,7),(3,23),(4,6),(4,22),(5,15),(6,16),(7,8),(7,24),(8,20),(9,19),(10,4),(10,19),(11,14),(11,18),(12,26),(13,26),(14,25),(15,1),(16,21),(17,13),(17,25),(18,12),(18,25),(19,3),(19,22),(20,12),(20,13),(21,14),(21,17),(22,16),(22,23),(23,11),(23,21),(23,24),(24,17),(24,18),(24,20),(25,5),(25,26),(26,15)],27)
=> ? ∊ {1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[2,4,5,3,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0]]
=> [[1,1,1,1,5],[2,2,4,5],[3,4,5],[4,5],[5]]
=> ([(0,17),(0,18),(1,11),(1,12),(2,53),(3,50),(4,33),(5,16),(5,65),(6,13),(6,61),(7,14),(7,62),(8,15),(8,51),(9,25),(9,63),(10,24),(10,52),(11,54),(12,22),(12,23),(12,54),(13,36),(14,35),(15,40),(16,59),(17,1),(17,48),(18,7),(18,48),(19,32),(19,45),(20,29),(20,38),(21,55),(21,58),(22,49),(22,60),(23,42),(23,49),(24,31),(24,46),(25,21),(25,60),(25,64),(27,70),(28,70),(29,67),(30,66),(31,68),(32,3),(32,69),(33,8),(34,39),(35,57),(36,39),(37,32),(37,66),(38,19),(38,37),(38,67),(39,26),(40,26),(41,30),(41,67),(42,52),(43,44),(43,68),(44,28),(44,69),(45,27),(45,69),(46,53),(46,68),(47,61),(48,9),(48,62),(49,5),(49,56),(50,34),(51,40),(52,2),(52,46),(53,6),(53,47),(54,10),(54,42),(55,31),(55,43),(56,43),(56,65),(57,29),(57,41),(58,30),(58,37),(59,27),(59,28),(60,55),(60,56),(61,34),(61,36),(62,35),(62,63),(63,20),(63,57),(63,64),(64,38),(64,41),(64,58),(65,44),(65,45),(65,59),(66,33),(67,4),(67,66),(68,47),(69,50),(69,70),(70,51)],71)
=> ? ∊ {1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[2,5,1,3,4] => [[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0]]
=> [[1,1,1,1,3],[2,3,3,3],[3,4,4],[4,5],[5]]
=> ([(0,4),(0,8),(1,11),(3,10),(4,9),(5,2),(6,3),(6,12),(7,5),(8,6),(8,9),(9,12),(10,11),(11,7),(12,1),(12,10)],13)
=> ? ∊ {1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[2,5,1,4,3] => [[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,1,0,0,0]]
=> [[1,1,1,1,3],[2,3,3,3],[3,4,5],[4,5],[5]]
=> ([(0,7),(0,8),(0,11),(1,23),(2,5),(2,20),(3,10),(3,21),(4,9),(4,22),(5,6),(5,15),(6,12),(7,4),(7,17),(8,3),(8,16),(9,14),(9,18),(10,13),(10,14),(11,16),(11,17),(13,24),(14,24),(15,12),(16,21),(17,22),(18,23),(18,24),(19,20),(20,15),(21,13),(22,1),(22,18),(23,2),(23,19),(24,19)],25)
=> ? ∊ {1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[2,5,3,1,4] => [[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,1,0,0,0]]
=> [[1,1,1,1,4],[2,3,3,4],[3,4,4],[4,5],[5]]
=> ([(0,17),(0,18),(1,21),(2,20),(3,28),(4,27),(5,23),(6,24),(7,2),(7,30),(8,1),(8,31),(9,15),(10,16),(11,13),(11,32),(12,14),(12,33),(13,3),(13,22),(14,4),(14,22),(15,5),(15,25),(16,6),(16,26),(17,7),(17,19),(18,8),(18,19),(19,30),(19,31),(20,32),(21,33),(22,27),(22,28),(23,29),(24,29),(25,23),(26,24),(27,25),(28,26),(30,11),(30,20),(31,12),(31,21),(32,9),(33,10)],34)
=> ? ∊ {1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[2,5,3,4,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0]]
=> [[1,1,1,1,5],[2,3,3,5],[3,4,5],[4,5],[5]]
=> ([(0,20),(0,21),(1,11),(2,10),(3,27),(3,36),(4,16),(4,34),(5,17),(5,35),(6,26),(6,52),(7,12),(7,53),(8,13),(8,55),(9,15),(9,25),(9,54),(10,51),(11,14),(11,63),(12,38),(13,39),(14,60),(15,19),(15,50),(16,18),(16,64),(17,56),(18,59),(19,22),(19,57),(20,9),(20,58),(21,8),(21,58),(22,47),(22,61),(23,31),(23,46),(24,45),(24,48),(25,37),(25,50),(26,24),(26,49),(26,62),(27,23),(27,61),(27,64),(29,68),(30,68),(31,67),(32,65),(33,69),(34,1),(35,2),(36,6),(37,36),(38,35),(39,34),(40,32),(40,67),(41,51),(41,69),(42,28),(43,28),(44,29),(45,41),(45,66),(46,63),(46,67),(47,52),(48,30),(48,66),(49,48),(49,65),(50,7),(50,57),(51,43),(52,62),(53,5),(53,38),(54,3),(54,37),(55,4),(55,39),(56,33),(56,41),(57,47),(57,53),(58,54),(58,55),(59,32),(59,49),(60,29),(60,30),(61,31),(61,40),(62,45),(62,56),(62,65),(63,44),(63,60),(64,40),(64,46),(64,59),(65,33),(65,66),(66,68),(66,69),(67,44),(68,42),(69,42),(69,43)],70)
=> ? ∊ {1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[2,5,4,1,3] => [[0,0,0,1,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0]]
=> [[1,1,1,1,4],[2,3,4,4],[3,4,5],[4,5],[5]]
=> ([(0,15),(0,16),(0,20),(1,9),(2,13),(2,47),(3,38),(4,14),(4,46),(5,40),(6,10),(6,26),(7,18),(7,45),(8,19),(8,39),(9,25),(10,32),(11,31),(12,44),(13,12),(13,48),(14,11),(14,42),(15,8),(15,28),(16,7),(16,37),(17,24),(17,33),(18,21),(18,36),(19,21),(19,35),(20,28),(20,37),(21,51),(22,49),(23,49),(24,3),(24,50),(26,1),(27,25),(28,39),(29,46),(30,27),(31,30),(32,27),(33,23),(33,50),(34,22),(34,50),(35,40),(35,51),(36,41),(36,51),(37,2),(37,45),(38,43),(39,5),(39,35),(40,4),(40,29),(41,24),(41,34),(42,31),(42,43),(43,30),(43,32),(44,22),(44,23),(45,36),(45,47),(46,38),(46,42),(47,17),(47,41),(47,48),(48,33),(48,34),(48,44),(49,26),(50,6),(50,49),(51,29)],52)
=> ? ∊ {1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[2,5,4,3,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0]]
=> [[1,1,1,1,5],[2,3,4,5],[3,4,5],[4,5],[5]]
=> ([(0,21),(0,22),(0,44),(1,103),(2,102),(3,19),(3,105),(4,20),(4,104),(5,40),(5,60),(6,37),(6,101),(7,42),(7,129),(8,36),(8,132),(9,34),(9,126),(10,41),(10,127),(11,38),(11,108),(12,32),(12,107),(13,43),(13,59),(14,35),(14,100),(15,18),(15,33),(15,106),(16,39),(16,128),(17,119),(18,26),(18,27),(18,99),(19,17),(19,130),(20,113),(21,16),(21,97),(22,11),(22,116),(23,78),(23,120),(24,58),(24,74),(25,57),(25,70),(26,98),(26,124),(27,23),(27,98),(27,115),(28,63),(28,117),(29,64),(29,73),(30,93),(30,95),(31,111),(31,118),(32,50),(32,89),(33,88),(33,99),(34,85),(34,87),(35,30),(35,86),(35,122),(36,84),(36,125),(37,55),(37,96),(38,51),(38,90),(39,51),(39,91),(40,49),(40,92),(41,28),(41,121),(41,123),(42,31),(42,124),(42,131),(43,29),(43,120),(43,130),(44,15),(44,97),(44,116),(45,145),(46,135),(47,133),(48,146),(49,134),(50,141),(51,142),(52,139),(53,136),(54,136),(55,138),(56,147),(57,2),(57,143),(58,1),(58,140),(59,14),(60,6),(61,77),(62,49),(62,145),(63,75),(64,84),(64,144),(65,54),(66,57),(66,133),(67,65),(68,53),(69,53),(70,62),(70,143),(71,126),(72,127),(73,121),(73,144),(74,25),(74,66),(74,140),(75,94),(76,52),(76,146),(77,52),(77,134),(78,100),(79,85),(79,147),(80,81),(80,141),(81,45),(81,143),(82,46),(82,144),(83,47),(83,140),(84,61),(85,68),(86,95),(86,135),(87,65),(88,59),(89,104),(89,141),(90,105),(90,142),(91,110),(91,142),(92,94),(92,134),(93,79),(93,137),(94,55),(94,139),(95,48),(95,137),(96,54),(96,138),(97,7),(97,128),(98,8),(98,114),(99,12),(99,115),(100,122),(101,96),(102,67),(103,109),(104,9),(104,71),(105,10),(105,72),(106,13),(106,88),(107,4),(107,89),(108,3),(108,90),(109,75),(109,92),(110,58),(110,83),(111,50),(111,80),(112,76),(112,77),(113,56),(113,79),(114,80),(114,132),(115,78),(115,107),(116,106),(116,108),(117,48),(117,76),(118,47),(118,66),(119,46),(119,86),(120,64),(120,82),(121,112),(121,117),(122,93),(122,113),(122,135),(123,63),(123,109),(124,111),(124,114),(125,45),(125,62),(126,67),(126,87),(127,103),(127,123),(128,91),(128,129),(129,24),(129,110),(129,131),(130,73),(130,82),(130,119),(131,74),(131,83),(131,118),(132,70),(132,81),(132,125),(133,60),(134,139),(135,56),(135,137),(137,146),(137,147),(138,136),(139,138),(140,5),(140,133),(141,71),(142,72),(143,102),(143,145),(144,61),(144,112),(145,101),(146,69),(147,68),(147,69)],148)
=> ? ∊ {1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[3,1,2,4,5] => [[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,2,2],[2,2,2,3],[3,3,3],[4,4],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[3,1,2,5,4] => [[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,2,2],[2,2,2,3],[3,3,3],[4,5],[5]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> 2
[3,1,4,2,5] => [[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,2,2],[2,2,2,4],[3,3,4],[4,4],[5]]
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> 3
[3,1,4,5,2] => [[0,1,0,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [[1,1,1,2,2],[2,2,2,5],[3,3,5],[4,5],[5]]
=> ([(0,1),(1,3),(1,4),(2,5),(2,18),(3,6),(3,19),(4,7),(4,8),(4,19),(5,13),(6,17),(7,12),(7,15),(8,12),(8,14),(10,21),(11,21),(12,2),(12,20),(13,9),(14,10),(14,20),(15,11),(15,20),(16,9),(17,10),(17,11),(18,13),(18,16),(19,14),(19,15),(19,17),(20,18),(20,21),(21,16)],22)
=> ? ∊ {1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[3,1,5,2,4] => [[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0]]
=> [[1,1,1,2,2],[2,2,2,4],[3,4,4],[4,5],[5]]
=> ([(0,6),(0,7),(1,5),(1,15),(2,4),(2,14),(3,13),(4,12),(5,3),(5,16),(6,9),(7,2),(7,9),(9,1),(9,14),(10,11),(11,8),(12,10),(13,8),(14,12),(14,15),(15,10),(15,16),(16,11),(16,13)],17)
=> ? ∊ {1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[3,1,5,4,2] => [[0,1,0,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]
=> [[1,1,1,2,2],[2,2,2,5],[3,4,5],[4,5],[5]]
=> ([(0,4),(0,5),(1,10),(1,33),(2,9),(2,34),(3,6),(3,7),(3,38),(4,25),(5,3),(5,8),(5,25),(6,19),(6,30),(7,12),(7,19),(7,37),(8,13),(8,35),(8,38),(9,18),(9,31),(10,11),(10,29),(10,36),(11,26),(11,28),(12,22),(12,29),(13,21),(13,32),(14,45),(15,44),(16,46),(17,41),(18,42),(19,2),(19,39),(20,27),(21,20),(22,17),(22,40),(23,17),(23,46),(24,14),(25,1),(25,35),(26,18),(26,43),(27,15),(27,41),(28,15),(28,43),(29,26),(29,40),(30,16),(30,39),(31,14),(31,42),(32,16),(32,23),(33,20),(33,36),(34,24),(34,31),(35,21),(35,33),(36,27),(36,28),(36,40),(37,22),(37,23),(37,39),(38,30),(38,32),(38,37),(39,34),(39,46),(40,41),(40,43),(41,44),(42,45),(43,42),(43,44),(44,45),(46,24)],47)
=> ? ∊ {1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[3,2,1,4,5] => [[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,2,3],[2,2,2,3],[3,3,3],[4,4],[5]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ? ∊ {1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[3,2,1,5,4] => [[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,2,3],[2,2,2,3],[3,3,3],[4,5],[5]]
=> ([(0,3),(0,6),(0,7),(1,8),(1,12),(2,8),(2,11),(3,9),(3,10),(4,2),(4,13),(5,1),(5,14),(6,4),(6,9),(7,5),(7,10),(8,15),(9,13),(10,14),(11,15),(12,15),(13,11),(14,12)],16)
=> ? ∊ {1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[3,2,4,1,5] => [[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,2,4],[2,2,2,4],[3,3,4],[4,4],[5]]
=> ([(0,7),(0,8),(1,12),(2,11),(3,10),(4,10),(4,11),(5,3),(6,1),(6,13),(7,9),(8,5),(9,2),(9,4),(10,14),(11,6),(11,14),(13,12),(14,13)],15)
=> ? ∊ {1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[3,2,4,5,1] => [[0,0,0,0,1],[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [[1,1,1,2,5],[2,2,2,5],[3,3,5],[4,5],[5]]
=> ([(0,11),(0,12),(1,22),(2,19),(3,18),(3,23),(4,14),(4,20),(5,15),(6,15),(6,17),(7,5),(8,1),(8,17),(9,3),(9,21),(9,26),(10,2),(10,16),(11,13),(12,7),(13,6),(13,8),(14,29),(15,24),(16,19),(17,9),(17,22),(17,24),(18,20),(18,28),(20,10),(20,29),(21,23),(21,27),(22,25),(22,26),(23,28),(24,21),(24,25),(25,27),(26,4),(26,18),(26,27),(27,14),(27,28),(28,29),(29,16)],30)
=> ? ∊ {1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[1,2,3,4,5,6] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,3],[4,4,4],[5,5],[6]]
=> ([],1)
=> 1
[1,2,3,4,6,5] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,3],[4,4,4],[5,6],[6]]
=> ([(0,1)],2)
=> 1
[1,2,3,5,4,6] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,3],[4,4,5],[5,5],[6]]
=> ([(0,1)],2)
=> 1
[1,2,3,5,6,4] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,3],[4,4,6],[5,6],[6]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,2,4,3,5,6] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,4],[4,4,4],[5,5],[6]]
=> ([(0,1)],2)
=> 1
[1,2,4,3,6,5] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,4],[4,4,4],[5,6],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,2,4,5,3,6] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,5],[4,4,5],[5,5],[6]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,3,2,4,5,6] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,3],[3,3,3,3],[4,4,4],[5,5],[6]]
=> ([(0,1)],2)
=> 1
[1,3,2,4,6,5] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,3],[3,3,3,3],[4,4,4],[5,6],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
Description
The width of the poset.
This is the size of the poset's longest antichain, also called Dilworth number.
Matching statistic: St000454
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 4% ●values known / values provided: 7%●distinct values known / distinct values provided: 4%
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 4% ●values known / values provided: 7%●distinct values known / distinct values provided: 4%
Values
[1] => [1] => [1] => ([],1)
=> 0 = 1 - 1
[1,2] => [1,2] => [2] => ([],2)
=> 0 = 1 - 1
[2,1] => [2,1] => [2] => ([],2)
=> 0 = 1 - 1
[1,2,3] => [1,2,3] => [3] => ([],3)
=> 0 = 1 - 1
[1,3,2] => [1,3,2] => [1,2] => ([(1,2)],3)
=> 1 = 2 - 1
[2,1,3] => [2,1,3] => [3] => ([],3)
=> 0 = 1 - 1
[2,3,1] => [3,2,1] => [2,1] => ([(0,2),(1,2)],3)
=> ? ∊ {1,1,1} - 1
[3,1,2] => [3,2,1] => [2,1] => ([(0,2),(1,2)],3)
=> ? ∊ {1,1,1} - 1
[3,2,1] => [3,2,1] => [2,1] => ([(0,2),(1,2)],3)
=> ? ∊ {1,1,1} - 1
[1,2,3,4] => [1,2,3,4] => [4] => ([],4)
=> 0 = 1 - 1
[1,2,4,3] => [1,2,4,3] => [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,5,5,6,16} - 1
[1,3,2,4] => [1,3,2,4] => [1,3] => ([(2,3)],4)
=> 1 = 2 - 1
[1,3,4,2] => [1,4,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,5,5,6,16} - 1
[1,4,2,3] => [1,4,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,5,5,6,16} - 1
[1,4,3,2] => [1,4,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,5,5,6,16} - 1
[2,1,3,4] => [2,1,3,4] => [4] => ([],4)
=> 0 = 1 - 1
[2,1,4,3] => [2,1,4,3] => [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,5,5,6,16} - 1
[2,3,1,4] => [3,2,1,4] => [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,5,5,6,16} - 1
[2,3,4,1] => [4,2,3,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,5,5,6,16} - 1
[2,4,1,3] => [3,4,1,2] => [4] => ([],4)
=> 0 = 1 - 1
[2,4,3,1] => [4,3,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,5,5,6,16} - 1
[3,1,2,4] => [3,2,1,4] => [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,5,5,6,16} - 1
[3,1,4,2] => [4,2,3,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,5,5,6,16} - 1
[3,2,1,4] => [3,2,1,4] => [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,5,5,6,16} - 1
[3,2,4,1] => [4,2,3,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,5,5,6,16} - 1
[3,4,1,2] => [4,3,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,5,5,6,16} - 1
[3,4,2,1] => [4,3,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,5,5,6,16} - 1
[4,1,2,3] => [4,2,3,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,5,5,6,16} - 1
[4,1,3,2] => [4,2,3,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,5,5,6,16} - 1
[4,2,1,3] => [4,3,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,5,5,6,16} - 1
[4,2,3,1] => [4,3,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,5,5,6,16} - 1
[4,3,1,2] => [4,3,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,5,5,6,16} - 1
[4,3,2,1] => [4,3,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,5,5,6,16} - 1
[1,2,3,4,5] => [1,2,3,4,5] => [5] => ([],5)
=> 0 = 1 - 1
[1,2,3,5,4] => [1,2,3,5,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
[1,2,4,3,5] => [1,2,4,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
[1,2,4,5,3] => [1,2,5,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
[1,2,5,3,4] => [1,2,5,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
[1,2,5,4,3] => [1,2,5,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
[1,3,2,4,5] => [1,3,2,4,5] => [1,4] => ([(3,4)],5)
=> 1 = 2 - 1
[1,3,2,5,4] => [1,3,2,5,4] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
[1,3,4,2,5] => [1,4,3,2,5] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
[1,3,4,5,2] => [1,5,3,4,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
[1,3,5,2,4] => [1,4,5,2,3] => [1,4] => ([(3,4)],5)
=> 1 = 2 - 1
[1,3,5,4,2] => [1,5,4,3,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
[1,4,2,3,5] => [1,4,3,2,5] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
[1,4,2,5,3] => [1,5,3,4,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
[1,4,3,2,5] => [1,4,3,2,5] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
[1,4,3,5,2] => [1,5,3,4,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
[1,4,5,2,3] => [1,5,4,3,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
[1,4,5,3,2] => [1,5,4,3,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
[1,5,2,3,4] => [1,5,3,4,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
[1,5,2,4,3] => [1,5,3,4,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
[1,5,3,2,4] => [1,5,4,3,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
[1,5,3,4,2] => [1,5,4,3,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
[1,5,4,2,3] => [1,5,4,3,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
[1,5,4,3,2] => [1,5,4,3,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
[2,1,3,4,5] => [2,1,3,4,5] => [5] => ([],5)
=> 0 = 1 - 1
[2,1,3,5,4] => [2,1,3,5,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
[2,1,4,3,5] => [2,1,4,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
[2,1,4,5,3] => [2,1,5,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
[2,1,5,3,4] => [2,1,5,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
[2,1,5,4,3] => [2,1,5,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
[2,3,1,4,5] => [3,2,1,4,5] => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
[2,3,4,5,1] => [5,2,3,4,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 3 - 1
[2,4,1,3,5] => [3,4,1,2,5] => [5] => ([],5)
=> 0 = 1 - 1
[2,4,1,5,3] => [3,5,1,4,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 3 - 1
[2,5,1,3,4] => [3,5,1,4,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 3 - 1
[2,5,1,4,3] => [3,5,1,4,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 3 - 1
[3,1,4,5,2] => [5,2,3,4,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 3 - 1
[3,2,4,5,1] => [5,2,3,4,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 3 - 1
[4,1,2,5,3] => [5,2,3,4,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 3 - 1
[4,1,3,5,2] => [5,2,3,4,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 3 - 1
[5,1,2,3,4] => [5,2,3,4,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 3 - 1
[5,1,2,4,3] => [5,2,3,4,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,2,3,4,5,6] => [1,2,3,4,5,6] => [6] => ([],6)
=> 0 = 1 - 1
[1,2,3,4,6,5] => [1,2,3,4,6,5] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 3 - 1
[1,3,2,4,5,6] => [1,3,2,4,5,6] => [1,5] => ([(4,5)],6)
=> 1 = 2 - 1
[1,3,5,2,4,6] => [1,4,5,2,3,6] => [1,5] => ([(4,5)],6)
=> 1 = 2 - 1
[2,1,3,4,5,6] => [2,1,3,4,5,6] => [6] => ([],6)
=> 0 = 1 - 1
[2,1,3,4,6,5] => [2,1,3,4,6,5] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 3 - 1
[2,3,4,1,6,5] => [4,2,3,1,6,5] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 4 - 1
[2,3,4,5,1,6] => [5,2,3,4,1,6] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 3 - 1
[2,4,1,3,5,6] => [3,4,1,2,5,6] => [6] => ([],6)
=> 0 = 1 - 1
[2,4,1,3,6,5] => [3,4,1,2,6,5] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 3 - 1
[2,4,1,5,3,6] => [3,5,1,4,2,6] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 3 - 1
[2,4,5,6,1,3] => [5,6,3,4,1,2] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 3 - 1
[2,4,6,1,3,5] => [4,5,6,1,2,3] => [6] => ([],6)
=> 0 = 1 - 1
[2,5,1,3,4,6] => [3,5,1,4,2,6] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 3 - 1
[2,5,1,4,3,6] => [3,5,1,4,2,6] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 3 - 1
[2,5,3,6,1,4] => [5,6,3,4,1,2] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 3 - 1
[2,5,4,6,1,3] => [5,6,3,4,1,2] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 3 - 1
[2,6,3,4,1,5] => [5,6,3,4,1,2] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 3 - 1
[2,6,3,5,1,4] => [5,6,3,4,1,2] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 3 - 1
[3,1,4,2,6,5] => [4,2,3,1,6,5] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 4 - 1
[3,1,4,5,2,6] => [5,2,3,4,1,6] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 3 - 1
[3,2,4,1,6,5] => [4,2,3,1,6,5] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 4 - 1
[3,2,4,5,1,6] => [5,2,3,4,1,6] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 3 - 1
[3,5,1,6,2,4] => [5,6,3,4,1,2] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 3 - 1
[3,5,2,6,1,4] => [5,6,3,4,1,2] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 3 - 1
Description
The largest eigenvalue of a graph if it is integral.
If a graph is $d$-regular, then its largest eigenvalue equals $d$. One can show that the largest eigenvalue always lies between the average degree and the maximal degree.
This statistic is undefined if the largest eigenvalue of the graph is not integral.
Matching statistic: St001438
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00123: Dyck paths —Barnabei-Castronuovo involution⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
St001438: Skew partitions ⟶ ℤResult quality: 5% ●values known / values provided: 6%●distinct values known / distinct values provided: 5%
Mp00123: Dyck paths —Barnabei-Castronuovo involution⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
St001438: Skew partitions ⟶ ℤResult quality: 5% ●values known / values provided: 6%●distinct values known / distinct values provided: 5%
Values
[1] => [1,0]
=> [1,0]
=> [[1],[]]
=> 0 = 1 - 1
[1,2] => [1,0,1,0]
=> [1,0,1,0]
=> [[1,1],[]]
=> 0 = 1 - 1
[2,1] => [1,1,0,0]
=> [1,1,0,0]
=> [[2],[]]
=> 0 = 1 - 1
[1,2,3] => [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> [[3],[]]
=> 0 = 1 - 1
[1,3,2] => [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [[2,2],[1]]
=> 1 = 2 - 1
[2,1,3] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [[2,1],[]]
=> 0 = 1 - 1
[2,3,1] => [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [[1,1,1],[]]
=> 0 = 1 - 1
[3,1,2] => [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [[2,2],[]]
=> 0 = 1 - 1
[3,2,1] => [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [[2,2],[]]
=> 0 = 1 - 1
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [[1,1,1,1],[]]
=> 0 = 1 - 1
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [[2,1,1],[]]
=> 0 = 1 - 1
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [[2,2,2],[]]
=> ? ∊ {1,1,3,5,5,6,16} - 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> 0 = 1 - 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> 1 = 2 - 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> 1 = 2 - 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> 2 = 3 - 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> 1 = 2 - 1
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [[3,3],[1]]
=> 1 = 2 - 1
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [[4],[]]
=> 0 = 1 - 1
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> 2 = 3 - 1
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> 2 = 3 - 1
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> 0 = 1 - 1
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [[3,1],[]]
=> 0 = 1 - 1
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> 0 = 1 - 1
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [[3,1],[]]
=> 0 = 1 - 1
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> 1 = 2 - 1
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> 1 = 2 - 1
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> ? ∊ {1,1,3,5,5,6,16} - 1
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> ? ∊ {1,1,3,5,5,6,16} - 1
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> ? ∊ {1,1,3,5,5,6,16} - 1
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> ? ∊ {1,1,3,5,5,6,16} - 1
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> ? ∊ {1,1,3,5,5,6,16} - 1
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> ? ∊ {1,1,3,5,5,6,16} - 1
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [[5],[]]
=> 0 = 1 - 1
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [[4,4],[2]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [[4,4],[3]]
=> 3 = 4 - 1
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3],[2,2]]
=> 4 = 5 - 1
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [[4,3],[2]]
=> 2 = 3 - 1
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [[4,3],[2]]
=> 2 = 3 - 1
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [[4,1],[]]
=> 0 = 1 - 1
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1],[1]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1],[2]]
=> 2 = 3 - 1
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1],[1,1]]
=> 2 = 3 - 1
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> 1 = 2 - 1
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> 1 = 2 - 1
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [[4,4],[]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [[4,4],[]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
[1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
[1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
[1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
[1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
[1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
[1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
[2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [[4,2],[]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
[2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
[2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2],[2]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
[2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
[2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
[2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
[2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [[3,1,1],[]]
=> 0 = 1 - 1
[2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1],[]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
[2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1],[1]]
=> 1 = 2 - 1
[2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [[1,1,1,1,1],[]]
=> 0 = 1 - 1
[2,3,5,1,4] => [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1],[]]
=> 0 = 1 - 1
[2,3,5,4,1] => [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1],[]]
=> 0 = 1 - 1
[2,4,1,3,5] => [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [[3,3,2],[]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
[2,4,1,5,3] => [1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [[2,2,2,2],[]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
[2,4,3,1,5] => [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [[3,3,2],[]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
[2,4,3,5,1] => [1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [[2,2,2,2],[]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
[2,4,5,1,3] => [1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [[3,2,2],[]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
[2,4,5,3,1] => [1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [[3,2,2],[]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
[2,5,1,3,4] => [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[2,2,2,2],[1]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
[2,5,1,4,3] => [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[2,2,2,2],[1]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
[2,5,3,1,4] => [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[2,2,2,2],[1]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
[2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[2,2,2,2],[1]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
[2,5,4,1,3] => [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[2,2,2,2],[1]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
[2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[2,2,2,2],[1]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
[3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [[4,2],[1]]
=> 1 = 2 - 1
[3,1,2,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
[3,1,4,2,5] => [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> 3 = 4 - 1
[3,1,4,5,2] => [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2],[1,1,1]]
=> 3 = 4 - 1
[3,1,5,2,4] => [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2],[1,1]]
=> 2 = 3 - 1
[3,1,5,4,2] => [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2],[1,1]]
=> 2 = 3 - 1
[3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [[4,2],[1]]
=> 1 = 2 - 1
[3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
[3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> 3 = 4 - 1
[3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2],[1,1,1]]
=> 3 = 4 - 1
[3,2,5,1,4] => [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2],[1,1]]
=> 2 = 3 - 1
[3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [[4,4],[1]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
[3,4,1,5,2] => [1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [[3,3,3],[1,1]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
[3,4,2,1,5] => [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [[4,4],[1]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
[3,4,2,5,1] => [1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [[3,3,3],[1,1]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
[3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [[4,3],[1]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
[3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [[4,3],[1]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
[3,5,1,2,4] => [1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
[3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
Description
The number of missing boxes of a skew partition.
Matching statistic: St001879
Mp00252: Permutations —restriction⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001879: Posets ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 7%
Mp00065: Permutations —permutation poset⟶ Posets
St001879: Posets ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 7%
Values
[1] => [] => ([],0)
=> ? = 1
[1,2] => [1] => ([],1)
=> ? ∊ {1,1}
[2,1] => [1] => ([],1)
=> ? ∊ {1,1}
[1,2,3] => [1,2] => ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,2}
[1,3,2] => [1,2] => ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,2}
[2,1,3] => [2,1] => ([],2)
=> ? ∊ {1,1,1,1,1,2}
[2,3,1] => [2,1] => ([],2)
=> ? ∊ {1,1,1,1,1,2}
[3,1,2] => [1,2] => ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,2}
[3,2,1] => [2,1] => ([],2)
=> ? ∊ {1,1,1,1,1,2}
[1,2,3,4] => [1,2,3] => ([(0,2),(2,1)],3)
=> 2
[1,2,4,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 2
[1,3,2,4] => [1,3,2] => ([(0,1),(0,2)],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,5,5,6,16}
[1,3,4,2] => [1,3,2] => ([(0,1),(0,2)],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,5,5,6,16}
[1,4,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 2
[1,4,3,2] => [1,3,2] => ([(0,1),(0,2)],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,5,5,6,16}
[2,1,3,4] => [2,1,3] => ([(0,2),(1,2)],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,5,5,6,16}
[2,1,4,3] => [2,1,3] => ([(0,2),(1,2)],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,5,5,6,16}
[2,3,1,4] => [2,3,1] => ([(1,2)],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,5,5,6,16}
[2,3,4,1] => [2,3,1] => ([(1,2)],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,5,5,6,16}
[2,4,1,3] => [2,1,3] => ([(0,2),(1,2)],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,5,5,6,16}
[2,4,3,1] => [2,3,1] => ([(1,2)],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,5,5,6,16}
[3,1,2,4] => [3,1,2] => ([(1,2)],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,5,5,6,16}
[3,1,4,2] => [3,1,2] => ([(1,2)],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,5,5,6,16}
[3,2,1,4] => [3,2,1] => ([],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,5,5,6,16}
[3,2,4,1] => [3,2,1] => ([],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,5,5,6,16}
[3,4,1,2] => [3,1,2] => ([(1,2)],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,5,5,6,16}
[3,4,2,1] => [3,2,1] => ([],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,5,5,6,16}
[4,1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 2
[4,1,3,2] => [1,3,2] => ([(0,1),(0,2)],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,5,5,6,16}
[4,2,1,3] => [2,1,3] => ([(0,2),(1,2)],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,5,5,6,16}
[4,2,3,1] => [2,3,1] => ([(1,2)],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,5,5,6,16}
[4,3,1,2] => [3,1,2] => ([(1,2)],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,5,5,6,16}
[4,3,2,1] => [3,2,1] => ([],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,5,5,6,16}
[1,2,3,4,5] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 3
[1,2,3,5,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 3
[1,2,4,3,5] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[1,2,4,5,3] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[1,2,5,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 3
[1,2,5,4,3] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[1,3,2,4,5] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[1,3,2,5,4] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[1,3,4,2,5] => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[1,3,4,5,2] => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[1,3,5,2,4] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[1,3,5,4,2] => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[1,4,2,3,5] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[1,4,2,5,3] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[1,4,3,2,5] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[1,4,3,5,2] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[1,4,5,2,3] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[1,4,5,3,2] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[1,5,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 3
[1,5,2,4,3] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[1,5,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[1,5,3,4,2] => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[1,5,4,2,3] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[1,5,4,3,2] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[2,1,3,4,5] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[2,1,3,5,4] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[2,1,4,3,5] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[2,1,4,5,3] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[2,1,5,3,4] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[5,1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 3
[5,1,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[1,2,3,4,5,6] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[1,2,3,4,6,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[1,2,3,6,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[1,2,4,3,5,6] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 5
[1,2,4,3,6,5] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 5
[1,2,4,6,3,5] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 5
[1,2,6,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[1,2,6,4,3,5] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 5
[1,3,2,4,5,6] => [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 5
[1,3,2,4,6,5] => [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 5
[1,3,2,6,4,5] => [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 5
[1,3,4,2,5,6] => [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 6
[1,3,4,2,6,5] => [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 6
[1,3,4,6,2,5] => [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 6
[1,3,6,2,4,5] => [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 5
[1,3,6,4,2,5] => [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 6
[1,4,2,3,5,6] => [1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 6
[1,4,2,3,6,5] => [1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 6
[1,4,2,6,3,5] => [1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 6
[1,4,3,2,5,6] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 9
[1,4,3,2,6,5] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 9
[1,4,3,6,2,5] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 9
[1,4,6,2,3,5] => [1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 6
[1,4,6,3,2,5] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 9
[1,6,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[1,6,2,4,3,5] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 5
[1,6,3,2,4,5] => [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 5
[1,6,3,4,2,5] => [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 6
[1,6,4,2,3,5] => [1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 6
[1,6,4,3,2,5] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 9
[6,1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[6,1,2,4,3,5] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 5
[6,1,3,2,4,5] => [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 5
[6,1,3,4,2,5] => [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 6
[6,1,4,2,3,5] => [1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 6
[6,1,4,3,2,5] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 9
Description
The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice.
Matching statistic: St001435
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
St001435: Skew partitions ⟶ ℤResult quality: 4% ●values known / values provided: 6%●distinct values known / distinct values provided: 4%
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
St001435: Skew partitions ⟶ ℤResult quality: 4% ●values known / values provided: 6%●distinct values known / distinct values provided: 4%
Values
[1] => [1,0]
=> [1,0]
=> [[1],[]]
=> 0 = 1 - 1
[1,2] => [1,0,1,0]
=> [1,1,0,0]
=> [[2],[]]
=> 0 = 1 - 1
[2,1] => [1,1,0,0]
=> [1,0,1,0]
=> [[1,1],[]]
=> 0 = 1 - 1
[1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [[2,2],[]]
=> 0 = 1 - 1
[1,3,2] => [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [[2,2],[1]]
=> 1 = 2 - 1
[2,1,3] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [[2,1],[]]
=> 0 = 1 - 1
[2,3,1] => [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [[1,1,1],[]]
=> 0 = 1 - 1
[3,1,2] => [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> [[3],[]]
=> 0 = 1 - 1
[3,2,1] => [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> [[3],[]]
=> 0 = 1 - 1
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> ? ∊ {1,3,3,5,5,6,16} - 1
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> 1 = 2 - 1
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> 1 = 2 - 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> 1 = 2 - 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> 0 = 1 - 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> 0 = 1 - 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> 0 = 1 - 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> 1 = 2 - 1
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [[2,1,1],[]]
=> 0 = 1 - 1
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [[1,1,1,1],[]]
=> 0 = 1 - 1
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [[3,1],[]]
=> 0 = 1 - 1
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [[3,1],[]]
=> 0 = 1 - 1
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [[3,3],[1]]
=> 1 = 2 - 1
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> 2 = 3 - 1
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [[3,3],[1]]
=> 1 = 2 - 1
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> 2 = 3 - 1
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [[4],[]]
=> 0 = 1 - 1
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [[4],[]]
=> 0 = 1 - 1
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [[2,2,2],[]]
=> ? ∊ {1,3,3,5,5,6,16} - 1
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [[2,2,2],[]]
=> ? ∊ {1,3,3,5,5,6,16} - 1
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [[2,2,2],[]]
=> ? ∊ {1,3,3,5,5,6,16} - 1
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [[2,2,2],[]]
=> ? ∊ {1,3,3,5,5,6,16} - 1
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [[2,2,2],[]]
=> ? ∊ {1,3,3,5,5,6,16} - 1
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [[2,2,2],[]]
=> ? ∊ {1,3,3,5,5,6,16} - 1
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> 2 = 3 - 1
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2],[1,1]]
=> 1 = 2 - 1
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2],[1,1,1]]
=> 1 = 2 - 1
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [[4,2],[1]]
=> 1 = 2 - 1
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [[4,2],[1]]
=> 1 = 2 - 1
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2],[2]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2],[2]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [[4,2],[]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [[4,2],[]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
[1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
[1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
[1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
[1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
[1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
[1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
[2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
[2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
[2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> 1 = 2 - 1
[2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1],[1,1]]
=> 1 = 2 - 1
[2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1],[]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
[2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1],[]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
[2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1],[]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
[2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1],[1]]
=> 1 = 2 - 1
[2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1],[]]
=> 0 = 1 - 1
[2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [[1,1,1,1,1],[]]
=> 0 = 1 - 1
[2,3,5,1,4] => [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [[3,1,1],[]]
=> 0 = 1 - 1
[2,3,5,4,1] => [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [[3,1,1],[]]
=> 0 = 1 - 1
[2,4,1,3,5] => [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1],[1]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
[2,4,1,5,3] => [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1],[2]]
=> 2 = 3 - 1
[2,4,3,1,5] => [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1],[1]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
[2,4,3,5,1] => [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1],[2]]
=> 2 = 3 - 1
[2,4,5,1,3] => [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [[4,1],[]]
=> 0 = 1 - 1
[2,4,5,3,1] => [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [[4,1],[]]
=> 0 = 1 - 1
[2,5,1,3,4] => [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1],[]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
[2,5,1,4,3] => [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1],[]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
[2,5,3,1,4] => [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1],[]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
[2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1],[]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
[2,5,4,1,3] => [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1],[]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
[2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1],[]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
[3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [[4,4],[1]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
[3,1,2,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
[3,1,4,2,5] => [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [[4,3],[2]]
=> 2 = 3 - 1
[3,1,4,5,2] => [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3],[2,2]]
=> 2 = 3 - 1
[3,1,5,2,4] => [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [[4,3],[1]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
[3,1,5,4,2] => [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [[4,3],[1]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
[3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [[4,4],[1]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
[3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
[3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [[4,3],[2]]
=> 2 = 3 - 1
[3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3],[2,2]]
=> 2 = 3 - 1
[3,2,5,1,4] => [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [[4,3],[1]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
[3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [[4,3],[1]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
[3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [[4,4],[2]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
[3,4,1,5,2] => [1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [[4,4],[3]]
=> 3 = 4 - 1
[3,4,2,1,5] => [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [[4,4],[2]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
[3,4,2,5,1] => [1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [[4,4],[3]]
=> 3 = 4 - 1
[3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [[5],[]]
=> 0 = 1 - 1
[3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [[5],[]]
=> 0 = 1 - 1
[3,5,1,2,4] => [1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [[3,3,3],[1,1]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
Description
The number of missing boxes in the first row.
Matching statistic: St001633
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
St001633: Posets ⟶ ℤResult quality: 2% ●values known / values provided: 6%●distinct values known / distinct values provided: 2%
Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
St001633: Posets ⟶ ℤResult quality: 2% ●values known / values provided: 6%●distinct values known / distinct values provided: 2%
Values
[1] => [[1]]
=> [[1]]
=> ([],1)
=> 0 = 1 - 1
[1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> ([],1)
=> 0 = 1 - 1
[2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> ([(0,1)],2)
=> 0 = 1 - 1
[1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ([],1)
=> 0 = 1 - 1
[1,3,2] => [[1,0,0],[0,0,1],[0,1,0]]
=> [[1,1,1],[2,3],[3]]
=> ([(0,1)],2)
=> 0 = 1 - 1
[2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
=> [[1,1,2],[2,2],[3]]
=> ([(0,1)],2)
=> 0 = 1 - 1
[2,3,1] => [[0,0,1],[1,0,0],[0,1,0]]
=> [[1,1,3],[2,3],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[3,1,2] => [[0,1,0],[0,0,1],[1,0,0]]
=> [[1,2,2],[2,3],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[3,2,1] => [[0,0,1],[0,1,0],[1,0,0]]
=> [[1,2,3],[2,3],[3]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ? = 2 - 1
[1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,2],[3,3],[4]]
=> ([],1)
=> 0 = 1 - 1
[1,2,4,3] => [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [[1,1,1,1],[2,2,2],[3,4],[4]]
=> ([(0,1)],2)
=> 0 = 1 - 1
[1,3,2,4] => [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,3],[3,3],[4]]
=> ([(0,1)],2)
=> 0 = 1 - 1
[1,3,4,2] => [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> [[1,1,1,1],[2,2,4],[3,4],[4]]
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[1,4,2,3] => [[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]]
=> [[1,1,1,1],[2,3,3],[3,4],[4]]
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[1,4,3,2] => [[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> [[1,1,1,1],[2,3,4],[3,4],[4]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ? ∊ {1,1,2,2,2,2,3,3,3,3,5,5,6,16} - 1
[2,1,3,4] => [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,2],[2,2,2],[3,3],[4]]
=> ([(0,1)],2)
=> 0 = 1 - 1
[2,1,4,3] => [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> [[1,1,1,2],[2,2,2],[3,4],[4]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[2,3,1,4] => [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> [[1,1,1,3],[2,2,3],[3,3],[4]]
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[2,3,4,1] => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [[1,1,1,4],[2,2,4],[3,4],[4]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ? ∊ {1,1,2,2,2,2,3,3,3,3,5,5,6,16} - 1
[2,4,1,3] => [[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
=> [[1,1,1,3],[2,3,3],[3,4],[4]]
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> 1 = 2 - 1
[2,4,3,1] => [[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
=> [[1,1,1,4],[2,3,4],[3,4],[4]]
=> ([(0,5),(0,10),(1,16),(2,15),(3,14),(4,13),(5,12),(6,2),(6,13),(7,4),(7,14),(8,1),(9,6),(10,11),(10,12),(11,3),(11,7),(12,9),(13,15),(14,8),(15,16)],17)
=> ? ∊ {1,1,2,2,2,2,3,3,3,3,5,5,6,16} - 1
[3,1,2,4] => [[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]]
=> [[1,1,2,2],[2,2,3],[3,3],[4]]
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[3,1,4,2] => [[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
=> [[1,1,2,2],[2,2,4],[3,4],[4]]
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> ? ∊ {1,1,2,2,2,2,3,3,3,3,5,5,6,16} - 1
[3,2,1,4] => [[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> [[1,1,2,3],[2,2,3],[3,3],[4]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ? ∊ {1,1,2,2,2,2,3,3,3,3,5,5,6,16} - 1
[3,2,4,1] => [[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
=> [[1,1,2,4],[2,2,4],[3,4],[4]]
=> ([(0,7),(0,8),(1,12),(2,11),(3,10),(4,10),(4,11),(5,3),(6,1),(6,13),(7,9),(8,5),(9,2),(9,4),(10,14),(11,6),(11,14),(13,12),(14,13)],15)
=> ? ∊ {1,1,2,2,2,2,3,3,3,3,5,5,6,16} - 1
[3,4,1,2] => [[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> [[1,1,3,3],[2,3,4],[3,4],[4]]
=> ([(0,8),(2,11),(2,12),(3,10),(4,9),(5,4),(5,14),(6,3),(6,14),(7,1),(8,5),(8,6),(9,11),(9,13),(10,12),(10,13),(11,15),(12,15),(13,15),(14,2),(14,9),(14,10),(15,7)],16)
=> ? ∊ {1,1,2,2,2,2,3,3,3,3,5,5,6,16} - 1
[3,4,2,1] => [[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
=> [[1,1,3,4],[2,3,4],[3,4],[4]]
=> ([(0,13),(0,15),(1,17),(2,16),(2,17),(3,19),(4,16),(4,18),(5,22),(6,21),(7,20),(8,23),(8,29),(9,4),(9,28),(10,3),(10,28),(11,6),(12,7),(12,24),(13,14),(14,1),(14,2),(15,9),(15,10),(16,25),(17,12),(17,25),(18,26),(18,29),(19,23),(19,26),(20,27),(22,27),(23,30),(24,20),(24,22),(25,24),(26,30),(27,21),(28,8),(28,18),(28,19),(29,5),(29,30),(30,11)],31)
=> ? ∊ {1,1,2,2,2,2,3,3,3,3,5,5,6,16} - 1
[4,1,2,3] => [[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> [[1,2,2,2],[2,3,3],[3,4],[4]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ? ∊ {1,1,2,2,2,2,3,3,3,3,5,5,6,16} - 1
[4,1,3,2] => [[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]]
=> [[1,2,2,2],[2,3,4],[3,4],[4]]
=> ([(0,7),(0,8),(1,12),(2,11),(3,10),(4,10),(4,11),(5,3),(6,1),(6,13),(7,9),(8,5),(9,2),(9,4),(10,14),(11,6),(11,14),(13,12),(14,13)],15)
=> ? ∊ {1,1,2,2,2,2,3,3,3,3,5,5,6,16} - 1
[4,2,1,3] => [[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]]
=> [[1,2,2,3],[2,3,3],[3,4],[4]]
=> ([(0,5),(0,10),(1,16),(2,15),(3,14),(4,13),(5,12),(6,2),(6,13),(7,4),(7,14),(8,1),(9,6),(10,11),(10,12),(11,3),(11,7),(12,9),(13,15),(14,8),(15,16)],17)
=> ? ∊ {1,1,2,2,2,2,3,3,3,3,5,5,6,16} - 1
[4,2,3,1] => [[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
=> [[1,2,2,4],[2,3,4],[3,4],[4]]
=> ([(0,14),(0,15),(1,19),(2,18),(3,29),(4,30),(5,22),(6,23),(7,24),(7,25),(8,9),(9,7),(9,18),(9,19),(10,5),(11,6),(12,2),(12,29),(13,1),(13,30),(14,16),(14,28),(15,17),(15,28),(16,3),(16,12),(17,4),(17,13),(18,24),(18,27),(19,25),(19,27),(20,26),(21,26),(22,20),(23,21),(24,22),(24,31),(25,23),(25,31),(27,31),(28,8),(29,10),(30,11),(31,20),(31,21)],32)
=> ? ∊ {1,1,2,2,2,2,3,3,3,3,5,5,6,16} - 1
[4,3,1,2] => [[0,0,1,0],[0,0,0,1],[0,1,0,0],[1,0,0,0]]
=> [[1,2,3,3],[2,3,4],[3,4],[4]]
=> ([(0,13),(0,15),(1,17),(2,16),(2,17),(3,19),(4,16),(4,18),(5,22),(6,21),(7,20),(8,23),(8,29),(9,4),(9,28),(10,3),(10,28),(11,6),(12,7),(12,24),(13,14),(14,1),(14,2),(15,9),(15,10),(16,25),(17,12),(17,25),(18,26),(18,29),(19,23),(19,26),(20,27),(22,27),(23,30),(24,20),(24,22),(25,24),(26,30),(27,21),(28,8),(28,18),(28,19),(29,5),(29,30),(30,11)],31)
=> ? ∊ {1,1,2,2,2,2,3,3,3,3,5,5,6,16} - 1
[4,3,2,1] => [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> [[1,2,3,4],[2,3,4],[3,4],[4]]
=> ([(0,5),(0,16),(0,17),(1,23),(2,19),(3,11),(3,21),(4,10),(4,20),(5,12),(5,13),(6,50),(7,51),(8,24),(8,58),(9,25),(9,59),(10,14),(10,52),(11,15),(11,53),(12,26),(12,60),(13,27),(13,60),(14,54),(15,55),(16,4),(16,29),(17,3),(17,29),(18,48),(18,49),(19,32),(19,33),(20,46),(20,52),(21,47),(21,53),(22,34),(22,35),(23,18),(23,54),(23,55),(24,40),(24,42),(25,41),(25,43),(26,46),(26,56),(27,47),(27,57),(28,63),(29,1),(30,62),(31,61),(32,61),(33,61),(34,6),(34,62),(35,7),(35,62),(36,58),(37,59),(38,32),(39,33),(40,44),(41,45),(42,38),(43,39),(44,31),(45,31),(46,36),(47,37),(48,40),(48,63),(49,41),(49,63),(50,38),(51,39),(52,8),(52,36),(53,9),(53,37),(54,28),(54,48),(55,28),(55,49),(56,30),(56,34),(57,30),(57,35),(58,42),(58,50),(59,43),(59,51),(60,22),(60,56),(60,57),(62,2),(63,44),(63,45)],64)
=> ? ∊ {1,1,2,2,2,2,3,3,3,3,5,5,6,16} - 1
[1,2,3,4,5] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,2,2],[3,3,3],[4,4],[5]]
=> ([],1)
=> 0 = 1 - 1
[1,2,3,5,4] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,1,1],[2,2,2,2],[3,3,3],[4,5],[5]]
=> ([(0,1)],2)
=> 0 = 1 - 1
[1,2,4,3,5] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,2,2],[3,3,4],[4,4],[5]]
=> ([(0,1)],2)
=> 0 = 1 - 1
[1,2,4,5,3] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [[1,1,1,1,1],[2,2,2,2],[3,3,5],[4,5],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[1,2,5,3,4] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,0,1,0,0]]
=> [[1,1,1,1,1],[2,2,2,2],[3,4,4],[4,5],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[1,2,5,4,3] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [[1,1,1,1,1],[2,2,2,2],[3,4,5],[4,5],[5]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
[1,3,2,4,5] => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,2,3],[3,3,3],[4,4],[5]]
=> ([(0,1)],2)
=> 0 = 1 - 1
[1,3,2,5,4] => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,1,1],[2,2,2,3],[3,3,3],[4,5],[5]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[1,3,4,2,5] => [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,2,4],[3,3,4],[4,4],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[1,3,4,5,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [[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)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
[1,3,5,2,4] => [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0]]
=> [[1,1,1,1,1],[2,2,2,4],[3,4,4],[4,5],[5]]
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> 1 = 2 - 1
[1,3,5,4,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]
=> [[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)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
[1,4,2,3,5] => [[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,3,3],[3,3,4],[4,4],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[1,4,2,5,3] => [[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [[1,1,1,1,1],[2,2,3,3],[3,3,5],[4,5],[5]]
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
[1,4,3,2,5] => [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
[1,4,3,5,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [[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)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
[1,4,5,2,3] => [[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0]]
=> [[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)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
[1,4,5,3,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0]]
=> [[1,1,1,1,1],[2,2,4,5],[3,4,5],[4,5],[5]]
=> ([(0,13),(0,15),(1,17),(2,16),(2,17),(3,19),(4,16),(4,18),(5,22),(6,21),(7,20),(8,23),(8,29),(9,4),(9,28),(10,3),(10,28),(11,6),(12,7),(12,24),(13,14),(14,1),(14,2),(15,9),(15,10),(16,25),(17,12),(17,25),(18,26),(18,29),(19,23),(19,26),(20,27),(22,27),(23,30),(24,20),(24,22),(25,24),(26,30),(27,21),(28,8),(28,18),(28,19),(29,5),(29,30),(30,11)],31)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
[1,5,2,3,4] => [[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0]]
=> [[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)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
[1,5,2,4,3] => [[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,1,0,0,0]]
=> [[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)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
[1,5,3,2,4] => [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1],[0,1,0,0,0]]
=> [[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)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
[1,5,3,4,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0]]
=> [[1,1,1,1,1],[2,3,3,5],[3,4,5],[4,5],[5]]
=> ([(0,14),(0,15),(1,19),(2,18),(3,29),(4,30),(5,22),(6,23),(7,24),(7,25),(8,9),(9,7),(9,18),(9,19),(10,5),(11,6),(12,2),(12,29),(13,1),(13,30),(14,16),(14,28),(15,17),(15,28),(16,3),(16,12),(17,4),(17,13),(18,24),(18,27),(19,25),(19,27),(20,26),(21,26),(22,20),(23,21),(24,22),(24,31),(25,23),(25,31),(27,31),(28,8),(29,10),(30,11),(31,20),(31,21)],32)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
[1,5,4,2,3] => [[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0]]
=> [[1,1,1,1,1],[2,3,4,4],[3,4,5],[4,5],[5]]
=> ([(0,13),(0,15),(1,17),(2,16),(2,17),(3,19),(4,16),(4,18),(5,22),(6,21),(7,20),(8,23),(8,29),(9,4),(9,28),(10,3),(10,28),(11,6),(12,7),(12,24),(13,14),(14,1),(14,2),(15,9),(15,10),(16,25),(17,12),(17,25),(18,26),(18,29),(19,23),(19,26),(20,27),(22,27),(23,30),(24,20),(24,22),(25,24),(26,30),(27,21),(28,8),(28,18),(28,19),(29,5),(29,30),(30,11)],31)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
[1,5,4,3,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0]]
=> [[1,1,1,1,1],[2,3,4,5],[3,4,5],[4,5],[5]]
=> ([(0,5),(0,16),(0,17),(1,23),(2,19),(3,11),(3,21),(4,10),(4,20),(5,12),(5,13),(6,50),(7,51),(8,24),(8,58),(9,25),(9,59),(10,14),(10,52),(11,15),(11,53),(12,26),(12,60),(13,27),(13,60),(14,54),(15,55),(16,4),(16,29),(17,3),(17,29),(18,48),(18,49),(19,32),(19,33),(20,46),(20,52),(21,47),(21,53),(22,34),(22,35),(23,18),(23,54),(23,55),(24,40),(24,42),(25,41),(25,43),(26,46),(26,56),(27,47),(27,57),(28,63),(29,1),(30,62),(31,61),(32,61),(33,61),(34,6),(34,62),(35,7),(35,62),(36,58),(37,59),(38,32),(39,33),(40,44),(41,45),(42,38),(43,39),(44,31),(45,31),(46,36),(47,37),(48,40),(48,63),(49,41),(49,63),(50,38),(51,39),(52,8),(52,36),(53,9),(53,37),(54,28),(54,48),(55,28),(55,49),(56,30),(56,34),(57,30),(57,35),(58,42),(58,50),(59,43),(59,51),(60,22),(60,56),(60,57),(62,2),(63,44),(63,45)],64)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
[2,1,3,4,5] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,2],[2,2,2,2],[3,3,3],[4,4],[5]]
=> ([(0,1)],2)
=> 0 = 1 - 1
[2,1,3,5,4] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,1,2],[2,2,2,2],[3,3,3],[4,5],[5]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[2,1,4,3,5] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,2],[2,2,2,2],[3,3,4],[4,4],[5]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[2,1,4,5,3] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [[1,1,1,1,2],[2,2,2,2],[3,3,5],[4,5],[5]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
[2,1,5,3,4] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,0,1,0,0]]
=> [[1,1,1,1,2],[2,2,2,2],[3,4,4],[4,5],[5]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
[2,1,5,4,3] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [[1,1,1,1,2],[2,2,2,2],[3,4,5],[4,5],[5]]
=> ([(0,3),(0,6),(0,7),(1,8),(1,12),(2,8),(2,11),(3,9),(3,10),(4,2),(4,13),(5,1),(5,14),(6,4),(6,9),(7,5),(7,10),(8,15),(9,13),(10,14),(11,15),(12,15),(13,11),(14,12)],16)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
[2,3,1,4,5] => [[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,3],[2,2,2,3],[3,3,3],[4,4],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[2,3,1,5,4] => [[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,1,3],[2,2,2,3],[3,3,3],[4,5],[5]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
[2,3,4,1,5] => [[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,4],[2,2,2,4],[3,3,4],[4,4],[5]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
[2,3,4,5,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [[1,1,1,1,5],[2,2,2,5],[3,3,5],[4,5],[5]]
=> ([(0,9),(1,13),(2,14),(4,12),(5,11),(6,1),(6,12),(7,3),(8,5),(8,15),(9,10),(10,4),(10,6),(11,14),(12,8),(12,13),(13,15),(14,7),(15,2),(15,11)],16)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
[2,3,5,1,4] => [[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0]]
=> [[1,1,1,1,4],[2,2,2,4],[3,4,4],[4,5],[5]]
=> ([(0,4),(0,8),(1,11),(3,10),(4,9),(5,2),(6,3),(6,12),(7,5),(8,6),(8,9),(9,12),(10,11),(11,7),(12,1),(12,10)],13)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
[2,3,5,4,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]
=> [[1,1,1,1,5],[2,2,2,5],[3,4,5],[4,5],[5]]
=> ([(0,8),(0,16),(1,19),(1,20),(2,21),(3,23),(4,26),(5,24),(6,22),(7,27),(8,18),(9,10),(9,20),(10,4),(10,31),(11,2),(11,30),(12,3),(13,7),(13,29),(14,11),(14,28),(15,6),(15,25),(16,17),(16,18),(17,1),(17,9),(17,33),(18,33),(19,24),(20,13),(20,31),(21,32),(22,32),(24,14),(25,12),(26,28),(27,25),(28,30),(29,15),(29,27),(30,21),(30,22),(31,26),(31,29),(32,23),(33,5),(33,19)],34)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
[2,4,1,3,5] => [[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,3],[2,2,3,3],[3,3,4],[4,4],[5]]
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> 1 = 2 - 1
[2,4,1,5,3] => [[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [[1,1,1,1,3],[2,2,3,3],[3,3,5],[4,5],[5]]
=> ([(0,6),(0,7),(1,5),(1,15),(2,4),(2,14),(3,13),(4,12),(5,3),(5,16),(6,9),(7,2),(7,9),(9,1),(9,14),(10,11),(11,8),(12,10),(13,8),(14,12),(14,15),(15,10),(15,16),(16,11),(16,13)],17)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
[2,4,3,1,5] => [[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,4],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ([(0,5),(0,10),(1,16),(2,15),(3,14),(4,13),(5,12),(6,2),(6,13),(7,4),(7,14),(8,1),(9,6),(10,11),(10,12),(11,3),(11,7),(12,9),(13,15),(14,8),(15,16)],17)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
[2,4,3,5,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [[1,1,1,1,5],[2,2,3,5],[3,3,5],[4,5],[5]]
=> ([(0,8),(0,13),(1,19),(2,16),(2,18),(3,21),(4,26),(5,24),(6,16),(6,22),(7,20),(7,23),(8,17),(9,4),(9,18),(10,9),(11,7),(11,30),(12,3),(12,25),(13,14),(13,17),(14,1),(14,15),(15,2),(15,6),(15,19),(16,28),(17,10),(18,26),(18,28),(19,11),(19,22),(20,24),(20,31),(22,30),(23,31),(24,12),(24,27),(25,21),(26,23),(26,29),(27,25),(28,29),(29,31),(30,5),(30,20),(31,27)],32)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
[2,4,5,1,3] => [[0,0,0,1,0],[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0]]
=> [[1,1,1,1,4],[2,2,4,4],[3,4,5],[4,5],[5]]
=> ([(0,9),(0,10),(1,2),(3,7),(3,23),(4,6),(4,22),(5,15),(6,16),(7,8),(7,24),(8,20),(9,19),(10,4),(10,19),(11,14),(11,18),(12,26),(13,26),(14,25),(15,1),(16,21),(17,13),(17,25),(18,12),(18,25),(19,3),(19,22),(20,12),(20,13),(21,14),(21,17),(22,16),(22,23),(23,11),(23,21),(23,24),(24,17),(24,18),(24,20),(25,5),(25,26),(26,15)],27)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
[2,4,5,3,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0]]
=> [[1,1,1,1,5],[2,2,4,5],[3,4,5],[4,5],[5]]
=> ([(0,17),(0,18),(1,11),(1,12),(2,53),(3,50),(4,33),(5,16),(5,65),(6,13),(6,61),(7,14),(7,62),(8,15),(8,51),(9,25),(9,63),(10,24),(10,52),(11,54),(12,22),(12,23),(12,54),(13,36),(14,35),(15,40),(16,59),(17,1),(17,48),(18,7),(18,48),(19,32),(19,45),(20,29),(20,38),(21,55),(21,58),(22,49),(22,60),(23,42),(23,49),(24,31),(24,46),(25,21),(25,60),(25,64),(27,70),(28,70),(29,67),(30,66),(31,68),(32,3),(32,69),(33,8),(34,39),(35,57),(36,39),(37,32),(37,66),(38,19),(38,37),(38,67),(39,26),(40,26),(41,30),(41,67),(42,52),(43,44),(43,68),(44,28),(44,69),(45,27),(45,69),(46,53),(46,68),(47,61),(48,9),(48,62),(49,5),(49,56),(50,34),(51,40),(52,2),(52,46),(53,6),(53,47),(54,10),(54,42),(55,31),(55,43),(56,43),(56,65),(57,29),(57,41),(58,30),(58,37),(59,27),(59,28),(60,55),(60,56),(61,34),(61,36),(62,35),(62,63),(63,20),(63,57),(63,64),(64,38),(64,41),(64,58),(65,44),(65,45),(65,59),(66,33),(67,4),(67,66),(68,47),(69,50),(69,70),(70,51)],71)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
[2,5,1,3,4] => [[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0]]
=> [[1,1,1,1,3],[2,3,3,3],[3,4,4],[4,5],[5]]
=> ([(0,4),(0,8),(1,11),(3,10),(4,9),(5,2),(6,3),(6,12),(7,5),(8,6),(8,9),(9,12),(10,11),(11,7),(12,1),(12,10)],13)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
[2,5,1,4,3] => [[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,1,0,0,0]]
=> [[1,1,1,1,3],[2,3,3,3],[3,4,5],[4,5],[5]]
=> ([(0,7),(0,8),(0,11),(1,23),(2,5),(2,20),(3,10),(3,21),(4,9),(4,22),(5,6),(5,15),(6,12),(7,4),(7,17),(8,3),(8,16),(9,14),(9,18),(10,13),(10,14),(11,16),(11,17),(13,24),(14,24),(15,12),(16,21),(17,22),(18,23),(18,24),(19,20),(20,15),(21,13),(22,1),(22,18),(23,2),(23,19),(24,19)],25)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
[2,5,3,1,4] => [[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,1,0,0,0]]
=> [[1,1,1,1,4],[2,3,3,4],[3,4,4],[4,5],[5]]
=> ([(0,17),(0,18),(1,21),(2,20),(3,28),(4,27),(5,23),(6,24),(7,2),(7,30),(8,1),(8,31),(9,15),(10,16),(11,13),(11,32),(12,14),(12,33),(13,3),(13,22),(14,4),(14,22),(15,5),(15,25),(16,6),(16,26),(17,7),(17,19),(18,8),(18,19),(19,30),(19,31),(20,32),(21,33),(22,27),(22,28),(23,29),(24,29),(25,23),(26,24),(27,25),(28,26),(30,11),(30,20),(31,12),(31,21),(32,9),(33,10)],34)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
[2,5,3,4,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0]]
=> [[1,1,1,1,5],[2,3,3,5],[3,4,5],[4,5],[5]]
=> ([(0,20),(0,21),(1,11),(2,10),(3,27),(3,36),(4,16),(4,34),(5,17),(5,35),(6,26),(6,52),(7,12),(7,53),(8,13),(8,55),(9,15),(9,25),(9,54),(10,51),(11,14),(11,63),(12,38),(13,39),(14,60),(15,19),(15,50),(16,18),(16,64),(17,56),(18,59),(19,22),(19,57),(20,9),(20,58),(21,8),(21,58),(22,47),(22,61),(23,31),(23,46),(24,45),(24,48),(25,37),(25,50),(26,24),(26,49),(26,62),(27,23),(27,61),(27,64),(29,68),(30,68),(31,67),(32,65),(33,69),(34,1),(35,2),(36,6),(37,36),(38,35),(39,34),(40,32),(40,67),(41,51),(41,69),(42,28),(43,28),(44,29),(45,41),(45,66),(46,63),(46,67),(47,52),(48,30),(48,66),(49,48),(49,65),(50,7),(50,57),(51,43),(52,62),(53,5),(53,38),(54,3),(54,37),(55,4),(55,39),(56,33),(56,41),(57,47),(57,53),(58,54),(58,55),(59,32),(59,49),(60,29),(60,30),(61,31),(61,40),(62,45),(62,56),(62,65),(63,44),(63,60),(64,40),(64,46),(64,59),(65,33),(65,66),(66,68),(66,69),(67,44),(68,42),(69,42),(69,43)],70)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
[2,5,4,1,3] => [[0,0,0,1,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0]]
=> [[1,1,1,1,4],[2,3,4,4],[3,4,5],[4,5],[5]]
=> ([(0,15),(0,16),(0,20),(1,9),(2,13),(2,47),(3,38),(4,14),(4,46),(5,40),(6,10),(6,26),(7,18),(7,45),(8,19),(8,39),(9,25),(10,32),(11,31),(12,44),(13,12),(13,48),(14,11),(14,42),(15,8),(15,28),(16,7),(16,37),(17,24),(17,33),(18,21),(18,36),(19,21),(19,35),(20,28),(20,37),(21,51),(22,49),(23,49),(24,3),(24,50),(26,1),(27,25),(28,39),(29,46),(30,27),(31,30),(32,27),(33,23),(33,50),(34,22),(34,50),(35,40),(35,51),(36,41),(36,51),(37,2),(37,45),(38,43),(39,5),(39,35),(40,4),(40,29),(41,24),(41,34),(42,31),(42,43),(43,30),(43,32),(44,22),(44,23),(45,36),(45,47),(46,38),(46,42),(47,17),(47,41),(47,48),(48,33),(48,34),(48,44),(49,26),(50,6),(50,49),(51,29)],52)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
[2,5,4,3,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0]]
=> [[1,1,1,1,5],[2,3,4,5],[3,4,5],[4,5],[5]]
=> ([(0,21),(0,22),(0,44),(1,103),(2,102),(3,19),(3,105),(4,20),(4,104),(5,40),(5,60),(6,37),(6,101),(7,42),(7,129),(8,36),(8,132),(9,34),(9,126),(10,41),(10,127),(11,38),(11,108),(12,32),(12,107),(13,43),(13,59),(14,35),(14,100),(15,18),(15,33),(15,106),(16,39),(16,128),(17,119),(18,26),(18,27),(18,99),(19,17),(19,130),(20,113),(21,16),(21,97),(22,11),(22,116),(23,78),(23,120),(24,58),(24,74),(25,57),(25,70),(26,98),(26,124),(27,23),(27,98),(27,115),(28,63),(28,117),(29,64),(29,73),(30,93),(30,95),(31,111),(31,118),(32,50),(32,89),(33,88),(33,99),(34,85),(34,87),(35,30),(35,86),(35,122),(36,84),(36,125),(37,55),(37,96),(38,51),(38,90),(39,51),(39,91),(40,49),(40,92),(41,28),(41,121),(41,123),(42,31),(42,124),(42,131),(43,29),(43,120),(43,130),(44,15),(44,97),(44,116),(45,145),(46,135),(47,133),(48,146),(49,134),(50,141),(51,142),(52,139),(53,136),(54,136),(55,138),(56,147),(57,2),(57,143),(58,1),(58,140),(59,14),(60,6),(61,77),(62,49),(62,145),(63,75),(64,84),(64,144),(65,54),(66,57),(66,133),(67,65),(68,53),(69,53),(70,62),(70,143),(71,126),(72,127),(73,121),(73,144),(74,25),(74,66),(74,140),(75,94),(76,52),(76,146),(77,52),(77,134),(78,100),(79,85),(79,147),(80,81),(80,141),(81,45),(81,143),(82,46),(82,144),(83,47),(83,140),(84,61),(85,68),(86,95),(86,135),(87,65),(88,59),(89,104),(89,141),(90,105),(90,142),(91,110),(91,142),(92,94),(92,134),(93,79),(93,137),(94,55),(94,139),(95,48),(95,137),(96,54),(96,138),(97,7),(97,128),(98,8),(98,114),(99,12),(99,115),(100,122),(101,96),(102,67),(103,109),(104,9),(104,71),(105,10),(105,72),(106,13),(106,88),(107,4),(107,89),(108,3),(108,90),(109,75),(109,92),(110,58),(110,83),(111,50),(111,80),(112,76),(112,77),(113,56),(113,79),(114,80),(114,132),(115,78),(115,107),(116,106),(116,108),(117,48),(117,76),(118,47),(118,66),(119,46),(119,86),(120,64),(120,82),(121,112),(121,117),(122,93),(122,113),(122,135),(123,63),(123,109),(124,111),(124,114),(125,45),(125,62),(126,67),(126,87),(127,103),(127,123),(128,91),(128,129),(129,24),(129,110),(129,131),(130,73),(130,82),(130,119),(131,74),(131,83),(131,118),(132,70),(132,81),(132,125),(133,60),(134,139),(135,56),(135,137),(137,146),(137,147),(138,136),(139,138),(140,5),(140,133),(141,71),(142,72),(143,102),(143,145),(144,61),(144,112),(145,101),(146,69),(147,68),(147,69)],148)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
[3,1,2,4,5] => [[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,2,2],[2,2,2,3],[3,3,3],[4,4],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[3,1,2,5,4] => [[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,2,2],[2,2,2,3],[3,3,3],[4,5],[5]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
[3,1,4,2,5] => [[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,2,2],[2,2,2,4],[3,3,4],[4,4],[5]]
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768} - 1
[1,2,3,4,5,6] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,3],[4,4,4],[5,5],[6]]
=> ([],1)
=> 0 = 1 - 1
[1,2,3,4,6,5] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,3],[4,4,4],[5,6],[6]]
=> ([(0,1)],2)
=> 0 = 1 - 1
[1,2,3,5,4,6] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,3],[4,4,5],[5,5],[6]]
=> ([(0,1)],2)
=> 0 = 1 - 1
[1,2,3,5,6,4] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,3],[4,4,6],[5,6],[6]]
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[1,2,4,3,5,6] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,4],[4,4,4],[5,5],[6]]
=> ([(0,1)],2)
=> 0 = 1 - 1
[1,2,4,3,6,5] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,4],[4,4,4],[5,6],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[1,2,4,5,3,6] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,5],[4,4,5],[5,5],[6]]
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[1,3,2,4,5,6] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,3],[3,3,3,3],[4,4,4],[5,5],[6]]
=> ([(0,1)],2)
=> 0 = 1 - 1
[1,3,2,4,6,5] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,3],[3,3,3,3],[4,4,4],[5,6],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[1,3,2,5,4,6] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,3],[3,3,3,3],[4,4,5],[5,5],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[1,3,4,2,5,6] => [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,4],[3,3,3,4],[4,4,4],[5,5],[6]]
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[2,1,3,4,5,6] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,3],[4,4,4],[5,5],[6]]
=> ([(0,1)],2)
=> 0 = 1 - 1
[2,1,3,4,6,5] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,3],[4,4,4],[5,6],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[2,1,3,5,4,6] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,3],[4,4,5],[5,5],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[2,1,4,3,5,6] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,4],[4,4,4],[5,5],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[2,3,1,4,5,6] => [[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,3],[2,2,2,2,3],[3,3,3,3],[4,4,4],[5,5],[6]]
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
Description
The number of simple modules with projective dimension two in the incidence algebra of the poset.
Matching statistic: St000181
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
St000181: Posets ⟶ ℤResult quality: 1% ●values known / values provided: 5%●distinct values known / distinct values provided: 1%
Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
St000181: Posets ⟶ ℤResult quality: 1% ●values known / values provided: 5%●distinct values known / distinct values provided: 1%
Values
[1] => [[1]]
=> [[1]]
=> ([],1)
=> 1
[1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> ([],1)
=> 1
[2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> ([(0,1)],2)
=> 1
[1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ([],1)
=> 1
[1,3,2] => [[1,0,0],[0,0,1],[0,1,0]]
=> [[1,1,1],[2,3],[3]]
=> ([(0,1)],2)
=> 1
[2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
=> [[1,1,2],[2,2],[3]]
=> ([(0,1)],2)
=> 1
[2,3,1] => [[0,0,1],[1,0,0],[0,1,0]]
=> [[1,1,3],[2,3],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[3,1,2] => [[0,1,0],[0,0,1],[1,0,0]]
=> [[1,2,2],[2,3],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[3,2,1] => [[0,0,1],[0,1,0],[1,0,0]]
=> [[1,2,3],[2,3],[3]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ? = 2
[1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,2],[3,3],[4]]
=> ([],1)
=> 1
[1,2,4,3] => [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [[1,1,1,1],[2,2,2],[3,4],[4]]
=> ([(0,1)],2)
=> 1
[1,3,2,4] => [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,3],[3,3],[4]]
=> ([(0,1)],2)
=> 1
[1,3,4,2] => [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> [[1,1,1,1],[2,2,4],[3,4],[4]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,4,2,3] => [[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]]
=> [[1,1,1,1],[2,3,3],[3,4],[4]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,4,3,2] => [[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> [[1,1,1,1],[2,3,4],[3,4],[4]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,5,5,6,16}
[2,1,3,4] => [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,2],[2,2,2],[3,3],[4]]
=> ([(0,1)],2)
=> 1
[2,1,4,3] => [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> [[1,1,1,2],[2,2,2],[3,4],[4]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[2,3,1,4] => [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> [[1,1,1,3],[2,2,3],[3,3],[4]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[2,3,4,1] => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [[1,1,1,4],[2,2,4],[3,4],[4]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,5,5,6,16}
[2,4,1,3] => [[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
=> [[1,1,1,3],[2,3,3],[3,4],[4]]
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,5,5,6,16}
[2,4,3,1] => [[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
=> [[1,1,1,4],[2,3,4],[3,4],[4]]
=> ([(0,5),(0,10),(1,16),(2,15),(3,14),(4,13),(5,12),(6,2),(6,13),(7,4),(7,14),(8,1),(9,6),(10,11),(10,12),(11,3),(11,7),(12,9),(13,15),(14,8),(15,16)],17)
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,5,5,6,16}
[3,1,2,4] => [[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]]
=> [[1,1,2,2],[2,2,3],[3,3],[4]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[3,1,4,2] => [[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
=> [[1,1,2,2],[2,2,4],[3,4],[4]]
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,5,5,6,16}
[3,2,1,4] => [[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> [[1,1,2,3],[2,2,3],[3,3],[4]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,5,5,6,16}
[3,2,4,1] => [[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
=> [[1,1,2,4],[2,2,4],[3,4],[4]]
=> ([(0,7),(0,8),(1,12),(2,11),(3,10),(4,10),(4,11),(5,3),(6,1),(6,13),(7,9),(8,5),(9,2),(9,4),(10,14),(11,6),(11,14),(13,12),(14,13)],15)
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,5,5,6,16}
[3,4,1,2] => [[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> [[1,1,3,3],[2,3,4],[3,4],[4]]
=> ([(0,8),(2,11),(2,12),(3,10),(4,9),(5,4),(5,14),(6,3),(6,14),(7,1),(8,5),(8,6),(9,11),(9,13),(10,12),(10,13),(11,15),(12,15),(13,15),(14,2),(14,9),(14,10),(15,7)],16)
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,5,5,6,16}
[3,4,2,1] => [[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
=> [[1,1,3,4],[2,3,4],[3,4],[4]]
=> ([(0,13),(0,15),(1,17),(2,16),(2,17),(3,19),(4,16),(4,18),(5,22),(6,21),(7,20),(8,23),(8,29),(9,4),(9,28),(10,3),(10,28),(11,6),(12,7),(12,24),(13,14),(14,1),(14,2),(15,9),(15,10),(16,25),(17,12),(17,25),(18,26),(18,29),(19,23),(19,26),(20,27),(22,27),(23,30),(24,20),(24,22),(25,24),(26,30),(27,21),(28,8),(28,18),(28,19),(29,5),(29,30),(30,11)],31)
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,5,5,6,16}
[4,1,2,3] => [[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> [[1,2,2,2],[2,3,3],[3,4],[4]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,5,5,6,16}
[4,1,3,2] => [[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]]
=> [[1,2,2,2],[2,3,4],[3,4],[4]]
=> ([(0,7),(0,8),(1,12),(2,11),(3,10),(4,10),(4,11),(5,3),(6,1),(6,13),(7,9),(8,5),(9,2),(9,4),(10,14),(11,6),(11,14),(13,12),(14,13)],15)
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,5,5,6,16}
[4,2,1,3] => [[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]]
=> [[1,2,2,3],[2,3,3],[3,4],[4]]
=> ([(0,5),(0,10),(1,16),(2,15),(3,14),(4,13),(5,12),(6,2),(6,13),(7,4),(7,14),(8,1),(9,6),(10,11),(10,12),(11,3),(11,7),(12,9),(13,15),(14,8),(15,16)],17)
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,5,5,6,16}
[4,2,3,1] => [[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
=> [[1,2,2,4],[2,3,4],[3,4],[4]]
=> ([(0,14),(0,15),(1,19),(2,18),(3,29),(4,30),(5,22),(6,23),(7,24),(7,25),(8,9),(9,7),(9,18),(9,19),(10,5),(11,6),(12,2),(12,29),(13,1),(13,30),(14,16),(14,28),(15,17),(15,28),(16,3),(16,12),(17,4),(17,13),(18,24),(18,27),(19,25),(19,27),(20,26),(21,26),(22,20),(23,21),(24,22),(24,31),(25,23),(25,31),(27,31),(28,8),(29,10),(30,11),(31,20),(31,21)],32)
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,5,5,6,16}
[4,3,1,2] => [[0,0,1,0],[0,0,0,1],[0,1,0,0],[1,0,0,0]]
=> [[1,2,3,3],[2,3,4],[3,4],[4]]
=> ([(0,13),(0,15),(1,17),(2,16),(2,17),(3,19),(4,16),(4,18),(5,22),(6,21),(7,20),(8,23),(8,29),(9,4),(9,28),(10,3),(10,28),(11,6),(12,7),(12,24),(13,14),(14,1),(14,2),(15,9),(15,10),(16,25),(17,12),(17,25),(18,26),(18,29),(19,23),(19,26),(20,27),(22,27),(23,30),(24,20),(24,22),(25,24),(26,30),(27,21),(28,8),(28,18),(28,19),(29,5),(29,30),(30,11)],31)
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,5,5,6,16}
[4,3,2,1] => [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> [[1,2,3,4],[2,3,4],[3,4],[4]]
=> ([(0,5),(0,16),(0,17),(1,23),(2,19),(3,11),(3,21),(4,10),(4,20),(5,12),(5,13),(6,50),(7,51),(8,24),(8,58),(9,25),(9,59),(10,14),(10,52),(11,15),(11,53),(12,26),(12,60),(13,27),(13,60),(14,54),(15,55),(16,4),(16,29),(17,3),(17,29),(18,48),(18,49),(19,32),(19,33),(20,46),(20,52),(21,47),(21,53),(22,34),(22,35),(23,18),(23,54),(23,55),(24,40),(24,42),(25,41),(25,43),(26,46),(26,56),(27,47),(27,57),(28,63),(29,1),(30,62),(31,61),(32,61),(33,61),(34,6),(34,62),(35,7),(35,62),(36,58),(37,59),(38,32),(39,33),(40,44),(41,45),(42,38),(43,39),(44,31),(45,31),(46,36),(47,37),(48,40),(48,63),(49,41),(49,63),(50,38),(51,39),(52,8),(52,36),(53,9),(53,37),(54,28),(54,48),(55,28),(55,49),(56,30),(56,34),(57,30),(57,35),(58,42),(58,50),(59,43),(59,51),(60,22),(60,56),(60,57),(62,2),(63,44),(63,45)],64)
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,3,5,5,6,16}
[1,2,3,4,5] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,2,2],[3,3,3],[4,4],[5]]
=> ([],1)
=> 1
[1,2,3,5,4] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,1,1],[2,2,2,2],[3,3,3],[4,5],[5]]
=> ([(0,1)],2)
=> 1
[1,2,4,3,5] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,2,2],[3,3,4],[4,4],[5]]
=> ([(0,1)],2)
=> 1
[1,2,4,5,3] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [[1,1,1,1,1],[2,2,2,2],[3,3,5],[4,5],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,2,5,3,4] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,0,1,0,0]]
=> [[1,1,1,1,1],[2,2,2,2],[3,4,4],[4,5],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,2,5,4,3] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [[1,1,1,1,1],[2,2,2,2],[3,4,5],[4,5],[5]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[1,3,2,4,5] => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,2,3],[3,3,3],[4,4],[5]]
=> ([(0,1)],2)
=> 1
[1,3,2,5,4] => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,1,1],[2,2,2,3],[3,3,3],[4,5],[5]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[1,3,4,2,5] => [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,2,4],[3,3,4],[4,4],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,3,4,5,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [[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)
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[1,3,5,2,4] => [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0]]
=> [[1,1,1,1,1],[2,2,2,4],[3,4,4],[4,5],[5]]
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[1,3,5,4,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]
=> [[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)
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[1,4,2,3,5] => [[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,3,3],[3,3,4],[4,4],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,4,2,5,3] => [[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [[1,1,1,1,1],[2,2,3,3],[3,3,5],[4,5],[5]]
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[1,4,3,2,5] => [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[1,4,3,5,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [[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)
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[1,4,5,2,3] => [[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0]]
=> [[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)
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[1,4,5,3,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0]]
=> [[1,1,1,1,1],[2,2,4,5],[3,4,5],[4,5],[5]]
=> ([(0,13),(0,15),(1,17),(2,16),(2,17),(3,19),(4,16),(4,18),(5,22),(6,21),(7,20),(8,23),(8,29),(9,4),(9,28),(10,3),(10,28),(11,6),(12,7),(12,24),(13,14),(14,1),(14,2),(15,9),(15,10),(16,25),(17,12),(17,25),(18,26),(18,29),(19,23),(19,26),(20,27),(22,27),(23,30),(24,20),(24,22),(25,24),(26,30),(27,21),(28,8),(28,18),(28,19),(29,5),(29,30),(30,11)],31)
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[1,5,2,3,4] => [[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0]]
=> [[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)
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[1,5,2,4,3] => [[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,1,0,0,0]]
=> [[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)
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[1,5,3,2,4] => [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1],[0,1,0,0,0]]
=> [[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)
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[1,5,3,4,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0]]
=> [[1,1,1,1,1],[2,3,3,5],[3,4,5],[4,5],[5]]
=> ([(0,14),(0,15),(1,19),(2,18),(3,29),(4,30),(5,22),(6,23),(7,24),(7,25),(8,9),(9,7),(9,18),(9,19),(10,5),(11,6),(12,2),(12,29),(13,1),(13,30),(14,16),(14,28),(15,17),(15,28),(16,3),(16,12),(17,4),(17,13),(18,24),(18,27),(19,25),(19,27),(20,26),(21,26),(22,20),(23,21),(24,22),(24,31),(25,23),(25,31),(27,31),(28,8),(29,10),(30,11),(31,20),(31,21)],32)
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[1,5,4,2,3] => [[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0]]
=> [[1,1,1,1,1],[2,3,4,4],[3,4,5],[4,5],[5]]
=> ([(0,13),(0,15),(1,17),(2,16),(2,17),(3,19),(4,16),(4,18),(5,22),(6,21),(7,20),(8,23),(8,29),(9,4),(9,28),(10,3),(10,28),(11,6),(12,7),(12,24),(13,14),(14,1),(14,2),(15,9),(15,10),(16,25),(17,12),(17,25),(18,26),(18,29),(19,23),(19,26),(20,27),(22,27),(23,30),(24,20),(24,22),(25,24),(26,30),(27,21),(28,8),(28,18),(28,19),(29,5),(29,30),(30,11)],31)
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[1,5,4,3,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0]]
=> [[1,1,1,1,1],[2,3,4,5],[3,4,5],[4,5],[5]]
=> ([(0,5),(0,16),(0,17),(1,23),(2,19),(3,11),(3,21),(4,10),(4,20),(5,12),(5,13),(6,50),(7,51),(8,24),(8,58),(9,25),(9,59),(10,14),(10,52),(11,15),(11,53),(12,26),(12,60),(13,27),(13,60),(14,54),(15,55),(16,4),(16,29),(17,3),(17,29),(18,48),(18,49),(19,32),(19,33),(20,46),(20,52),(21,47),(21,53),(22,34),(22,35),(23,18),(23,54),(23,55),(24,40),(24,42),(25,41),(25,43),(26,46),(26,56),(27,47),(27,57),(28,63),(29,1),(30,62),(31,61),(32,61),(33,61),(34,6),(34,62),(35,7),(35,62),(36,58),(37,59),(38,32),(39,33),(40,44),(41,45),(42,38),(43,39),(44,31),(45,31),(46,36),(47,37),(48,40),(48,63),(49,41),(49,63),(50,38),(51,39),(52,8),(52,36),(53,9),(53,37),(54,28),(54,48),(55,28),(55,49),(56,30),(56,34),(57,30),(57,35),(58,42),(58,50),(59,43),(59,51),(60,22),(60,56),(60,57),(62,2),(63,44),(63,45)],64)
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[2,1,3,4,5] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,2],[2,2,2,2],[3,3,3],[4,4],[5]]
=> ([(0,1)],2)
=> 1
[2,1,3,5,4] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,1,2],[2,2,2,2],[3,3,3],[4,5],[5]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[2,1,4,3,5] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,2],[2,2,2,2],[3,3,4],[4,4],[5]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[2,1,4,5,3] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [[1,1,1,1,2],[2,2,2,2],[3,3,5],[4,5],[5]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[2,1,5,3,4] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,0,1,0,0]]
=> [[1,1,1,1,2],[2,2,2,2],[3,4,4],[4,5],[5]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[2,1,5,4,3] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [[1,1,1,1,2],[2,2,2,2],[3,4,5],[4,5],[5]]
=> ([(0,3),(0,6),(0,7),(1,8),(1,12),(2,8),(2,11),(3,9),(3,10),(4,2),(4,13),(5,1),(5,14),(6,4),(6,9),(7,5),(7,10),(8,15),(9,13),(10,14),(11,15),(12,15),(13,11),(14,12)],16)
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[2,3,1,4,5] => [[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,3],[2,2,2,3],[3,3,3],[4,4],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[2,3,1,5,4] => [[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,1,3],[2,2,2,3],[3,3,3],[4,5],[5]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[2,3,4,1,5] => [[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,4],[2,2,2,4],[3,3,4],[4,4],[5]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[2,3,4,5,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [[1,1,1,1,5],[2,2,2,5],[3,3,5],[4,5],[5]]
=> ([(0,9),(1,13),(2,14),(4,12),(5,11),(6,1),(6,12),(7,3),(8,5),(8,15),(9,10),(10,4),(10,6),(11,14),(12,8),(12,13),(13,15),(14,7),(15,2),(15,11)],16)
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[2,3,5,1,4] => [[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0]]
=> [[1,1,1,1,4],[2,2,2,4],[3,4,4],[4,5],[5]]
=> ([(0,4),(0,8),(1,11),(3,10),(4,9),(5,2),(6,3),(6,12),(7,5),(8,6),(8,9),(9,12),(10,11),(11,7),(12,1),(12,10)],13)
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[2,3,5,4,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]
=> [[1,1,1,1,5],[2,2,2,5],[3,4,5],[4,5],[5]]
=> ([(0,8),(0,16),(1,19),(1,20),(2,21),(3,23),(4,26),(5,24),(6,22),(7,27),(8,18),(9,10),(9,20),(10,4),(10,31),(11,2),(11,30),(12,3),(13,7),(13,29),(14,11),(14,28),(15,6),(15,25),(16,17),(16,18),(17,1),(17,9),(17,33),(18,33),(19,24),(20,13),(20,31),(21,32),(22,32),(24,14),(25,12),(26,28),(27,25),(28,30),(29,15),(29,27),(30,21),(30,22),(31,26),(31,29),(32,23),(33,5),(33,19)],34)
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[2,4,1,3,5] => [[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,3],[2,2,3,3],[3,3,4],[4,4],[5]]
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[2,4,1,5,3] => [[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [[1,1,1,1,3],[2,2,3,3],[3,3,5],[4,5],[5]]
=> ([(0,6),(0,7),(1,5),(1,15),(2,4),(2,14),(3,13),(4,12),(5,3),(5,16),(6,9),(7,2),(7,9),(9,1),(9,14),(10,11),(11,8),(12,10),(13,8),(14,12),(14,15),(15,10),(15,16),(16,11),(16,13)],17)
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[2,4,3,1,5] => [[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,4],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ([(0,5),(0,10),(1,16),(2,15),(3,14),(4,13),(5,12),(6,2),(6,13),(7,4),(7,14),(8,1),(9,6),(10,11),(10,12),(11,3),(11,7),(12,9),(13,15),(14,8),(15,16)],17)
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[2,4,3,5,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [[1,1,1,1,5],[2,2,3,5],[3,3,5],[4,5],[5]]
=> ([(0,8),(0,13),(1,19),(2,16),(2,18),(3,21),(4,26),(5,24),(6,16),(6,22),(7,20),(7,23),(8,17),(9,4),(9,18),(10,9),(11,7),(11,30),(12,3),(12,25),(13,14),(13,17),(14,1),(14,15),(15,2),(15,6),(15,19),(16,28),(17,10),(18,26),(18,28),(19,11),(19,22),(20,24),(20,31),(22,30),(23,31),(24,12),(24,27),(25,21),(26,23),(26,29),(27,25),(28,29),(29,31),(30,5),(30,20),(31,27)],32)
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[2,4,5,1,3] => [[0,0,0,1,0],[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0]]
=> [[1,1,1,1,4],[2,2,4,4],[3,4,5],[4,5],[5]]
=> ([(0,9),(0,10),(1,2),(3,7),(3,23),(4,6),(4,22),(5,15),(6,16),(7,8),(7,24),(8,20),(9,19),(10,4),(10,19),(11,14),(11,18),(12,26),(13,26),(14,25),(15,1),(16,21),(17,13),(17,25),(18,12),(18,25),(19,3),(19,22),(20,12),(20,13),(21,14),(21,17),(22,16),(22,23),(23,11),(23,21),(23,24),(24,17),(24,18),(24,20),(25,5),(25,26),(26,15)],27)
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[2,4,5,3,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0]]
=> [[1,1,1,1,5],[2,2,4,5],[3,4,5],[4,5],[5]]
=> ([(0,17),(0,18),(1,11),(1,12),(2,53),(3,50),(4,33),(5,16),(5,65),(6,13),(6,61),(7,14),(7,62),(8,15),(8,51),(9,25),(9,63),(10,24),(10,52),(11,54),(12,22),(12,23),(12,54),(13,36),(14,35),(15,40),(16,59),(17,1),(17,48),(18,7),(18,48),(19,32),(19,45),(20,29),(20,38),(21,55),(21,58),(22,49),(22,60),(23,42),(23,49),(24,31),(24,46),(25,21),(25,60),(25,64),(27,70),(28,70),(29,67),(30,66),(31,68),(32,3),(32,69),(33,8),(34,39),(35,57),(36,39),(37,32),(37,66),(38,19),(38,37),(38,67),(39,26),(40,26),(41,30),(41,67),(42,52),(43,44),(43,68),(44,28),(44,69),(45,27),(45,69),(46,53),(46,68),(47,61),(48,9),(48,62),(49,5),(49,56),(50,34),(51,40),(52,2),(52,46),(53,6),(53,47),(54,10),(54,42),(55,31),(55,43),(56,43),(56,65),(57,29),(57,41),(58,30),(58,37),(59,27),(59,28),(60,55),(60,56),(61,34),(61,36),(62,35),(62,63),(63,20),(63,57),(63,64),(64,38),(64,41),(64,58),(65,44),(65,45),(65,59),(66,33),(67,4),(67,66),(68,47),(69,50),(69,70),(70,51)],71)
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[2,5,1,3,4] => [[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0]]
=> [[1,1,1,1,3],[2,3,3,3],[3,4,4],[4,5],[5]]
=> ([(0,4),(0,8),(1,11),(3,10),(4,9),(5,2),(6,3),(6,12),(7,5),(8,6),(8,9),(9,12),(10,11),(11,7),(12,1),(12,10)],13)
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[2,5,1,4,3] => [[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,1,0,0,0]]
=> [[1,1,1,1,3],[2,3,3,3],[3,4,5],[4,5],[5]]
=> ([(0,7),(0,8),(0,11),(1,23),(2,5),(2,20),(3,10),(3,21),(4,9),(4,22),(5,6),(5,15),(6,12),(7,4),(7,17),(8,3),(8,16),(9,14),(9,18),(10,13),(10,14),(11,16),(11,17),(13,24),(14,24),(15,12),(16,21),(17,22),(18,23),(18,24),(19,20),(20,15),(21,13),(22,1),(22,18),(23,2),(23,19),(24,19)],25)
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[2,5,3,1,4] => [[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,1,0,0,0]]
=> [[1,1,1,1,4],[2,3,3,4],[3,4,4],[4,5],[5]]
=> ([(0,17),(0,18),(1,21),(2,20),(3,28),(4,27),(5,23),(6,24),(7,2),(7,30),(8,1),(8,31),(9,15),(10,16),(11,13),(11,32),(12,14),(12,33),(13,3),(13,22),(14,4),(14,22),(15,5),(15,25),(16,6),(16,26),(17,7),(17,19),(18,8),(18,19),(19,30),(19,31),(20,32),(21,33),(22,27),(22,28),(23,29),(24,29),(25,23),(26,24),(27,25),(28,26),(30,11),(30,20),(31,12),(31,21),(32,9),(33,10)],34)
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[2,5,3,4,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0]]
=> [[1,1,1,1,5],[2,3,3,5],[3,4,5],[4,5],[5]]
=> ([(0,20),(0,21),(1,11),(2,10),(3,27),(3,36),(4,16),(4,34),(5,17),(5,35),(6,26),(6,52),(7,12),(7,53),(8,13),(8,55),(9,15),(9,25),(9,54),(10,51),(11,14),(11,63),(12,38),(13,39),(14,60),(15,19),(15,50),(16,18),(16,64),(17,56),(18,59),(19,22),(19,57),(20,9),(20,58),(21,8),(21,58),(22,47),(22,61),(23,31),(23,46),(24,45),(24,48),(25,37),(25,50),(26,24),(26,49),(26,62),(27,23),(27,61),(27,64),(29,68),(30,68),(31,67),(32,65),(33,69),(34,1),(35,2),(36,6),(37,36),(38,35),(39,34),(40,32),(40,67),(41,51),(41,69),(42,28),(43,28),(44,29),(45,41),(45,66),(46,63),(46,67),(47,52),(48,30),(48,66),(49,48),(49,65),(50,7),(50,57),(51,43),(52,62),(53,5),(53,38),(54,3),(54,37),(55,4),(55,39),(56,33),(56,41),(57,47),(57,53),(58,54),(58,55),(59,32),(59,49),(60,29),(60,30),(61,31),(61,40),(62,45),(62,56),(62,65),(63,44),(63,60),(64,40),(64,46),(64,59),(65,33),(65,66),(66,68),(66,69),(67,44),(68,42),(69,42),(69,43)],70)
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[2,5,4,1,3] => [[0,0,0,1,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0]]
=> [[1,1,1,1,4],[2,3,4,4],[3,4,5],[4,5],[5]]
=> ([(0,15),(0,16),(0,20),(1,9),(2,13),(2,47),(3,38),(4,14),(4,46),(5,40),(6,10),(6,26),(7,18),(7,45),(8,19),(8,39),(9,25),(10,32),(11,31),(12,44),(13,12),(13,48),(14,11),(14,42),(15,8),(15,28),(16,7),(16,37),(17,24),(17,33),(18,21),(18,36),(19,21),(19,35),(20,28),(20,37),(21,51),(22,49),(23,49),(24,3),(24,50),(26,1),(27,25),(28,39),(29,46),(30,27),(31,30),(32,27),(33,23),(33,50),(34,22),(34,50),(35,40),(35,51),(36,41),(36,51),(37,2),(37,45),(38,43),(39,5),(39,35),(40,4),(40,29),(41,24),(41,34),(42,31),(42,43),(43,30),(43,32),(44,22),(44,23),(45,36),(45,47),(46,38),(46,42),(47,17),(47,41),(47,48),(48,33),(48,34),(48,44),(49,26),(50,6),(50,49),(51,29)],52)
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[3,1,2,4,5] => [[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,2,2],[2,2,2,3],[3,3,3],[4,4],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,2,3,4,5,6] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,3],[4,4,4],[5,5],[6]]
=> ([],1)
=> 1
[1,2,3,4,6,5] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,3],[4,4,4],[5,6],[6]]
=> ([(0,1)],2)
=> 1
[1,2,3,5,4,6] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,3],[4,4,5],[5,5],[6]]
=> ([(0,1)],2)
=> 1
[1,2,3,5,6,4] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,3],[4,4,6],[5,6],[6]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,2,4,3,5,6] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,4],[4,4,4],[5,5],[6]]
=> ([(0,1)],2)
=> 1
[1,2,4,3,6,5] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,4],[4,4,4],[5,6],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[1,2,4,5,3,6] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,5],[4,4,5],[5,5],[6]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,3,2,4,5,6] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,3],[3,3,3,3],[4,4,4],[5,5],[6]]
=> ([(0,1)],2)
=> 1
[1,3,2,4,6,5] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,3],[3,3,3,3],[4,4,4],[5,6],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[1,3,2,5,4,6] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,3],[3,3,3,3],[4,4,5],[5,5],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[1,3,4,2,5,6] => [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,4],[3,3,3,4],[4,4,4],[5,5],[6]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[2,1,3,4,5,6] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,3],[4,4,4],[5,5],[6]]
=> ([(0,1)],2)
=> 1
[2,1,3,4,6,5] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,3],[4,4,4],[5,6],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[2,1,3,5,4,6] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,3],[4,4,5],[5,5],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[2,1,4,3,5,6] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,4],[4,4,4],[5,5],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[2,3,1,4,5,6] => [[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,3],[2,2,2,2,3],[3,3,3,3],[4,4,4],[5,5],[6]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
Description
The number of connected components of the Hasse diagram for the poset.
Matching statistic: St000298
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
St000298: Posets ⟶ ℤResult quality: 2% ●values known / values provided: 5%●distinct values known / distinct values provided: 2%
Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
St000298: Posets ⟶ ℤResult quality: 2% ●values known / values provided: 5%●distinct values known / distinct values provided: 2%
Values
[1] => [[1]]
=> [[1]]
=> ([],1)
=> 1
[1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> ([],1)
=> 1
[2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> ([(0,1)],2)
=> 1
[1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ([],1)
=> 1
[1,3,2] => [[1,0,0],[0,0,1],[0,1,0]]
=> [[1,1,1],[2,3],[3]]
=> ([(0,1)],2)
=> 1
[2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
=> [[1,1,2],[2,2],[3]]
=> ([(0,1)],2)
=> 1
[2,3,1] => [[0,0,1],[1,0,0],[0,1,0]]
=> [[1,1,3],[2,3],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[3,1,2] => [[0,1,0],[0,0,1],[1,0,0]]
=> [[1,2,2],[2,3],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[3,2,1] => [[0,0,1],[0,1,0],[1,0,0]]
=> [[1,2,3],[2,3],[3]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ? = 2
[1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,2],[3,3],[4]]
=> ([],1)
=> 1
[1,2,4,3] => [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [[1,1,1,1],[2,2,2],[3,4],[4]]
=> ([(0,1)],2)
=> 1
[1,3,2,4] => [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,3],[3,3],[4]]
=> ([(0,1)],2)
=> 1
[1,3,4,2] => [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> [[1,1,1,1],[2,2,4],[3,4],[4]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,4,2,3] => [[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]]
=> [[1,1,1,1],[2,3,3],[3,4],[4]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,4,3,2] => [[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> [[1,1,1,1],[2,3,4],[3,4],[4]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ? ∊ {1,1,2,2,2,2,2,3,3,3,3,5,5,6,16}
[2,1,3,4] => [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,2],[2,2,2],[3,3],[4]]
=> ([(0,1)],2)
=> 1
[2,1,4,3] => [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> [[1,1,1,2],[2,2,2],[3,4],[4]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,3,1,4] => [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> [[1,1,1,3],[2,2,3],[3,3],[4]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[2,3,4,1] => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [[1,1,1,4],[2,2,4],[3,4],[4]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ? ∊ {1,1,2,2,2,2,2,3,3,3,3,5,5,6,16}
[2,4,1,3] => [[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
=> [[1,1,1,3],[2,3,3],[3,4],[4]]
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ? ∊ {1,1,2,2,2,2,2,3,3,3,3,5,5,6,16}
[2,4,3,1] => [[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
=> [[1,1,1,4],[2,3,4],[3,4],[4]]
=> ([(0,5),(0,10),(1,16),(2,15),(3,14),(4,13),(5,12),(6,2),(6,13),(7,4),(7,14),(8,1),(9,6),(10,11),(10,12),(11,3),(11,7),(12,9),(13,15),(14,8),(15,16)],17)
=> ? ∊ {1,1,2,2,2,2,2,3,3,3,3,5,5,6,16}
[3,1,2,4] => [[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]]
=> [[1,1,2,2],[2,2,3],[3,3],[4]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[3,1,4,2] => [[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
=> [[1,1,2,2],[2,2,4],[3,4],[4]]
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> ? ∊ {1,1,2,2,2,2,2,3,3,3,3,5,5,6,16}
[3,2,1,4] => [[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> [[1,1,2,3],[2,2,3],[3,3],[4]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ? ∊ {1,1,2,2,2,2,2,3,3,3,3,5,5,6,16}
[3,2,4,1] => [[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
=> [[1,1,2,4],[2,2,4],[3,4],[4]]
=> ([(0,7),(0,8),(1,12),(2,11),(3,10),(4,10),(4,11),(5,3),(6,1),(6,13),(7,9),(8,5),(9,2),(9,4),(10,14),(11,6),(11,14),(13,12),(14,13)],15)
=> ? ∊ {1,1,2,2,2,2,2,3,3,3,3,5,5,6,16}
[3,4,1,2] => [[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> [[1,1,3,3],[2,3,4],[3,4],[4]]
=> ([(0,8),(2,11),(2,12),(3,10),(4,9),(5,4),(5,14),(6,3),(6,14),(7,1),(8,5),(8,6),(9,11),(9,13),(10,12),(10,13),(11,15),(12,15),(13,15),(14,2),(14,9),(14,10),(15,7)],16)
=> ? ∊ {1,1,2,2,2,2,2,3,3,3,3,5,5,6,16}
[3,4,2,1] => [[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
=> [[1,1,3,4],[2,3,4],[3,4],[4]]
=> ([(0,13),(0,15),(1,17),(2,16),(2,17),(3,19),(4,16),(4,18),(5,22),(6,21),(7,20),(8,23),(8,29),(9,4),(9,28),(10,3),(10,28),(11,6),(12,7),(12,24),(13,14),(14,1),(14,2),(15,9),(15,10),(16,25),(17,12),(17,25),(18,26),(18,29),(19,23),(19,26),(20,27),(22,27),(23,30),(24,20),(24,22),(25,24),(26,30),(27,21),(28,8),(28,18),(28,19),(29,5),(29,30),(30,11)],31)
=> ? ∊ {1,1,2,2,2,2,2,3,3,3,3,5,5,6,16}
[4,1,2,3] => [[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> [[1,2,2,2],[2,3,3],[3,4],[4]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ? ∊ {1,1,2,2,2,2,2,3,3,3,3,5,5,6,16}
[4,1,3,2] => [[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]]
=> [[1,2,2,2],[2,3,4],[3,4],[4]]
=> ([(0,7),(0,8),(1,12),(2,11),(3,10),(4,10),(4,11),(5,3),(6,1),(6,13),(7,9),(8,5),(9,2),(9,4),(10,14),(11,6),(11,14),(13,12),(14,13)],15)
=> ? ∊ {1,1,2,2,2,2,2,3,3,3,3,5,5,6,16}
[4,2,1,3] => [[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]]
=> [[1,2,2,3],[2,3,3],[3,4],[4]]
=> ([(0,5),(0,10),(1,16),(2,15),(3,14),(4,13),(5,12),(6,2),(6,13),(7,4),(7,14),(8,1),(9,6),(10,11),(10,12),(11,3),(11,7),(12,9),(13,15),(14,8),(15,16)],17)
=> ? ∊ {1,1,2,2,2,2,2,3,3,3,3,5,5,6,16}
[4,2,3,1] => [[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
=> [[1,2,2,4],[2,3,4],[3,4],[4]]
=> ([(0,14),(0,15),(1,19),(2,18),(3,29),(4,30),(5,22),(6,23),(7,24),(7,25),(8,9),(9,7),(9,18),(9,19),(10,5),(11,6),(12,2),(12,29),(13,1),(13,30),(14,16),(14,28),(15,17),(15,28),(16,3),(16,12),(17,4),(17,13),(18,24),(18,27),(19,25),(19,27),(20,26),(21,26),(22,20),(23,21),(24,22),(24,31),(25,23),(25,31),(27,31),(28,8),(29,10),(30,11),(31,20),(31,21)],32)
=> ? ∊ {1,1,2,2,2,2,2,3,3,3,3,5,5,6,16}
[4,3,1,2] => [[0,0,1,0],[0,0,0,1],[0,1,0,0],[1,0,0,0]]
=> [[1,2,3,3],[2,3,4],[3,4],[4]]
=> ([(0,13),(0,15),(1,17),(2,16),(2,17),(3,19),(4,16),(4,18),(5,22),(6,21),(7,20),(8,23),(8,29),(9,4),(9,28),(10,3),(10,28),(11,6),(12,7),(12,24),(13,14),(14,1),(14,2),(15,9),(15,10),(16,25),(17,12),(17,25),(18,26),(18,29),(19,23),(19,26),(20,27),(22,27),(23,30),(24,20),(24,22),(25,24),(26,30),(27,21),(28,8),(28,18),(28,19),(29,5),(29,30),(30,11)],31)
=> ? ∊ {1,1,2,2,2,2,2,3,3,3,3,5,5,6,16}
[4,3,2,1] => [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> [[1,2,3,4],[2,3,4],[3,4],[4]]
=> ([(0,5),(0,16),(0,17),(1,23),(2,19),(3,11),(3,21),(4,10),(4,20),(5,12),(5,13),(6,50),(7,51),(8,24),(8,58),(9,25),(9,59),(10,14),(10,52),(11,15),(11,53),(12,26),(12,60),(13,27),(13,60),(14,54),(15,55),(16,4),(16,29),(17,3),(17,29),(18,48),(18,49),(19,32),(19,33),(20,46),(20,52),(21,47),(21,53),(22,34),(22,35),(23,18),(23,54),(23,55),(24,40),(24,42),(25,41),(25,43),(26,46),(26,56),(27,47),(27,57),(28,63),(29,1),(30,62),(31,61),(32,61),(33,61),(34,6),(34,62),(35,7),(35,62),(36,58),(37,59),(38,32),(39,33),(40,44),(41,45),(42,38),(43,39),(44,31),(45,31),(46,36),(47,37),(48,40),(48,63),(49,41),(49,63),(50,38),(51,39),(52,8),(52,36),(53,9),(53,37),(54,28),(54,48),(55,28),(55,49),(56,30),(56,34),(57,30),(57,35),(58,42),(58,50),(59,43),(59,51),(60,22),(60,56),(60,57),(62,2),(63,44),(63,45)],64)
=> ? ∊ {1,1,2,2,2,2,2,3,3,3,3,5,5,6,16}
[1,2,3,4,5] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,2,2],[3,3,3],[4,4],[5]]
=> ([],1)
=> 1
[1,2,3,5,4] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,1,1],[2,2,2,2],[3,3,3],[4,5],[5]]
=> ([(0,1)],2)
=> 1
[1,2,4,3,5] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,2,2],[3,3,4],[4,4],[5]]
=> ([(0,1)],2)
=> 1
[1,2,4,5,3] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [[1,1,1,1,1],[2,2,2,2],[3,3,5],[4,5],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,2,5,3,4] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,0,1,0,0]]
=> [[1,1,1,1,1],[2,2,2,2],[3,4,4],[4,5],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,2,5,4,3] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [[1,1,1,1,1],[2,2,2,2],[3,4,5],[4,5],[5]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[1,3,2,4,5] => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,2,3],[3,3,3],[4,4],[5]]
=> ([(0,1)],2)
=> 1
[1,3,2,5,4] => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,1,1],[2,2,2,3],[3,3,3],[4,5],[5]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,3,4,2,5] => [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,2,4],[3,3,4],[4,4],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,3,4,5,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [[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)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[1,3,5,2,4] => [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0]]
=> [[1,1,1,1,1],[2,2,2,4],[3,4,4],[4,5],[5]]
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[1,3,5,4,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]
=> [[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)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[1,4,2,3,5] => [[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,3,3],[3,3,4],[4,4],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,4,2,5,3] => [[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [[1,1,1,1,1],[2,2,3,3],[3,3,5],[4,5],[5]]
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[1,4,3,2,5] => [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[1,4,3,5,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [[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)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[1,4,5,2,3] => [[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0]]
=> [[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)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[1,4,5,3,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0]]
=> [[1,1,1,1,1],[2,2,4,5],[3,4,5],[4,5],[5]]
=> ([(0,13),(0,15),(1,17),(2,16),(2,17),(3,19),(4,16),(4,18),(5,22),(6,21),(7,20),(8,23),(8,29),(9,4),(9,28),(10,3),(10,28),(11,6),(12,7),(12,24),(13,14),(14,1),(14,2),(15,9),(15,10),(16,25),(17,12),(17,25),(18,26),(18,29),(19,23),(19,26),(20,27),(22,27),(23,30),(24,20),(24,22),(25,24),(26,30),(27,21),(28,8),(28,18),(28,19),(29,5),(29,30),(30,11)],31)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[1,5,2,3,4] => [[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0]]
=> [[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)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[1,5,2,4,3] => [[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,1,0,0,0]]
=> [[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)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[1,5,3,2,4] => [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1],[0,1,0,0,0]]
=> [[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)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[1,5,3,4,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0]]
=> [[1,1,1,1,1],[2,3,3,5],[3,4,5],[4,5],[5]]
=> ([(0,14),(0,15),(1,19),(2,18),(3,29),(4,30),(5,22),(6,23),(7,24),(7,25),(8,9),(9,7),(9,18),(9,19),(10,5),(11,6),(12,2),(12,29),(13,1),(13,30),(14,16),(14,28),(15,17),(15,28),(16,3),(16,12),(17,4),(17,13),(18,24),(18,27),(19,25),(19,27),(20,26),(21,26),(22,20),(23,21),(24,22),(24,31),(25,23),(25,31),(27,31),(28,8),(29,10),(30,11),(31,20),(31,21)],32)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[1,5,4,2,3] => [[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0]]
=> [[1,1,1,1,1],[2,3,4,4],[3,4,5],[4,5],[5]]
=> ([(0,13),(0,15),(1,17),(2,16),(2,17),(3,19),(4,16),(4,18),(5,22),(6,21),(7,20),(8,23),(8,29),(9,4),(9,28),(10,3),(10,28),(11,6),(12,7),(12,24),(13,14),(14,1),(14,2),(15,9),(15,10),(16,25),(17,12),(17,25),(18,26),(18,29),(19,23),(19,26),(20,27),(22,27),(23,30),(24,20),(24,22),(25,24),(26,30),(27,21),(28,8),(28,18),(28,19),(29,5),(29,30),(30,11)],31)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[1,5,4,3,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0]]
=> [[1,1,1,1,1],[2,3,4,5],[3,4,5],[4,5],[5]]
=> ([(0,5),(0,16),(0,17),(1,23),(2,19),(3,11),(3,21),(4,10),(4,20),(5,12),(5,13),(6,50),(7,51),(8,24),(8,58),(9,25),(9,59),(10,14),(10,52),(11,15),(11,53),(12,26),(12,60),(13,27),(13,60),(14,54),(15,55),(16,4),(16,29),(17,3),(17,29),(18,48),(18,49),(19,32),(19,33),(20,46),(20,52),(21,47),(21,53),(22,34),(22,35),(23,18),(23,54),(23,55),(24,40),(24,42),(25,41),(25,43),(26,46),(26,56),(27,47),(27,57),(28,63),(29,1),(30,62),(31,61),(32,61),(33,61),(34,6),(34,62),(35,7),(35,62),(36,58),(37,59),(38,32),(39,33),(40,44),(41,45),(42,38),(43,39),(44,31),(45,31),(46,36),(47,37),(48,40),(48,63),(49,41),(49,63),(50,38),(51,39),(52,8),(52,36),(53,9),(53,37),(54,28),(54,48),(55,28),(55,49),(56,30),(56,34),(57,30),(57,35),(58,42),(58,50),(59,43),(59,51),(60,22),(60,56),(60,57),(62,2),(63,44),(63,45)],64)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[2,1,3,4,5] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,2],[2,2,2,2],[3,3,3],[4,4],[5]]
=> ([(0,1)],2)
=> 1
[2,1,3,5,4] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,1,2],[2,2,2,2],[3,3,3],[4,5],[5]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,1,4,3,5] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,2],[2,2,2,2],[3,3,4],[4,4],[5]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,1,4,5,3] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [[1,1,1,1,2],[2,2,2,2],[3,3,5],[4,5],[5]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[2,1,5,3,4] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,0,1,0,0]]
=> [[1,1,1,1,2],[2,2,2,2],[3,4,4],[4,5],[5]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[2,1,5,4,3] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [[1,1,1,1,2],[2,2,2,2],[3,4,5],[4,5],[5]]
=> ([(0,3),(0,6),(0,7),(1,8),(1,12),(2,8),(2,11),(3,9),(3,10),(4,2),(4,13),(5,1),(5,14),(6,4),(6,9),(7,5),(7,10),(8,15),(9,13),(10,14),(11,15),(12,15),(13,11),(14,12)],16)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[2,3,1,4,5] => [[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,3],[2,2,2,3],[3,3,3],[4,4],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[2,3,1,5,4] => [[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,1,3],[2,2,2,3],[3,3,3],[4,5],[5]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[2,3,4,1,5] => [[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,4],[2,2,2,4],[3,3,4],[4,4],[5]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[2,3,4,5,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [[1,1,1,1,5],[2,2,2,5],[3,3,5],[4,5],[5]]
=> ([(0,9),(1,13),(2,14),(4,12),(5,11),(6,1),(6,12),(7,3),(8,5),(8,15),(9,10),(10,4),(10,6),(11,14),(12,8),(12,13),(13,15),(14,7),(15,2),(15,11)],16)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[2,3,5,1,4] => [[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0]]
=> [[1,1,1,1,4],[2,2,2,4],[3,4,4],[4,5],[5]]
=> ([(0,4),(0,8),(1,11),(3,10),(4,9),(5,2),(6,3),(6,12),(7,5),(8,6),(8,9),(9,12),(10,11),(11,7),(12,1),(12,10)],13)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[2,3,5,4,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]
=> [[1,1,1,1,5],[2,2,2,5],[3,4,5],[4,5],[5]]
=> ([(0,8),(0,16),(1,19),(1,20),(2,21),(3,23),(4,26),(5,24),(6,22),(7,27),(8,18),(9,10),(9,20),(10,4),(10,31),(11,2),(11,30),(12,3),(13,7),(13,29),(14,11),(14,28),(15,6),(15,25),(16,17),(16,18),(17,1),(17,9),(17,33),(18,33),(19,24),(20,13),(20,31),(21,32),(22,32),(24,14),(25,12),(26,28),(27,25),(28,30),(29,15),(29,27),(30,21),(30,22),(31,26),(31,29),(32,23),(33,5),(33,19)],34)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[2,4,1,3,5] => [[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,3],[2,2,3,3],[3,3,4],[4,4],[5]]
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[2,4,1,5,3] => [[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [[1,1,1,1,3],[2,2,3,3],[3,3,5],[4,5],[5]]
=> ([(0,6),(0,7),(1,5),(1,15),(2,4),(2,14),(3,13),(4,12),(5,3),(5,16),(6,9),(7,2),(7,9),(9,1),(9,14),(10,11),(11,8),(12,10),(13,8),(14,12),(14,15),(15,10),(15,16),(16,11),(16,13)],17)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[2,4,3,1,5] => [[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,4],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ([(0,5),(0,10),(1,16),(2,15),(3,14),(4,13),(5,12),(6,2),(6,13),(7,4),(7,14),(8,1),(9,6),(10,11),(10,12),(11,3),(11,7),(12,9),(13,15),(14,8),(15,16)],17)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[2,4,3,5,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [[1,1,1,1,5],[2,2,3,5],[3,3,5],[4,5],[5]]
=> ([(0,8),(0,13),(1,19),(2,16),(2,18),(3,21),(4,26),(5,24),(6,16),(6,22),(7,20),(7,23),(8,17),(9,4),(9,18),(10,9),(11,7),(11,30),(12,3),(12,25),(13,14),(13,17),(14,1),(14,15),(15,2),(15,6),(15,19),(16,28),(17,10),(18,26),(18,28),(19,11),(19,22),(20,24),(20,31),(22,30),(23,31),(24,12),(24,27),(25,21),(26,23),(26,29),(27,25),(28,29),(29,31),(30,5),(30,20),(31,27)],32)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[2,4,5,1,3] => [[0,0,0,1,0],[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0]]
=> [[1,1,1,1,4],[2,2,4,4],[3,4,5],[4,5],[5]]
=> ([(0,9),(0,10),(1,2),(3,7),(3,23),(4,6),(4,22),(5,15),(6,16),(7,8),(7,24),(8,20),(9,19),(10,4),(10,19),(11,14),(11,18),(12,26),(13,26),(14,25),(15,1),(16,21),(17,13),(17,25),(18,12),(18,25),(19,3),(19,22),(20,12),(20,13),(21,14),(21,17),(22,16),(22,23),(23,11),(23,21),(23,24),(24,17),(24,18),(24,20),(25,5),(25,26),(26,15)],27)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[2,4,5,3,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0]]
=> [[1,1,1,1,5],[2,2,4,5],[3,4,5],[4,5],[5]]
=> ([(0,17),(0,18),(1,11),(1,12),(2,53),(3,50),(4,33),(5,16),(5,65),(6,13),(6,61),(7,14),(7,62),(8,15),(8,51),(9,25),(9,63),(10,24),(10,52),(11,54),(12,22),(12,23),(12,54),(13,36),(14,35),(15,40),(16,59),(17,1),(17,48),(18,7),(18,48),(19,32),(19,45),(20,29),(20,38),(21,55),(21,58),(22,49),(22,60),(23,42),(23,49),(24,31),(24,46),(25,21),(25,60),(25,64),(27,70),(28,70),(29,67),(30,66),(31,68),(32,3),(32,69),(33,8),(34,39),(35,57),(36,39),(37,32),(37,66),(38,19),(38,37),(38,67),(39,26),(40,26),(41,30),(41,67),(42,52),(43,44),(43,68),(44,28),(44,69),(45,27),(45,69),(46,53),(46,68),(47,61),(48,9),(48,62),(49,5),(49,56),(50,34),(51,40),(52,2),(52,46),(53,6),(53,47),(54,10),(54,42),(55,31),(55,43),(56,43),(56,65),(57,29),(57,41),(58,30),(58,37),(59,27),(59,28),(60,55),(60,56),(61,34),(61,36),(62,35),(62,63),(63,20),(63,57),(63,64),(64,38),(64,41),(64,58),(65,44),(65,45),(65,59),(66,33),(67,4),(67,66),(68,47),(69,50),(69,70),(70,51)],71)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[2,5,1,3,4] => [[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0]]
=> [[1,1,1,1,3],[2,3,3,3],[3,4,4],[4,5],[5]]
=> ([(0,4),(0,8),(1,11),(3,10),(4,9),(5,2),(6,3),(6,12),(7,5),(8,6),(8,9),(9,12),(10,11),(11,7),(12,1),(12,10)],13)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[2,5,1,4,3] => [[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,1,0,0,0]]
=> [[1,1,1,1,3],[2,3,3,3],[3,4,5],[4,5],[5]]
=> ([(0,7),(0,8),(0,11),(1,23),(2,5),(2,20),(3,10),(3,21),(4,9),(4,22),(5,6),(5,15),(6,12),(7,4),(7,17),(8,3),(8,16),(9,14),(9,18),(10,13),(10,14),(11,16),(11,17),(13,24),(14,24),(15,12),(16,21),(17,22),(18,23),(18,24),(19,20),(20,15),(21,13),(22,1),(22,18),(23,2),(23,19),(24,19)],25)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[2,5,3,1,4] => [[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,1,0,0,0]]
=> [[1,1,1,1,4],[2,3,3,4],[3,4,4],[4,5],[5]]
=> ([(0,17),(0,18),(1,21),(2,20),(3,28),(4,27),(5,23),(6,24),(7,2),(7,30),(8,1),(8,31),(9,15),(10,16),(11,13),(11,32),(12,14),(12,33),(13,3),(13,22),(14,4),(14,22),(15,5),(15,25),(16,6),(16,26),(17,7),(17,19),(18,8),(18,19),(19,30),(19,31),(20,32),(21,33),(22,27),(22,28),(23,29),(24,29),(25,23),(26,24),(27,25),(28,26),(30,11),(30,20),(31,12),(31,21),(32,9),(33,10)],34)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[2,5,3,4,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0]]
=> [[1,1,1,1,5],[2,3,3,5],[3,4,5],[4,5],[5]]
=> ([(0,20),(0,21),(1,11),(2,10),(3,27),(3,36),(4,16),(4,34),(5,17),(5,35),(6,26),(6,52),(7,12),(7,53),(8,13),(8,55),(9,15),(9,25),(9,54),(10,51),(11,14),(11,63),(12,38),(13,39),(14,60),(15,19),(15,50),(16,18),(16,64),(17,56),(18,59),(19,22),(19,57),(20,9),(20,58),(21,8),(21,58),(22,47),(22,61),(23,31),(23,46),(24,45),(24,48),(25,37),(25,50),(26,24),(26,49),(26,62),(27,23),(27,61),(27,64),(29,68),(30,68),(31,67),(32,65),(33,69),(34,1),(35,2),(36,6),(37,36),(38,35),(39,34),(40,32),(40,67),(41,51),(41,69),(42,28),(43,28),(44,29),(45,41),(45,66),(46,63),(46,67),(47,52),(48,30),(48,66),(49,48),(49,65),(50,7),(50,57),(51,43),(52,62),(53,5),(53,38),(54,3),(54,37),(55,4),(55,39),(56,33),(56,41),(57,47),(57,53),(58,54),(58,55),(59,32),(59,49),(60,29),(60,30),(61,31),(61,40),(62,45),(62,56),(62,65),(63,44),(63,60),(64,40),(64,46),(64,59),(65,33),(65,66),(66,68),(66,69),(67,44),(68,42),(69,42),(69,43)],70)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[2,5,4,1,3] => [[0,0,0,1,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0]]
=> [[1,1,1,1,4],[2,3,4,4],[3,4,5],[4,5],[5]]
=> ([(0,15),(0,16),(0,20),(1,9),(2,13),(2,47),(3,38),(4,14),(4,46),(5,40),(6,10),(6,26),(7,18),(7,45),(8,19),(8,39),(9,25),(10,32),(11,31),(12,44),(13,12),(13,48),(14,11),(14,42),(15,8),(15,28),(16,7),(16,37),(17,24),(17,33),(18,21),(18,36),(19,21),(19,35),(20,28),(20,37),(21,51),(22,49),(23,49),(24,3),(24,50),(26,1),(27,25),(28,39),(29,46),(30,27),(31,30),(32,27),(33,23),(33,50),(34,22),(34,50),(35,40),(35,51),(36,41),(36,51),(37,2),(37,45),(38,43),(39,5),(39,35),(40,4),(40,29),(41,24),(41,34),(42,31),(42,43),(43,30),(43,32),(44,22),(44,23),(45,36),(45,47),(46,38),(46,42),(47,17),(47,41),(47,48),(48,33),(48,34),(48,44),(49,26),(50,6),(50,49),(51,29)],52)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,8,8,9,9,9,9,10,10,10,10,11,11,11,11,14,14,16,16,16,16,16,16,16,16,19,19,20,21,21,21,21,35,35,35,35,35,35,35,35,42,56,56,70,70,70,70,90,90,168,168,216,216,768}
[3,1,2,4,5] => [[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,2,2],[2,2,2,3],[3,3,3],[4,4],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,2,3,4,5,6] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,3],[4,4,4],[5,5],[6]]
=> ([],1)
=> 1
[1,2,3,4,6,5] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,3],[4,4,4],[5,6],[6]]
=> ([(0,1)],2)
=> 1
[1,2,3,5,4,6] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,3],[4,4,5],[5,5],[6]]
=> ([(0,1)],2)
=> 1
[1,2,3,5,6,4] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,3],[4,4,6],[5,6],[6]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,2,4,3,5,6] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,4],[4,4,4],[5,5],[6]]
=> ([(0,1)],2)
=> 1
[1,2,4,3,6,5] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,4],[4,4,4],[5,6],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,2,4,5,3,6] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,5],[4,4,5],[5,5],[6]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,3,2,4,5,6] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,3],[3,3,3,3],[4,4,4],[5,5],[6]]
=> ([(0,1)],2)
=> 1
[1,3,2,4,6,5] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,3],[3,3,3,3],[4,4,4],[5,6],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,3,2,5,4,6] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,3],[3,3,3,3],[4,4,5],[5,5],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,3,4,2,5,6] => [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,4],[3,3,3,4],[4,4,4],[5,5],[6]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[2,1,3,4,5,6] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,3],[4,4,4],[5,5],[6]]
=> ([(0,1)],2)
=> 1
[2,1,3,4,6,5] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,3],[4,4,4],[5,6],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,1,3,5,4,6] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,3],[4,4,5],[5,5],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,1,4,3,5,6] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,4],[4,4,4],[5,5],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,3,1,4,5,6] => [[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,3],[2,2,2,2,3],[3,3,3,3],[4,4,4],[5,5],[6]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
Description
The order dimension or Dushnik-Miller dimension of a poset.
This is the minimal number of linear orderings whose intersection is the given poset.
The following 36 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000307The number of rowmotion orbits of a poset. St000908The length of the shortest maximal antichain in a poset. St001268The size of the largest ordinal summand in the poset. St001399The distinguishing number of a poset. St001510The number of self-evacuating linear extensions of a finite poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001533The largest coefficient of the Poincare polynomial of the poset cone. St001779The order of promotion on the set of linear extensions of a poset. St000632The jump number of the poset. St001301The first Betti number of the order complex associated with the poset. St001396Number of triples of incomparable elements in a finite poset. St001397Number of pairs of incomparable elements in a finite poset. St001398Number of subsets of size 3 of elements in a poset that form a "v". St001902The number of potential covers of a poset. St001964The interval resolution global dimension of a poset. St001472The permanent of the Coxeter matrix of the poset. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. 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. St001875The number of simple modules with projective dimension at most 1. St000524The number of posets with the same order polynomial. St000525The number of posets with the same zeta polynomial. St000526The number of posets with combinatorially isomorphic order polytopes. St000633The size of the automorphism group of a poset. St000640The rank of the largest boolean interval in a poset. St000910The number of maximal chains of minimal length in a poset. St000914The sum of the values of the Möbius function of a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001890The maximum magnitude of the Möbius function of a poset. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001768The number of reduced words of a signed permutation. St001487The number of inner corners of a skew partition. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset.
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