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Your data matches 96 different statistics following compositions of up to 3 maps.
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Matching statistic: St001060
Mp00002: Alternating sign matrices —to left key permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001060: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001060: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [4,1,2,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 3
[[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
=> [4,2,1,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 3
[[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
=> [3,1,4,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> 2
[[0,0,1,0],[0,1,0,0],[1,-1,0,1],[0,1,0,0]]
=> [3,1,4,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> 2
[[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
=> [4,1,3,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 2
[[0,0,0,1],[0,1,0,0],[1,-1,1,0],[0,1,0,0]]
=> [4,1,3,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 2
[[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
=> [4,3,1,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 2
[[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]]
=> [3,2,4,1] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> 2
[[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
=> [4,2,3,1] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 2
[[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> [4,3,2,1] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 2
[[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]]
=> [5,1,2,3,4] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[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]]
=> [5,2,1,3,4] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],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]]
=> [3,1,5,2,4] => [1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[0,0,1,0,0],[0,1,0,0,0],[1,-1,0,0,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [3,1,5,2,4] => [1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[0,0,0,1,0],[1,0,0,0,0],[0,0,1,-1,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [3,1,5,2,4] => [1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[0,0,0,1,0],[0,1,0,0,0],[1,-1,1,-1,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [3,1,5,2,4] => [1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[0,0,1,0,0],[0,0,0,1,0],[1,0,0,-1,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [3,1,5,2,4] => [1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[0,0,0,1,0],[0,0,1,0,0],[1,0,0,-1,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [3,1,5,2,4] => [1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[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]]
=> [5,1,3,2,4] => [1,5,4,2,3] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[0,0,0,0,1],[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [5,1,3,2,4] => [1,5,4,2,3] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[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]]
=> [5,3,1,2,4] => [1,5,4,2,3] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,0,1,0]]
=> [3,2,5,1,4] => [1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[0,0,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[1,0,0,0,0],[0,0,0,1,0]]
=> [3,2,5,1,4] => [1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[0,0,1,0,0],[0,0,0,1,0],[0,1,0,-1,1],[1,0,0,0,0],[0,0,0,1,0]]
=> [3,2,5,1,4] => [1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[0,0,0,1,0],[0,0,1,0,0],[0,1,0,-1,1],[1,0,0,0,0],[0,0,0,1,0]]
=> [3,2,5,1,4] => [1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0]]
=> [5,2,3,1,4] => [1,5,4,2,3] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0]]
=> [5,3,2,1,4] => [1,5,4,2,3] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[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]]
=> [4,1,2,5,3] => [1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[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]]
=> [4,2,1,5,3] => [1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,0,1],[0,0,1,0,0]]
=> [4,1,2,5,3] => [1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[0,0,0,1,0],[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,0,1],[0,0,1,0,0]]
=> [4,1,2,5,3] => [1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,0,1],[0,0,1,0,0]]
=> [4,2,1,5,3] => [1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[1,0,-1,0,1],[0,0,1,0,0]]
=> [4,2,1,5,3] => [1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[1,0,-1,0,1],[0,0,1,0,0]]
=> [4,2,1,5,3] => [1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[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]]
=> [5,1,2,4,3] => [1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[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]]
=> [5,2,1,4,3] => [1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0]]
=> [5,1,2,4,3] => [1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[0,0,0,0,1],[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,0,0]]
=> [5,1,2,4,3] => [1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[0,0,0,0,1],[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,1,0],[0,0,1,0,0]]
=> [5,2,1,4,3] => [1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[1,0,-1,1,0],[0,0,1,0,0]]
=> [5,2,1,4,3] => [1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[1,0,-1,1,0],[0,0,1,0,0]]
=> [5,2,1,4,3] => [1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[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]]
=> [5,1,4,2,3] => [1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[0,0,0,0,1],[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,1,0,0]]
=> [5,1,4,2,3] => [1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[0,0,0,0,1],[0,0,1,0,0],[1,0,-1,1,0],[0,1,0,0,0],[0,0,1,0,0]]
=> [5,1,4,2,3] => [1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 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]]
=> [4,5,1,2,3] => [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> 2
[[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]]
=> [5,4,1,2,3] => [1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0]]
=> [5,2,4,1,3] => [1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[0,0,0,0,1],[0,0,1,0,0],[0,1,-1,1,0],[1,0,0,0,0],[0,0,1,0,0]]
=> [5,2,4,1,3] => [1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0]]
=> [4,5,2,1,3] => [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> 2
[[0,0,0,0,1],[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0]]
=> [5,4,2,1,3] => [1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
Description
The distinguishing index of a graph.
This is the smallest number of colours such that there is a colouring of the edges which is not preserved by any automorphism.
If the graph has a connected component which is a single edge, or at least two isolated vertices, this statistic is undefined.
Matching statistic: St000264
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00007: Alternating sign matrices —to Dyck path⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000264: Graphs ⟶ ℤResult quality: 33% ●values known / values provided: 59%●distinct values known / distinct values provided: 33%
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000264: Graphs ⟶ ℤResult quality: 33% ●values known / values provided: 59%●distinct values known / distinct values provided: 33%
Values
[[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [1,1,1,1,0,0,0,0]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ? = 3 + 2
[[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
=> [1,1,1,1,0,0,0,0]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ? = 3 + 2
[[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> ? = 2 + 2
[[0,0,1,0],[0,1,0,0],[1,-1,0,1],[0,1,0,0]]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> ? = 2 + 2
[[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
=> [1,1,1,1,0,0,0,0]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ? = 2 + 2
[[0,0,0,1],[0,1,0,0],[1,-1,1,0],[0,1,0,0]]
=> [1,1,1,1,0,0,0,0]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ? = 2 + 2
[[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
=> [1,1,1,1,0,0,0,0]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ? = 2 + 2
[[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> ? = 2 + 2
[[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
=> [1,1,1,1,0,0,0,0]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ? = 2 + 2
[[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> [1,1,1,1,0,0,0,0]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ? = 2 + 2
[[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]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? = 3 + 2
[[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]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? = 3 + 2
[[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]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2 + 2
[[0,0,1,0,0],[0,1,0,0,0],[1,-1,0,0,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2 + 2
[[0,0,0,1,0],[1,0,0,0,0],[0,0,1,-1,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 4 = 2 + 2
[[0,0,0,1,0],[0,1,0,0,0],[1,-1,1,-1,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 4 = 2 + 2
[[0,0,1,0,0],[0,0,0,1,0],[1,0,0,-1,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 4 = 2 + 2
[[0,0,0,1,0],[0,0,1,0,0],[1,0,0,-1,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 4 = 2 + 2
[[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]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? = 2 + 2
[[0,0,0,0,1],[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? = 2 + 2
[[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]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? = 2 + 2
[[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,0,1,0]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2 + 2
[[0,0,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[1,0,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 4 = 2 + 2
[[0,0,1,0,0],[0,0,0,1,0],[0,1,0,-1,1],[1,0,0,0,0],[0,0,0,1,0]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 4 = 2 + 2
[[0,0,0,1,0],[0,0,1,0,0],[0,1,0,-1,1],[1,0,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 4 = 2 + 2
[[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? = 2 + 2
[[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? = 2 + 2
[[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]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 2 + 2
[[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]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 2 + 2
[[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,0,1],[0,0,1,0,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 2 + 2
[[0,0,0,1,0],[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,0,1],[0,0,1,0,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 2 + 2
[[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,0,1],[0,0,1,0,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 2 + 2
[[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[1,0,-1,0,1],[0,0,1,0,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 2 + 2
[[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[1,0,-1,0,1],[0,0,1,0,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 2 + 2
[[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]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? = 2 + 2
[[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]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? = 2 + 2
[[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? = 2 + 2
[[0,0,0,0,1],[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,0,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? = 2 + 2
[[0,0,0,0,1],[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,1,0],[0,0,1,0,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? = 2 + 2
[[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[1,0,-1,1,0],[0,0,1,0,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? = 2 + 2
[[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[1,0,-1,1,0],[0,0,1,0,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? = 2 + 2
[[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]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? = 2 + 2
[[0,0,0,0,1],[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,1,0,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? = 2 + 2
[[0,0,0,0,1],[0,0,1,0,0],[1,0,-1,1,0],[0,1,0,0,0],[0,0,1,0,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? = 2 + 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]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 4 = 2 + 2
[[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]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? = 2 + 2
[[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? = 2 + 2
[[0,0,0,0,1],[0,0,1,0,0],[0,1,-1,1,0],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? = 2 + 2
[[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 4 = 2 + 2
[[0,0,0,0,1],[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? = 2 + 2
[[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]]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 3 + 2
[[0,0,1,0,0],[0,1,0,0,0],[1,-1,0,1,0],[0,0,0,0,1],[0,1,0,0,0]]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 3 + 2
[[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]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 3 + 2
[[0,0,0,1,0],[0,1,0,0,0],[1,-1,1,0,0],[0,0,0,0,1],[0,1,0,0,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 3 + 2
[[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 3 + 2
[[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[1,-1,0,0,1],[0,1,0,0,0]]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 3 + 2
[[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[1,-1,0,0,1],[0,1,0,0,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 3 + 2
[[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[1,-1,0,0,1],[0,1,0,0,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 3 + 2
[[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]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2 + 2
[[0,0,0,1,0],[1,0,0,0,0],[0,0,1,-1,1],[0,0,0,1,0],[0,1,0,0,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 4 = 2 + 2
[[0,0,0,1,0],[0,1,0,0,0],[1,-1,1,-1,1],[0,0,0,1,0],[0,1,0,0,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 4 = 2 + 2
[[0,0,1,0,0],[0,0,0,1,0],[1,0,0,-1,1],[0,0,0,1,0],[0,1,0,0,0]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 4 = 2 + 2
[[0,0,0,1,0],[0,0,1,0,0],[1,0,0,-1,1],[0,0,0,1,0],[0,1,0,0,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 4 = 2 + 2
[[0,0,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[1,-1,0,1,0],[0,1,0,0,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 4 = 2 + 2
[[0,0,1,0,0],[0,0,0,1,0],[0,1,0,-1,1],[1,-1,0,1,0],[0,1,0,0,0]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 4 = 2 + 2
[[0,0,0,1,0],[0,0,1,0,0],[0,1,0,-1,1],[1,-1,0,1,0],[0,1,0,0,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 4 = 2 + 2
[[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]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 4 = 2 + 2
[[0,0,0,1,0],[0,1,0,0,0],[1,-1,0,0,1],[0,0,1,0,0],[0,1,0,0,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 4 = 2 + 2
[[0,0,0,1,0],[0,0,1,0,0],[1,0,-1,0,1],[0,0,1,0,0],[0,1,0,0,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 4 = 2 + 2
[[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1],[1,-1,1,0,0],[0,1,0,0,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 4 = 2 + 2
[[0,0,0,1,0],[0,0,1,0,0],[0,1,-1,0,1],[1,-1,1,0,0],[0,1,0,0,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 4 = 2 + 2
[[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 4 = 2 + 2
[[0,0,0,1,0],[0,0,0,0,1],[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 4 = 2 + 2
[[0,0,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0],[1,0,0,0,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 4 = 2 + 2
[[0,0,1,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,0,1,0],[1,0,0,0,0]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 4 = 2 + 2
[[0,0,0,1,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0],[1,0,0,0,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 4 = 2 + 2
[[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[1,0,0,0,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 4 = 2 + 2
[[0,0,0,1,0],[0,0,1,0,0],[0,1,-1,0,1],[0,0,1,0,0],[1,0,0,0,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 4 = 2 + 2
[[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0],[1,0,0,0,0]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 4 = 2 + 2
[[0,0,0,1,0],[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 4 = 2 + 2
[[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,1,-1,0,1],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [4,1,6,2,3,5] => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> 4 = 2 + 2
[[0,0,0,1,0,0],[0,1,0,0,0,0],[1,-1,1,-1,0,1],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [4,1,6,2,3,5] => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> 4 = 2 + 2
[[0,0,1,0,0,0],[0,0,0,1,0,0],[1,0,0,-1,0,1],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [3,4,6,1,2,5] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> 4 = 2 + 2
[[0,0,0,1,0,0],[0,0,1,0,0,0],[1,0,0,-1,0,1],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [4,1,6,2,3,5] => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> 4 = 2 + 2
[[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,1,0,-1,1],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [5,1,6,2,3,4] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 4 = 2 + 2
[[0,0,0,0,1,0],[0,1,0,0,0,0],[1,-1,1,0,-1,1],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [5,1,6,2,3,4] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 4 = 2 + 2
[[0,0,1,0,0,0],[0,0,0,0,1,0],[1,0,0,0,-1,1],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [3,5,6,1,2,4] => ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> 4 = 2 + 2
[[0,0,0,1,0,0],[0,0,1,-1,1,0],[1,0,0,0,-1,1],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [4,5,6,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> 4 = 2 + 2
[[0,0,0,0,1,0],[0,0,1,0,0,0],[1,0,0,0,-1,1],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [5,1,6,2,3,4] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 4 = 2 + 2
[[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,1,-1,0,1],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [4,1,6,2,3,5] => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> 4 = 2 + 2
[[0,0,1,0,0,0],[0,0,0,1,0,0],[0,1,0,-1,0,1],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [3,4,6,1,2,5] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> 4 = 2 + 2
[[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,-1,0,1],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [4,1,6,2,3,5] => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> 4 = 2 + 2
[[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,1,0,-1,1],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [5,1,6,2,3,4] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 4 = 2 + 2
[[0,0,1,0,0,0],[0,0,0,0,1,0],[0,1,0,0,-1,1],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [3,5,6,1,2,4] => ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> 4 = 2 + 2
[[0,0,0,1,0,0],[0,0,1,-1,1,0],[0,1,0,0,-1,1],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [4,5,6,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> 4 = 2 + 2
[[0,0,0,0,1,0],[0,0,1,0,0,0],[0,1,0,0,-1,1],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [5,1,6,2,3,4] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 4 = 2 + 2
[[0,0,0,0,1,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,-1,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [5,1,2,6,3,4] => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4 = 2 + 2
[[0,0,0,0,1,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,-1,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [5,1,2,6,3,4] => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4 = 2 + 2
[[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,-1,1,-1,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [5,1,2,6,3,4] => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4 = 2 + 2
[[0,0,0,0,1,0],[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,-1,1,-1,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [5,1,2,6,3,4] => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4 = 2 + 2
Description
The girth of a graph, which is not a tree.
This is the length of the shortest cycle in the graph.
Matching statistic: St000618
Mp00007: Alternating sign matrices —to Dyck path⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000618: Integer partitions ⟶ ℤResult quality: 33% ●values known / values provided: 36%●distinct values known / distinct values provided: 33%
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000618: Integer partitions ⟶ ℤResult quality: 33% ●values known / values provided: 36%●distinct values known / distinct values provided: 33%
Values
[[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> ? = 3 - 1
[[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
=> [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> ? = 3 - 1
[[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
=> [1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> []
=> ? = 2 - 1
[[0,0,1,0],[0,1,0,0],[1,-1,0,1],[0,1,0,0]]
=> [1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> []
=> ? = 2 - 1
[[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
=> [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,1],[0,1,0,0],[1,-1,1,0],[0,1,0,0]]
=> [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
=> [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> ? = 2 - 1
[[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]]
=> [1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> []
=> ? = 2 - 1
[[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
=> [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> ? = 2 - 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]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 3 - 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]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 3 - 1
[[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]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,1,0,0],[0,1,0,0,0],[1,-1,0,0,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0],[1,0,0,0,0],[0,0,1,-1,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0],[0,1,0,0,0],[1,-1,1,-1,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,1,0,0],[0,0,0,1,0],[1,0,0,-1,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [[2,2,2,2],[]]
=> []
=> ? = 2 - 1
[[0,0,0,1,0],[0,0,1,0,0],[1,0,0,-1,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 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]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,0,1],[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 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]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 1
[[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,0,1,0]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[1,0,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,1,0,0],[0,0,0,1,0],[0,1,0,-1,1],[1,0,0,0,0],[0,0,0,1,0]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [[2,2,2,2],[]]
=> []
=> ? = 2 - 1
[[0,0,0,1,0],[0,0,1,0,0],[0,1,0,-1,1],[1,0,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 1
[[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]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> []
=> ? = 2 - 1
[[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]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,0,1],[0,0,1,0,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,1,0],[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,0,1],[0,0,1,0,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,0,1],[0,0,1,0,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[1,0,-1,0,1],[0,0,1,0,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[1,0,-1,0,1],[0,0,1,0,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> []
=> ? = 2 - 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]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 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]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,0,1],[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,0,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,0,1],[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,1,0],[0,0,1,0,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[1,0,-1,1,0],[0,0,1,0,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[1,0,-1,1,0],[0,0,1,0,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 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]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,0,1],[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,1,0,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,0,1],[0,0,1,0,0],[1,0,-1,1,0],[0,1,0,0,0],[0,0,1,0,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 1
[[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]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [[4,4],[]]
=> []
=> ? = 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]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,0,1],[0,0,1,0,0],[0,1,-1,1,0],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [[4,4],[]]
=> []
=> ? = 2 - 1
[[0,0,0,0,1],[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 1
[[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]]
=> [1,1,1,0,0,1,0,1,0,0]
=> [[4,2],[]]
=> []
=> ? = 3 - 1
[[0,0,1,0,0],[0,1,0,0,0],[1,-1,0,1,0],[0,0,0,0,1],[0,1,0,0,0]]
=> [1,1,1,0,0,1,0,1,0,0]
=> [[4,2],[]]
=> []
=> ? = 3 - 1
[[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]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> []
=> ? = 3 - 1
[[0,0,0,1,0],[0,1,0,0,0],[1,-1,1,0,0],[0,0,0,0,1],[0,1,0,0,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> []
=> ? = 3 - 1
[[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> []
=> ? = 3 - 1
[[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[1,-1,0,0,1],[0,1,0,0,0]]
=> [1,1,1,0,0,1,0,1,0,0]
=> [[4,2],[]]
=> []
=> ? = 3 - 1
[[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[1,-1,0,0,1],[0,1,0,0,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> []
=> ? = 3 - 1
[[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[1,-1,0,0,1],[0,1,0,0,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> []
=> ? = 3 - 1
[[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]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,1,0,0],[0,1,0,0,0],[1,-1,0,0,1],[0,0,0,1,0],[0,1,0,0,0]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0],[1,0,0,0,0],[0,0,1,-1,1],[0,0,0,1,0],[0,1,0,0,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0],[0,1,0,0,0],[1,-1,1,-1,1],[0,0,0,1,0],[0,1,0,0,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0],[0,0,1,0,0],[1,0,0,-1,1],[0,0,0,1,0],[0,1,0,0,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[1,-1,0,1,0],[0,1,0,0,0]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[1,-1,0,1,0],[0,1,0,0,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0],[0,0,1,0,0],[0,1,0,-1,1],[1,-1,0,1,0],[0,1,0,0,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[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]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0],[0,1,0,0,0],[1,-1,0,0,1],[0,0,1,0,0],[0,1,0,0,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0],[0,0,1,0,0],[1,0,-1,0,1],[0,0,1,0,0],[0,1,0,0,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1],[1,-1,1,0,0],[0,1,0,0,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0],[0,0,1,0,0],[0,1,-1,0,1],[1,-1,1,0,0],[0,1,0,0,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[1,0,0,0,0]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0],[1,0,0,0,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0],[1,0,0,0,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[1,0,0,0,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0],[0,0,1,0,0],[0,1,-1,0,1],[0,0,1,0,0],[1,0,0,0,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [[4,4,2],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,1,0,0,0],[0,1,0,0,0,0],[1,-1,0,0,0,1],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [[4,4,2],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,1,-1,0,1],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [[4,4,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0,0],[0,1,0,0,0,0],[1,-1,1,-1,0,1],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [[4,4,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0,0],[0,0,1,0,0,0],[1,0,0,-1,0,1],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [[4,4,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [[4,4,2],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,1,-1,0,1],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [[4,4,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,-1,0,1],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [[4,4,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [[4,4,3],[2]]
=> [2]
=> 1 = 2 - 1
[[0,0,0,1,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [[4,4,3],[2]]
=> [2]
=> 1 = 2 - 1
[[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,-1,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [[4,4,3],[2]]
=> [2]
=> 1 = 2 - 1
[[0,0,0,1,0,0],[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,-1,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [[4,4,3],[2]]
=> [2]
=> 1 = 2 - 1
[[0,0,0,1,0,0],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,-1,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [[4,4,3],[2]]
=> [2]
=> 1 = 2 - 1
[[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[1,0,-1,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [[4,4,3],[2]]
=> [2]
=> 1 = 2 - 1
[[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,-1,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [[4,4,3],[2]]
=> [2]
=> 1 = 2 - 1
[[0,0,0,0,1,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,-1,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [[3,3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,0,1,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,-1,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [[3,3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,-1,1,-1,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [[3,3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,0,1,0],[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,-1,1,-1,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [[3,3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,0,1,0],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,-1,1,-1,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [[3,3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[1,0,-1,1,-1,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [[3,3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,0,1,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,-1,1,-1,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [[3,3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,1,0,0,-1,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [[3,3,3,3],[1,1]]
=> [1,1]
=> 1 = 2 - 1
[[0,0,0,1,0,0],[0,1,0,0,0,0],[1,-1,0,0,1,0],[0,1,0,0,-1,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [[3,3,3,3],[1,1]]
=> [1,1]
=> 1 = 2 - 1
Description
The number of self-evacuating tableaux of given shape.
This is the same as the number of standard domino tableaux of the given shape.
Matching statistic: St000781
Mp00007: Alternating sign matrices —to Dyck path⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000781: Integer partitions ⟶ ℤResult quality: 33% ●values known / values provided: 36%●distinct values known / distinct values provided: 33%
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000781: Integer partitions ⟶ ℤResult quality: 33% ●values known / values provided: 36%●distinct values known / distinct values provided: 33%
Values
[[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> ? = 3 - 1
[[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
=> [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> ? = 3 - 1
[[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
=> [1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> []
=> ? = 2 - 1
[[0,0,1,0],[0,1,0,0],[1,-1,0,1],[0,1,0,0]]
=> [1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> []
=> ? = 2 - 1
[[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
=> [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,1],[0,1,0,0],[1,-1,1,0],[0,1,0,0]]
=> [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
=> [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> ? = 2 - 1
[[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]]
=> [1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> []
=> ? = 2 - 1
[[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
=> [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> ? = 2 - 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]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 3 - 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]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 3 - 1
[[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]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,1,0,0],[0,1,0,0,0],[1,-1,0,0,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0],[1,0,0,0,0],[0,0,1,-1,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0],[0,1,0,0,0],[1,-1,1,-1,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,1,0,0],[0,0,0,1,0],[1,0,0,-1,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [[2,2,2,2],[]]
=> []
=> ? = 2 - 1
[[0,0,0,1,0],[0,0,1,0,0],[1,0,0,-1,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 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]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,0,1],[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 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]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 1
[[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,0,1,0]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[1,0,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,1,0,0],[0,0,0,1,0],[0,1,0,-1,1],[1,0,0,0,0],[0,0,0,1,0]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [[2,2,2,2],[]]
=> []
=> ? = 2 - 1
[[0,0,0,1,0],[0,0,1,0,0],[0,1,0,-1,1],[1,0,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 1
[[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]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> []
=> ? = 2 - 1
[[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]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,0,1],[0,0,1,0,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,1,0],[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,0,1],[0,0,1,0,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,0,1],[0,0,1,0,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[1,0,-1,0,1],[0,0,1,0,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[1,0,-1,0,1],[0,0,1,0,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> []
=> ? = 2 - 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]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 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]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,0,1],[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,0,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,0,1],[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,1,0],[0,0,1,0,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[1,0,-1,1,0],[0,0,1,0,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[1,0,-1,1,0],[0,0,1,0,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 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]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,0,1],[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,1,0,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,0,1],[0,0,1,0,0],[1,0,-1,1,0],[0,1,0,0,0],[0,0,1,0,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 1
[[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]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [[4,4],[]]
=> []
=> ? = 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]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,0,1],[0,0,1,0,0],[0,1,-1,1,0],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [[4,4],[]]
=> []
=> ? = 2 - 1
[[0,0,0,0,1],[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 1
[[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]]
=> [1,1,1,0,0,1,0,1,0,0]
=> [[4,2],[]]
=> []
=> ? = 3 - 1
[[0,0,1,0,0],[0,1,0,0,0],[1,-1,0,1,0],[0,0,0,0,1],[0,1,0,0,0]]
=> [1,1,1,0,0,1,0,1,0,0]
=> [[4,2],[]]
=> []
=> ? = 3 - 1
[[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]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> []
=> ? = 3 - 1
[[0,0,0,1,0],[0,1,0,0,0],[1,-1,1,0,0],[0,0,0,0,1],[0,1,0,0,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> []
=> ? = 3 - 1
[[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> []
=> ? = 3 - 1
[[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[1,-1,0,0,1],[0,1,0,0,0]]
=> [1,1,1,0,0,1,0,1,0,0]
=> [[4,2],[]]
=> []
=> ? = 3 - 1
[[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[1,-1,0,0,1],[0,1,0,0,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> []
=> ? = 3 - 1
[[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[1,-1,0,0,1],[0,1,0,0,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> []
=> ? = 3 - 1
[[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]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,1,0,0],[0,1,0,0,0],[1,-1,0,0,1],[0,0,0,1,0],[0,1,0,0,0]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0],[1,0,0,0,0],[0,0,1,-1,1],[0,0,0,1,0],[0,1,0,0,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0],[0,1,0,0,0],[1,-1,1,-1,1],[0,0,0,1,0],[0,1,0,0,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0],[0,0,1,0,0],[1,0,0,-1,1],[0,0,0,1,0],[0,1,0,0,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[1,-1,0,1,0],[0,1,0,0,0]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[1,-1,0,1,0],[0,1,0,0,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0],[0,0,1,0,0],[0,1,0,-1,1],[1,-1,0,1,0],[0,1,0,0,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[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]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0],[0,1,0,0,0],[1,-1,0,0,1],[0,0,1,0,0],[0,1,0,0,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0],[0,0,1,0,0],[1,0,-1,0,1],[0,0,1,0,0],[0,1,0,0,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1],[1,-1,1,0,0],[0,1,0,0,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0],[0,0,1,0,0],[0,1,-1,0,1],[1,-1,1,0,0],[0,1,0,0,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[1,0,0,0,0]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0],[1,0,0,0,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0],[1,0,0,0,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[1,0,0,0,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0],[0,0,1,0,0],[0,1,-1,0,1],[0,0,1,0,0],[1,0,0,0,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [[4,4,2],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,1,0,0,0],[0,1,0,0,0,0],[1,-1,0,0,0,1],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [[4,4,2],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,1,-1,0,1],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [[4,4,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0,0],[0,1,0,0,0,0],[1,-1,1,-1,0,1],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [[4,4,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0,0],[0,0,1,0,0,0],[1,0,0,-1,0,1],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [[4,4,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [[4,4,2],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,1,-1,0,1],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [[4,4,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,-1,0,1],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [[4,4,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [[4,4,3],[2]]
=> [2]
=> 1 = 2 - 1
[[0,0,0,1,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [[4,4,3],[2]]
=> [2]
=> 1 = 2 - 1
[[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,-1,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [[4,4,3],[2]]
=> [2]
=> 1 = 2 - 1
[[0,0,0,1,0,0],[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,-1,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [[4,4,3],[2]]
=> [2]
=> 1 = 2 - 1
[[0,0,0,1,0,0],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,-1,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [[4,4,3],[2]]
=> [2]
=> 1 = 2 - 1
[[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[1,0,-1,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [[4,4,3],[2]]
=> [2]
=> 1 = 2 - 1
[[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,-1,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [[4,4,3],[2]]
=> [2]
=> 1 = 2 - 1
[[0,0,0,0,1,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,-1,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [[3,3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,0,1,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,-1,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [[3,3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,-1,1,-1,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [[3,3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,0,1,0],[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,-1,1,-1,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [[3,3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,0,1,0],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,-1,1,-1,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [[3,3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[1,0,-1,1,-1,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [[3,3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,0,1,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,-1,1,-1,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [[3,3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,1,0,0,-1,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [[3,3,3,3],[1,1]]
=> [1,1]
=> 1 = 2 - 1
[[0,0,0,1,0,0],[0,1,0,0,0,0],[1,-1,0,0,1,0],[0,1,0,0,-1,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [[3,3,3,3],[1,1]]
=> [1,1]
=> 1 = 2 - 1
Description
The number of proper colouring schemes of a Ferrers diagram.
A colouring of a Ferrers diagram is proper if no two cells in a row or in a column have the same colour. The minimal number of colours needed is the maximum of the length and the first part of the partition, because we can restrict a latin square to the shape. We can associate to each colouring the integer partition recording how often each colour is used, see [1].
This statistic is the number of distinct such integer partitions that occur.
Matching statistic: St001364
Mp00007: Alternating sign matrices —to Dyck path⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001364: Integer partitions ⟶ ℤResult quality: 33% ●values known / values provided: 36%●distinct values known / distinct values provided: 33%
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001364: Integer partitions ⟶ ℤResult quality: 33% ●values known / values provided: 36%●distinct values known / distinct values provided: 33%
Values
[[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> ? = 3 - 1
[[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
=> [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> ? = 3 - 1
[[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
=> [1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> []
=> ? = 2 - 1
[[0,0,1,0],[0,1,0,0],[1,-1,0,1],[0,1,0,0]]
=> [1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> []
=> ? = 2 - 1
[[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
=> [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,1],[0,1,0,0],[1,-1,1,0],[0,1,0,0]]
=> [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
=> [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> ? = 2 - 1
[[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]]
=> [1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> []
=> ? = 2 - 1
[[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
=> [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> ? = 2 - 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]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 3 - 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]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 3 - 1
[[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]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,1,0,0],[0,1,0,0,0],[1,-1,0,0,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0],[1,0,0,0,0],[0,0,1,-1,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0],[0,1,0,0,0],[1,-1,1,-1,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,1,0,0],[0,0,0,1,0],[1,0,0,-1,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [[2,2,2,2],[]]
=> []
=> ? = 2 - 1
[[0,0,0,1,0],[0,0,1,0,0],[1,0,0,-1,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 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]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,0,1],[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 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]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 1
[[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,0,1,0]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[1,0,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,1,0,0],[0,0,0,1,0],[0,1,0,-1,1],[1,0,0,0,0],[0,0,0,1,0]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [[2,2,2,2],[]]
=> []
=> ? = 2 - 1
[[0,0,0,1,0],[0,0,1,0,0],[0,1,0,-1,1],[1,0,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 1
[[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]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> []
=> ? = 2 - 1
[[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]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,0,1],[0,0,1,0,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,1,0],[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,0,1],[0,0,1,0,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,0,1],[0,0,1,0,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[1,0,-1,0,1],[0,0,1,0,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[1,0,-1,0,1],[0,0,1,0,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> []
=> ? = 2 - 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]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 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]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,0,1],[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,0,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,0,1],[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,1,0],[0,0,1,0,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[1,0,-1,1,0],[0,0,1,0,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[1,0,-1,1,0],[0,0,1,0,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 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]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,0,1],[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,1,0,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,0,1],[0,0,1,0,0],[1,0,-1,1,0],[0,1,0,0,0],[0,0,1,0,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 1
[[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]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [[4,4],[]]
=> []
=> ? = 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]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,0,1],[0,0,1,0,0],[0,1,-1,1,0],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [[4,4],[]]
=> []
=> ? = 2 - 1
[[0,0,0,0,1],[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 1
[[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]]
=> [1,1,1,0,0,1,0,1,0,0]
=> [[4,2],[]]
=> []
=> ? = 3 - 1
[[0,0,1,0,0],[0,1,0,0,0],[1,-1,0,1,0],[0,0,0,0,1],[0,1,0,0,0]]
=> [1,1,1,0,0,1,0,1,0,0]
=> [[4,2],[]]
=> []
=> ? = 3 - 1
[[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]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> []
=> ? = 3 - 1
[[0,0,0,1,0],[0,1,0,0,0],[1,-1,1,0,0],[0,0,0,0,1],[0,1,0,0,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> []
=> ? = 3 - 1
[[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> []
=> ? = 3 - 1
[[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[1,-1,0,0,1],[0,1,0,0,0]]
=> [1,1,1,0,0,1,0,1,0,0]
=> [[4,2],[]]
=> []
=> ? = 3 - 1
[[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[1,-1,0,0,1],[0,1,0,0,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> []
=> ? = 3 - 1
[[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[1,-1,0,0,1],[0,1,0,0,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> []
=> ? = 3 - 1
[[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]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,1,0,0],[0,1,0,0,0],[1,-1,0,0,1],[0,0,0,1,0],[0,1,0,0,0]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0],[1,0,0,0,0],[0,0,1,-1,1],[0,0,0,1,0],[0,1,0,0,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0],[0,1,0,0,0],[1,-1,1,-1,1],[0,0,0,1,0],[0,1,0,0,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0],[0,0,1,0,0],[1,0,0,-1,1],[0,0,0,1,0],[0,1,0,0,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[1,-1,0,1,0],[0,1,0,0,0]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[1,-1,0,1,0],[0,1,0,0,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0],[0,0,1,0,0],[0,1,0,-1,1],[1,-1,0,1,0],[0,1,0,0,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[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]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0],[0,1,0,0,0],[1,-1,0,0,1],[0,0,1,0,0],[0,1,0,0,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0],[0,0,1,0,0],[1,0,-1,0,1],[0,0,1,0,0],[0,1,0,0,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1],[1,-1,1,0,0],[0,1,0,0,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0],[0,0,1,0,0],[0,1,-1,0,1],[1,-1,1,0,0],[0,1,0,0,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[1,0,0,0,0]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0],[1,0,0,0,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0],[1,0,0,0,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[1,0,0,0,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0],[0,0,1,0,0],[0,1,-1,0,1],[0,0,1,0,0],[1,0,0,0,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [[4,4,2],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,1,0,0,0],[0,1,0,0,0,0],[1,-1,0,0,0,1],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [[4,4,2],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,1,-1,0,1],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [[4,4,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0,0],[0,1,0,0,0,0],[1,-1,1,-1,0,1],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [[4,4,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0,0],[0,0,1,0,0,0],[1,0,0,-1,0,1],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [[4,4,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [[4,4,2],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,1,-1,0,1],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [[4,4,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,-1,0,1],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [[4,4,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [[4,4,3],[2]]
=> [2]
=> 1 = 2 - 1
[[0,0,0,1,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [[4,4,3],[2]]
=> [2]
=> 1 = 2 - 1
[[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,-1,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [[4,4,3],[2]]
=> [2]
=> 1 = 2 - 1
[[0,0,0,1,0,0],[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,-1,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [[4,4,3],[2]]
=> [2]
=> 1 = 2 - 1
[[0,0,0,1,0,0],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,-1,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [[4,4,3],[2]]
=> [2]
=> 1 = 2 - 1
[[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[1,0,-1,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [[4,4,3],[2]]
=> [2]
=> 1 = 2 - 1
[[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,-1,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [[4,4,3],[2]]
=> [2]
=> 1 = 2 - 1
[[0,0,0,0,1,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,-1,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [[3,3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,0,1,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,-1,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [[3,3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,-1,1,-1,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [[3,3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,0,1,0],[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,-1,1,-1,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [[3,3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,0,1,0],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,-1,1,-1,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [[3,3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[1,0,-1,1,-1,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [[3,3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,0,1,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,-1,1,-1,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [[3,3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,1,0,0,-1,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [[3,3,3,3],[1,1]]
=> [1,1]
=> 1 = 2 - 1
[[0,0,0,1,0,0],[0,1,0,0,0,0],[1,-1,0,0,1,0],[0,1,0,0,-1,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [[3,3,3,3],[1,1]]
=> [1,1]
=> 1 = 2 - 1
Description
The number of permutations whose cube equals a fixed permutation of given cycle type.
For example, the permutation $\pi=412365$ has cycle type $(4,2)$ and $234165$ is the unique permutation whose cube is $\pi$.
Matching statistic: St001432
Mp00007: Alternating sign matrices —to Dyck path⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001432: Integer partitions ⟶ ℤResult quality: 33% ●values known / values provided: 36%●distinct values known / distinct values provided: 33%
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001432: Integer partitions ⟶ ℤResult quality: 33% ●values known / values provided: 36%●distinct values known / distinct values provided: 33%
Values
[[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> ? = 3 - 1
[[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
=> [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> ? = 3 - 1
[[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
=> [1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> []
=> ? = 2 - 1
[[0,0,1,0],[0,1,0,0],[1,-1,0,1],[0,1,0,0]]
=> [1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> []
=> ? = 2 - 1
[[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
=> [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,1],[0,1,0,0],[1,-1,1,0],[0,1,0,0]]
=> [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
=> [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> ? = 2 - 1
[[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]]
=> [1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> []
=> ? = 2 - 1
[[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
=> [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> ? = 2 - 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]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 3 - 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]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 3 - 1
[[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]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,1,0,0],[0,1,0,0,0],[1,-1,0,0,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0],[1,0,0,0,0],[0,0,1,-1,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0],[0,1,0,0,0],[1,-1,1,-1,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,1,0,0],[0,0,0,1,0],[1,0,0,-1,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [[2,2,2,2],[]]
=> []
=> ? = 2 - 1
[[0,0,0,1,0],[0,0,1,0,0],[1,0,0,-1,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 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]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,0,1],[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 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]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 1
[[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,0,1,0]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[1,0,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,1,0,0],[0,0,0,1,0],[0,1,0,-1,1],[1,0,0,0,0],[0,0,0,1,0]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [[2,2,2,2],[]]
=> []
=> ? = 2 - 1
[[0,0,0,1,0],[0,0,1,0,0],[0,1,0,-1,1],[1,0,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 1
[[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]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> []
=> ? = 2 - 1
[[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]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,0,1],[0,0,1,0,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,1,0],[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,0,1],[0,0,1,0,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,0,1],[0,0,1,0,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[1,0,-1,0,1],[0,0,1,0,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[1,0,-1,0,1],[0,0,1,0,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> []
=> ? = 2 - 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]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 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]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,0,1],[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,0,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,0,1],[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,1,0],[0,0,1,0,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[1,0,-1,1,0],[0,0,1,0,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[1,0,-1,1,0],[0,0,1,0,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 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]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,0,1],[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,1,0,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,0,1],[0,0,1,0,0],[1,0,-1,1,0],[0,1,0,0,0],[0,0,1,0,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 1
[[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]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [[4,4],[]]
=> []
=> ? = 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]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,0,1],[0,0,1,0,0],[0,1,-1,1,0],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [[4,4],[]]
=> []
=> ? = 2 - 1
[[0,0,0,0,1],[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 1
[[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]]
=> [1,1,1,0,0,1,0,1,0,0]
=> [[4,2],[]]
=> []
=> ? = 3 - 1
[[0,0,1,0,0],[0,1,0,0,0],[1,-1,0,1,0],[0,0,0,0,1],[0,1,0,0,0]]
=> [1,1,1,0,0,1,0,1,0,0]
=> [[4,2],[]]
=> []
=> ? = 3 - 1
[[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]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> []
=> ? = 3 - 1
[[0,0,0,1,0],[0,1,0,0,0],[1,-1,1,0,0],[0,0,0,0,1],[0,1,0,0,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> []
=> ? = 3 - 1
[[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> []
=> ? = 3 - 1
[[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[1,-1,0,0,1],[0,1,0,0,0]]
=> [1,1,1,0,0,1,0,1,0,0]
=> [[4,2],[]]
=> []
=> ? = 3 - 1
[[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[1,-1,0,0,1],[0,1,0,0,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> []
=> ? = 3 - 1
[[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[1,-1,0,0,1],[0,1,0,0,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> []
=> ? = 3 - 1
[[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]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,1,0,0],[0,1,0,0,0],[1,-1,0,0,1],[0,0,0,1,0],[0,1,0,0,0]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0],[1,0,0,0,0],[0,0,1,-1,1],[0,0,0,1,0],[0,1,0,0,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0],[0,1,0,0,0],[1,-1,1,-1,1],[0,0,0,1,0],[0,1,0,0,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0],[0,0,1,0,0],[1,0,0,-1,1],[0,0,0,1,0],[0,1,0,0,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[1,-1,0,1,0],[0,1,0,0,0]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[1,-1,0,1,0],[0,1,0,0,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0],[0,0,1,0,0],[0,1,0,-1,1],[1,-1,0,1,0],[0,1,0,0,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[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]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0],[0,1,0,0,0],[1,-1,0,0,1],[0,0,1,0,0],[0,1,0,0,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0],[0,0,1,0,0],[1,0,-1,0,1],[0,0,1,0,0],[0,1,0,0,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1],[1,-1,1,0,0],[0,1,0,0,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0],[0,0,1,0,0],[0,1,-1,0,1],[1,-1,1,0,0],[0,1,0,0,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[1,0,0,0,0]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0],[1,0,0,0,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0],[1,0,0,0,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[1,0,0,0,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0],[0,0,1,0,0],[0,1,-1,0,1],[0,0,1,0,0],[1,0,0,0,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [[4,4,2],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,1,0,0,0],[0,1,0,0,0,0],[1,-1,0,0,0,1],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [[4,4,2],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,1,-1,0,1],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [[4,4,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0,0],[0,1,0,0,0,0],[1,-1,1,-1,0,1],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [[4,4,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0,0],[0,0,1,0,0,0],[1,0,0,-1,0,1],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [[4,4,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [[4,4,2],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,1,-1,0,1],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [[4,4,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,-1,0,1],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [[4,4,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [[4,4,3],[2]]
=> [2]
=> 1 = 2 - 1
[[0,0,0,1,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [[4,4,3],[2]]
=> [2]
=> 1 = 2 - 1
[[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,-1,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [[4,4,3],[2]]
=> [2]
=> 1 = 2 - 1
[[0,0,0,1,0,0],[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,-1,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [[4,4,3],[2]]
=> [2]
=> 1 = 2 - 1
[[0,0,0,1,0,0],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,-1,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [[4,4,3],[2]]
=> [2]
=> 1 = 2 - 1
[[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[1,0,-1,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [[4,4,3],[2]]
=> [2]
=> 1 = 2 - 1
[[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,-1,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [[4,4,3],[2]]
=> [2]
=> 1 = 2 - 1
[[0,0,0,0,1,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,-1,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [[3,3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,0,1,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,-1,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [[3,3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,-1,1,-1,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [[3,3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,0,1,0],[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,-1,1,-1,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [[3,3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,0,1,0],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,-1,1,-1,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [[3,3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[1,0,-1,1,-1,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [[3,3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,0,1,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,-1,1,-1,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [[3,3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,1,0,0,-1,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [[3,3,3,3],[1,1]]
=> [1,1]
=> 1 = 2 - 1
[[0,0,0,1,0,0],[0,1,0,0,0,0],[1,-1,0,0,1,0],[0,1,0,0,-1,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [[3,3,3,3],[1,1]]
=> [1,1]
=> 1 = 2 - 1
Description
The order dimension of the partition.
Given a partition $\lambda$, let $I(\lambda)$ be the principal order ideal in the Young lattice generated by $\lambda$. The order dimension of a partition is defined as the order dimension of the poset $I(\lambda)$.
Matching statistic: St001599
Mp00007: Alternating sign matrices —to Dyck path⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001599: Integer partitions ⟶ ℤResult quality: 33% ●values known / values provided: 36%●distinct values known / distinct values provided: 33%
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001599: Integer partitions ⟶ ℤResult quality: 33% ●values known / values provided: 36%●distinct values known / distinct values provided: 33%
Values
[[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> ? = 3 - 1
[[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
=> [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> ? = 3 - 1
[[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
=> [1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> []
=> ? = 2 - 1
[[0,0,1,0],[0,1,0,0],[1,-1,0,1],[0,1,0,0]]
=> [1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> []
=> ? = 2 - 1
[[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
=> [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,1],[0,1,0,0],[1,-1,1,0],[0,1,0,0]]
=> [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
=> [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> ? = 2 - 1
[[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]]
=> [1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> []
=> ? = 2 - 1
[[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
=> [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> ? = 2 - 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]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 3 - 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]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 3 - 1
[[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]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,1,0,0],[0,1,0,0,0],[1,-1,0,0,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0],[1,0,0,0,0],[0,0,1,-1,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0],[0,1,0,0,0],[1,-1,1,-1,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,1,0,0],[0,0,0,1,0],[1,0,0,-1,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [[2,2,2,2],[]]
=> []
=> ? = 2 - 1
[[0,0,0,1,0],[0,0,1,0,0],[1,0,0,-1,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 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]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,0,1],[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 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]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 1
[[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,0,1,0]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[1,0,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,1,0,0],[0,0,0,1,0],[0,1,0,-1,1],[1,0,0,0,0],[0,0,0,1,0]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [[2,2,2,2],[]]
=> []
=> ? = 2 - 1
[[0,0,0,1,0],[0,0,1,0,0],[0,1,0,-1,1],[1,0,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 1
[[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]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> []
=> ? = 2 - 1
[[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]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,0,1],[0,0,1,0,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,1,0],[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,0,1],[0,0,1,0,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,0,1],[0,0,1,0,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[1,0,-1,0,1],[0,0,1,0,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[1,0,-1,0,1],[0,0,1,0,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> []
=> ? = 2 - 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]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 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]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,0,1],[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,0,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,0,1],[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,1,0],[0,0,1,0,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[1,0,-1,1,0],[0,0,1,0,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[1,0,-1,1,0],[0,0,1,0,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 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]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,0,1],[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,1,0,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,0,1],[0,0,1,0,0],[1,0,-1,1,0],[0,1,0,0,0],[0,0,1,0,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 1
[[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]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [[4,4],[]]
=> []
=> ? = 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]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,0,1],[0,0,1,0,0],[0,1,-1,1,0],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [[4,4],[]]
=> []
=> ? = 2 - 1
[[0,0,0,0,1],[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 1
[[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]]
=> [1,1,1,0,0,1,0,1,0,0]
=> [[4,2],[]]
=> []
=> ? = 3 - 1
[[0,0,1,0,0],[0,1,0,0,0],[1,-1,0,1,0],[0,0,0,0,1],[0,1,0,0,0]]
=> [1,1,1,0,0,1,0,1,0,0]
=> [[4,2],[]]
=> []
=> ? = 3 - 1
[[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]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> []
=> ? = 3 - 1
[[0,0,0,1,0],[0,1,0,0,0],[1,-1,1,0,0],[0,0,0,0,1],[0,1,0,0,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> []
=> ? = 3 - 1
[[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> []
=> ? = 3 - 1
[[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[1,-1,0,0,1],[0,1,0,0,0]]
=> [1,1,1,0,0,1,0,1,0,0]
=> [[4,2],[]]
=> []
=> ? = 3 - 1
[[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[1,-1,0,0,1],[0,1,0,0,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> []
=> ? = 3 - 1
[[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[1,-1,0,0,1],[0,1,0,0,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> []
=> ? = 3 - 1
[[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]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,1,0,0],[0,1,0,0,0],[1,-1,0,0,1],[0,0,0,1,0],[0,1,0,0,0]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0],[1,0,0,0,0],[0,0,1,-1,1],[0,0,0,1,0],[0,1,0,0,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0],[0,1,0,0,0],[1,-1,1,-1,1],[0,0,0,1,0],[0,1,0,0,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0],[0,0,1,0,0],[1,0,0,-1,1],[0,0,0,1,0],[0,1,0,0,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[1,-1,0,1,0],[0,1,0,0,0]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[1,-1,0,1,0],[0,1,0,0,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0],[0,0,1,0,0],[0,1,0,-1,1],[1,-1,0,1,0],[0,1,0,0,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[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]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0],[0,1,0,0,0],[1,-1,0,0,1],[0,0,1,0,0],[0,1,0,0,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0],[0,0,1,0,0],[1,0,-1,0,1],[0,0,1,0,0],[0,1,0,0,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1],[1,-1,1,0,0],[0,1,0,0,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0],[0,0,1,0,0],[0,1,-1,0,1],[1,-1,1,0,0],[0,1,0,0,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[1,0,0,0,0]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0],[1,0,0,0,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0],[1,0,0,0,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[1,0,0,0,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0],[0,0,1,0,0],[0,1,-1,0,1],[0,0,1,0,0],[1,0,0,0,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [[4,4,2],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,1,0,0,0],[0,1,0,0,0,0],[1,-1,0,0,0,1],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [[4,4,2],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,1,-1,0,1],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [[4,4,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0,0],[0,1,0,0,0,0],[1,-1,1,-1,0,1],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [[4,4,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0,0],[0,0,1,0,0,0],[1,0,0,-1,0,1],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [[4,4,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [[4,4,2],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,1,-1,0,1],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [[4,4,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,-1,0,1],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [[4,4,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [[4,4,3],[2]]
=> [2]
=> 1 = 2 - 1
[[0,0,0,1,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [[4,4,3],[2]]
=> [2]
=> 1 = 2 - 1
[[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,-1,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [[4,4,3],[2]]
=> [2]
=> 1 = 2 - 1
[[0,0,0,1,0,0],[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,-1,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [[4,4,3],[2]]
=> [2]
=> 1 = 2 - 1
[[0,0,0,1,0,0],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,-1,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [[4,4,3],[2]]
=> [2]
=> 1 = 2 - 1
[[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[1,0,-1,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [[4,4,3],[2]]
=> [2]
=> 1 = 2 - 1
[[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,-1,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [[4,4,3],[2]]
=> [2]
=> 1 = 2 - 1
[[0,0,0,0,1,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,-1,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [[3,3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,0,1,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,-1,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [[3,3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,-1,1,-1,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [[3,3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,0,1,0],[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,-1,1,-1,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [[3,3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,0,1,0],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,-1,1,-1,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [[3,3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[1,0,-1,1,-1,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [[3,3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,0,1,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,-1,1,-1,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [[3,3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,1,0,0,-1,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [[3,3,3,3],[1,1]]
=> [1,1]
=> 1 = 2 - 1
[[0,0,0,1,0,0],[0,1,0,0,0,0],[1,-1,0,0,1,0],[0,1,0,0,-1,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [[3,3,3,3],[1,1]]
=> [1,1]
=> 1 = 2 - 1
Description
The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees.
Matching statistic: St001609
Mp00007: Alternating sign matrices —to Dyck path⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001609: Integer partitions ⟶ ℤResult quality: 33% ●values known / values provided: 36%●distinct values known / distinct values provided: 33%
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001609: Integer partitions ⟶ ℤResult quality: 33% ●values known / values provided: 36%●distinct values known / distinct values provided: 33%
Values
[[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> ? = 3 - 1
[[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
=> [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> ? = 3 - 1
[[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
=> [1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> []
=> ? = 2 - 1
[[0,0,1,0],[0,1,0,0],[1,-1,0,1],[0,1,0,0]]
=> [1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> []
=> ? = 2 - 1
[[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
=> [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,1],[0,1,0,0],[1,-1,1,0],[0,1,0,0]]
=> [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
=> [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> ? = 2 - 1
[[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]]
=> [1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> []
=> ? = 2 - 1
[[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
=> [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> ? = 2 - 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]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 3 - 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]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 3 - 1
[[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]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,1,0,0],[0,1,0,0,0],[1,-1,0,0,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0],[1,0,0,0,0],[0,0,1,-1,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0],[0,1,0,0,0],[1,-1,1,-1,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,1,0,0],[0,0,0,1,0],[1,0,0,-1,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [[2,2,2,2],[]]
=> []
=> ? = 2 - 1
[[0,0,0,1,0],[0,0,1,0,0],[1,0,0,-1,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 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]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,0,1],[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 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]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 1
[[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,0,1,0]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[1,0,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,1,0,0],[0,0,0,1,0],[0,1,0,-1,1],[1,0,0,0,0],[0,0,0,1,0]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [[2,2,2,2],[]]
=> []
=> ? = 2 - 1
[[0,0,0,1,0],[0,0,1,0,0],[0,1,0,-1,1],[1,0,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 1
[[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]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> []
=> ? = 2 - 1
[[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]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,0,1],[0,0,1,0,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,1,0],[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,0,1],[0,0,1,0,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,0,1],[0,0,1,0,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[1,0,-1,0,1],[0,0,1,0,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[1,0,-1,0,1],[0,0,1,0,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> []
=> ? = 2 - 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]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 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]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,0,1],[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,0,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,0,1],[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,1,0],[0,0,1,0,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[1,0,-1,1,0],[0,0,1,0,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[1,0,-1,1,0],[0,0,1,0,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 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]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,0,1],[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,1,0,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,0,1],[0,0,1,0,0],[1,0,-1,1,0],[0,1,0,0,0],[0,0,1,0,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 1
[[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]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [[4,4],[]]
=> []
=> ? = 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]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,0,1],[0,0,1,0,0],[0,1,-1,1,0],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [[4,4],[]]
=> []
=> ? = 2 - 1
[[0,0,0,0,1],[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 1
[[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]]
=> [1,1,1,0,0,1,0,1,0,0]
=> [[4,2],[]]
=> []
=> ? = 3 - 1
[[0,0,1,0,0],[0,1,0,0,0],[1,-1,0,1,0],[0,0,0,0,1],[0,1,0,0,0]]
=> [1,1,1,0,0,1,0,1,0,0]
=> [[4,2],[]]
=> []
=> ? = 3 - 1
[[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]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> []
=> ? = 3 - 1
[[0,0,0,1,0],[0,1,0,0,0],[1,-1,1,0,0],[0,0,0,0,1],[0,1,0,0,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> []
=> ? = 3 - 1
[[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> []
=> ? = 3 - 1
[[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[1,-1,0,0,1],[0,1,0,0,0]]
=> [1,1,1,0,0,1,0,1,0,0]
=> [[4,2],[]]
=> []
=> ? = 3 - 1
[[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[1,-1,0,0,1],[0,1,0,0,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> []
=> ? = 3 - 1
[[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[1,-1,0,0,1],[0,1,0,0,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> []
=> ? = 3 - 1
[[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]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,1,0,0],[0,1,0,0,0],[1,-1,0,0,1],[0,0,0,1,0],[0,1,0,0,0]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0],[1,0,0,0,0],[0,0,1,-1,1],[0,0,0,1,0],[0,1,0,0,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0],[0,1,0,0,0],[1,-1,1,-1,1],[0,0,0,1,0],[0,1,0,0,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0],[0,0,1,0,0],[1,0,0,-1,1],[0,0,0,1,0],[0,1,0,0,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[1,-1,0,1,0],[0,1,0,0,0]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[1,-1,0,1,0],[0,1,0,0,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0],[0,0,1,0,0],[0,1,0,-1,1],[1,-1,0,1,0],[0,1,0,0,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[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]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0],[0,1,0,0,0],[1,-1,0,0,1],[0,0,1,0,0],[0,1,0,0,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0],[0,0,1,0,0],[1,0,-1,0,1],[0,0,1,0,0],[0,1,0,0,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1],[1,-1,1,0,0],[0,1,0,0,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0],[0,0,1,0,0],[0,1,-1,0,1],[1,-1,1,0,0],[0,1,0,0,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[1,0,0,0,0]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0],[1,0,0,0,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0],[1,0,0,0,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[1,0,0,0,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0],[0,0,1,0,0],[0,1,-1,0,1],[0,0,1,0,0],[1,0,0,0,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [[4,4,2],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,1,0,0,0],[0,1,0,0,0,0],[1,-1,0,0,0,1],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [[4,4,2],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,1,-1,0,1],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [[4,4,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0,0],[0,1,0,0,0,0],[1,-1,1,-1,0,1],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [[4,4,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0,0],[0,0,1,0,0,0],[1,0,0,-1,0,1],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [[4,4,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [[4,4,2],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,1,-1,0,1],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [[4,4,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,-1,0,1],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [[4,4,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [[4,4,3],[2]]
=> [2]
=> 1 = 2 - 1
[[0,0,0,1,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [[4,4,3],[2]]
=> [2]
=> 1 = 2 - 1
[[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,-1,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [[4,4,3],[2]]
=> [2]
=> 1 = 2 - 1
[[0,0,0,1,0,0],[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,-1,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [[4,4,3],[2]]
=> [2]
=> 1 = 2 - 1
[[0,0,0,1,0,0],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,-1,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [[4,4,3],[2]]
=> [2]
=> 1 = 2 - 1
[[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[1,0,-1,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [[4,4,3],[2]]
=> [2]
=> 1 = 2 - 1
[[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,-1,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [[4,4,3],[2]]
=> [2]
=> 1 = 2 - 1
[[0,0,0,0,1,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,-1,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [[3,3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,0,1,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,-1,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [[3,3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,-1,1,-1,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [[3,3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,0,1,0],[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,-1,1,-1,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [[3,3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,0,1,0],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,-1,1,-1,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [[3,3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[1,0,-1,1,-1,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [[3,3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,0,1,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,-1,1,-1,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [[3,3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,1,0,0,-1,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [[3,3,3,3],[1,1]]
=> [1,1]
=> 1 = 2 - 1
[[0,0,0,1,0,0],[0,1,0,0,0,0],[1,-1,0,0,1,0],[0,1,0,0,-1,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [[3,3,3,3],[1,1]]
=> [1,1]
=> 1 = 2 - 1
Description
The number of coloured trees such that the multiplicities of colours are given by a partition.
In particular, the value on the partition $(n)$ is the number of unlabelled trees on $n$ vertices, [[oeis:A000055]], whereas the value on the partition $(1^n)$ is the number of labelled trees [[oeis:A000272]].
Matching statistic: St001627
Mp00007: Alternating sign matrices —to Dyck path⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001627: Integer partitions ⟶ ℤResult quality: 33% ●values known / values provided: 36%●distinct values known / distinct values provided: 33%
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001627: Integer partitions ⟶ ℤResult quality: 33% ●values known / values provided: 36%●distinct values known / distinct values provided: 33%
Values
[[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> ? = 3 - 1
[[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
=> [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> ? = 3 - 1
[[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
=> [1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> []
=> ? = 2 - 1
[[0,0,1,0],[0,1,0,0],[1,-1,0,1],[0,1,0,0]]
=> [1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> []
=> ? = 2 - 1
[[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
=> [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,1],[0,1,0,0],[1,-1,1,0],[0,1,0,0]]
=> [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
=> [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> ? = 2 - 1
[[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]]
=> [1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> []
=> ? = 2 - 1
[[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
=> [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> ? = 2 - 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]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 3 - 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]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 3 - 1
[[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]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,1,0,0],[0,1,0,0,0],[1,-1,0,0,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0],[1,0,0,0,0],[0,0,1,-1,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0],[0,1,0,0,0],[1,-1,1,-1,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,1,0,0],[0,0,0,1,0],[1,0,0,-1,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [[2,2,2,2],[]]
=> []
=> ? = 2 - 1
[[0,0,0,1,0],[0,0,1,0,0],[1,0,0,-1,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 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]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,0,1],[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 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]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 1
[[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,0,1,0]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[1,0,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,1,0,0],[0,0,0,1,0],[0,1,0,-1,1],[1,0,0,0,0],[0,0,0,1,0]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [[2,2,2,2],[]]
=> []
=> ? = 2 - 1
[[0,0,0,1,0],[0,0,1,0,0],[0,1,0,-1,1],[1,0,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 1
[[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]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> []
=> ? = 2 - 1
[[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]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,0,1],[0,0,1,0,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,1,0],[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,0,1],[0,0,1,0,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,0,1],[0,0,1,0,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[1,0,-1,0,1],[0,0,1,0,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[1,0,-1,0,1],[0,0,1,0,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> []
=> ? = 2 - 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]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 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]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,0,1],[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,0,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,0,1],[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,1,0],[0,0,1,0,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[1,0,-1,1,0],[0,0,1,0,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[1,0,-1,1,0],[0,0,1,0,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 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]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,0,1],[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,1,0,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,0,1],[0,0,1,0,0],[1,0,-1,1,0],[0,1,0,0,0],[0,0,1,0,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 1
[[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]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [[4,4],[]]
=> []
=> ? = 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]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,0,1],[0,0,1,0,0],[0,1,-1,1,0],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [[4,4],[]]
=> []
=> ? = 2 - 1
[[0,0,0,0,1],[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 1
[[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]]
=> [1,1,1,0,0,1,0,1,0,0]
=> [[4,2],[]]
=> []
=> ? = 3 - 1
[[0,0,1,0,0],[0,1,0,0,0],[1,-1,0,1,0],[0,0,0,0,1],[0,1,0,0,0]]
=> [1,1,1,0,0,1,0,1,0,0]
=> [[4,2],[]]
=> []
=> ? = 3 - 1
[[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]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> []
=> ? = 3 - 1
[[0,0,0,1,0],[0,1,0,0,0],[1,-1,1,0,0],[0,0,0,0,1],[0,1,0,0,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> []
=> ? = 3 - 1
[[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> []
=> ? = 3 - 1
[[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[1,-1,0,0,1],[0,1,0,0,0]]
=> [1,1,1,0,0,1,0,1,0,0]
=> [[4,2],[]]
=> []
=> ? = 3 - 1
[[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[1,-1,0,0,1],[0,1,0,0,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> []
=> ? = 3 - 1
[[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[1,-1,0,0,1],[0,1,0,0,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> []
=> ? = 3 - 1
[[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]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,1,0,0],[0,1,0,0,0],[1,-1,0,0,1],[0,0,0,1,0],[0,1,0,0,0]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0],[1,0,0,0,0],[0,0,1,-1,1],[0,0,0,1,0],[0,1,0,0,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0],[0,1,0,0,0],[1,-1,1,-1,1],[0,0,0,1,0],[0,1,0,0,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0],[0,0,1,0,0],[1,0,0,-1,1],[0,0,0,1,0],[0,1,0,0,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[1,-1,0,1,0],[0,1,0,0,0]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[1,-1,0,1,0],[0,1,0,0,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0],[0,0,1,0,0],[0,1,0,-1,1],[1,-1,0,1,0],[0,1,0,0,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[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]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0],[0,1,0,0,0],[1,-1,0,0,1],[0,0,1,0,0],[0,1,0,0,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0],[0,0,1,0,0],[1,0,-1,0,1],[0,0,1,0,0],[0,1,0,0,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1],[1,-1,1,0,0],[0,1,0,0,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0],[0,0,1,0,0],[0,1,-1,0,1],[1,-1,1,0,0],[0,1,0,0,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[1,0,0,0,0]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0],[1,0,0,0,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0],[1,0,0,0,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[1,0,0,0,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0],[0,0,1,0,0],[0,1,-1,0,1],[0,0,1,0,0],[1,0,0,0,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [[4,4,2],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,1,0,0,0],[0,1,0,0,0,0],[1,-1,0,0,0,1],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [[4,4,2],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,1,-1,0,1],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [[4,4,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0,0],[0,1,0,0,0,0],[1,-1,1,-1,0,1],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [[4,4,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0,0],[0,0,1,0,0,0],[1,0,0,-1,0,1],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [[4,4,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [[4,4,2],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,1,-1,0,1],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [[4,4,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,-1,0,1],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [[4,4,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [[4,4,3],[2]]
=> [2]
=> 1 = 2 - 1
[[0,0,0,1,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [[4,4,3],[2]]
=> [2]
=> 1 = 2 - 1
[[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,-1,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [[4,4,3],[2]]
=> [2]
=> 1 = 2 - 1
[[0,0,0,1,0,0],[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,-1,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [[4,4,3],[2]]
=> [2]
=> 1 = 2 - 1
[[0,0,0,1,0,0],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,-1,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [[4,4,3],[2]]
=> [2]
=> 1 = 2 - 1
[[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[1,0,-1,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [[4,4,3],[2]]
=> [2]
=> 1 = 2 - 1
[[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,-1,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [[4,4,3],[2]]
=> [2]
=> 1 = 2 - 1
[[0,0,0,0,1,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,-1,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [[3,3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,0,1,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,-1,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [[3,3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,-1,1,-1,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [[3,3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,0,1,0],[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,-1,1,-1,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [[3,3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,0,1,0],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,-1,1,-1,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [[3,3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[1,0,-1,1,-1,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [[3,3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,0,1,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,-1,1,-1,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [[3,3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,1,0,0,-1,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [[3,3,3,3],[1,1]]
=> [1,1]
=> 1 = 2 - 1
[[0,0,0,1,0,0],[0,1,0,0,0,0],[1,-1,0,0,1,0],[0,1,0,0,-1,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [[3,3,3,3],[1,1]]
=> [1,1]
=> 1 = 2 - 1
Description
The number of coloured connected graphs such that the multiplicities of colours are given by a partition.
In particular, the value on the partition $(n)$ is the number of unlabelled connected graphs on $n$ vertices, [[oeis:A001349]], whereas the value on the partition $(1^n)$ is the number of labelled connected graphs [[oeis:A001187]].
Matching statistic: St001763
Mp00007: Alternating sign matrices —to Dyck path⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001763: Integer partitions ⟶ ℤResult quality: 33% ●values known / values provided: 36%●distinct values known / distinct values provided: 33%
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001763: Integer partitions ⟶ ℤResult quality: 33% ●values known / values provided: 36%●distinct values known / distinct values provided: 33%
Values
[[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> ? = 3 - 1
[[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
=> [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> ? = 3 - 1
[[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
=> [1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> []
=> ? = 2 - 1
[[0,0,1,0],[0,1,0,0],[1,-1,0,1],[0,1,0,0]]
=> [1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> []
=> ? = 2 - 1
[[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
=> [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,1],[0,1,0,0],[1,-1,1,0],[0,1,0,0]]
=> [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
=> [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> ? = 2 - 1
[[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]]
=> [1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> []
=> ? = 2 - 1
[[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
=> [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> ? = 2 - 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]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 3 - 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]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 3 - 1
[[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]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,1,0,0],[0,1,0,0,0],[1,-1,0,0,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0],[1,0,0,0,0],[0,0,1,-1,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0],[0,1,0,0,0],[1,-1,1,-1,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,1,0,0],[0,0,0,1,0],[1,0,0,-1,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [[2,2,2,2],[]]
=> []
=> ? = 2 - 1
[[0,0,0,1,0],[0,0,1,0,0],[1,0,0,-1,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 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]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,0,1],[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 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]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 1
[[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,0,1,0]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[1,0,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,1,0,0],[0,0,0,1,0],[0,1,0,-1,1],[1,0,0,0,0],[0,0,0,1,0]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [[2,2,2,2],[]]
=> []
=> ? = 2 - 1
[[0,0,0,1,0],[0,0,1,0,0],[0,1,0,-1,1],[1,0,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 1
[[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]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> []
=> ? = 2 - 1
[[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]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,0,1],[0,0,1,0,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,1,0],[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,0,1],[0,0,1,0,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,0,1],[0,0,1,0,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[1,0,-1,0,1],[0,0,1,0,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[1,0,-1,0,1],[0,0,1,0,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> []
=> ? = 2 - 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]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 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]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,0,1],[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,0,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,0,1],[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,1,0],[0,0,1,0,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[1,0,-1,1,0],[0,0,1,0,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[1,0,-1,1,0],[0,0,1,0,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 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]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,0,1],[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,1,0,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,0,1],[0,0,1,0,0],[1,0,-1,1,0],[0,1,0,0,0],[0,0,1,0,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 1
[[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]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [[4,4],[]]
=> []
=> ? = 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]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,0,1],[0,0,1,0,0],[0,1,-1,1,0],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 1
[[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [[4,4],[]]
=> []
=> ? = 2 - 1
[[0,0,0,0,1],[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? = 2 - 1
[[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]]
=> [1,1,1,0,0,1,0,1,0,0]
=> [[4,2],[]]
=> []
=> ? = 3 - 1
[[0,0,1,0,0],[0,1,0,0,0],[1,-1,0,1,0],[0,0,0,0,1],[0,1,0,0,0]]
=> [1,1,1,0,0,1,0,1,0,0]
=> [[4,2],[]]
=> []
=> ? = 3 - 1
[[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]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> []
=> ? = 3 - 1
[[0,0,0,1,0],[0,1,0,0,0],[1,-1,1,0,0],[0,0,0,0,1],[0,1,0,0,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> []
=> ? = 3 - 1
[[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> []
=> ? = 3 - 1
[[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[1,-1,0,0,1],[0,1,0,0,0]]
=> [1,1,1,0,0,1,0,1,0,0]
=> [[4,2],[]]
=> []
=> ? = 3 - 1
[[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[1,-1,0,0,1],[0,1,0,0,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> []
=> ? = 3 - 1
[[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[1,-1,0,0,1],[0,1,0,0,0]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> []
=> ? = 3 - 1
[[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]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,1,0,0],[0,1,0,0,0],[1,-1,0,0,1],[0,0,0,1,0],[0,1,0,0,0]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0],[1,0,0,0,0],[0,0,1,-1,1],[0,0,0,1,0],[0,1,0,0,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0],[0,1,0,0,0],[1,-1,1,-1,1],[0,0,0,1,0],[0,1,0,0,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0],[0,0,1,0,0],[1,0,0,-1,1],[0,0,0,1,0],[0,1,0,0,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[1,-1,0,1,0],[0,1,0,0,0]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[1,-1,0,1,0],[0,1,0,0,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0],[0,0,1,0,0],[0,1,0,-1,1],[1,-1,0,1,0],[0,1,0,0,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[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]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0],[0,1,0,0,0],[1,-1,0,0,1],[0,0,1,0,0],[0,1,0,0,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0],[0,0,1,0,0],[1,0,-1,0,1],[0,0,1,0,0],[0,1,0,0,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1],[1,-1,1,0,0],[0,1,0,0,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0],[0,0,1,0,0],[0,1,-1,0,1],[1,-1,1,0,0],[0,1,0,0,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[1,0,0,0,0]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0],[1,0,0,0,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0],[1,0,0,0,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[1,0,0,0,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0],[0,0,1,0,0],[0,1,-1,0,1],[0,0,1,0,0],[1,0,0,0,0]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [[4,4,2],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,1,0,0,0],[0,1,0,0,0,0],[1,-1,0,0,0,1],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [[4,4,2],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,1,-1,0,1],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [[4,4,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0,0],[0,1,0,0,0,0],[1,-1,1,-1,0,1],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [[4,4,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0,0],[0,0,1,0,0,0],[1,0,0,-1,0,1],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [[4,4,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [[4,4,2],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,1,-1,0,1],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [[4,4,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,-1,0,1],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [[4,4,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [[4,4,3],[2]]
=> [2]
=> 1 = 2 - 1
[[0,0,0,1,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [[4,4,3],[2]]
=> [2]
=> 1 = 2 - 1
[[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,-1,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [[4,4,3],[2]]
=> [2]
=> 1 = 2 - 1
[[0,0,0,1,0,0],[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,-1,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [[4,4,3],[2]]
=> [2]
=> 1 = 2 - 1
[[0,0,0,1,0,0],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,-1,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [[4,4,3],[2]]
=> [2]
=> 1 = 2 - 1
[[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[1,0,-1,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [[4,4,3],[2]]
=> [2]
=> 1 = 2 - 1
[[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,-1,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [[4,4,3],[2]]
=> [2]
=> 1 = 2 - 1
[[0,0,0,0,1,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,-1,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [[3,3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,0,1,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,-1,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [[3,3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,-1,1,-1,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [[3,3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,0,1,0],[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,-1,1,-1,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [[3,3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,0,1,0],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,-1,1,-1,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [[3,3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[1,0,-1,1,-1,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [[3,3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,0,1,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,-1,1,-1,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [[3,3,3,3],[1]]
=> [1]
=> 1 = 2 - 1
[[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,1,0,0,-1,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [[3,3,3,3],[1,1]]
=> [1,1]
=> 1 = 2 - 1
[[0,0,0,1,0,0],[0,1,0,0,0,0],[1,-1,0,0,1,0],[0,1,0,0,-1,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [[3,3,3,3],[1,1]]
=> [1,1]
=> 1 = 2 - 1
Description
The Hurwitz number of an integer partition.
See [eq.(9),pg.21, 1].
The following 86 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001780The order of promotion on the set of standard tableaux of given shape. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001924The number of cells in an integer partition whose arm and leg length coincide. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St001938The number of transitive monotone factorizations of genus zero of a permutation of given cycle type. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000225Difference between largest and smallest parts in a partition. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St000944The 3-degree of an integer partition. St001175The size of a partition minus the hook length of the base cell. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001498The normalised height of a Nakayama algebra with magnitude 1. St001820The size of the image of the pop stack sorting operator. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000284The Plancherel distribution on integer partitions. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000681The Grundy value of Chomp on Ferrers diagrams. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000901The cube of the number of standard Young tableaux with shape given by the partition. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001128The exponens consonantiae of a partition. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St000941The number of characters of the symmetric group whose value on the partition is even. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000454The largest eigenvalue of a graph if it is integral. St000422The energy of a graph, if it is integral. St000007The number of saliances of the permutation. St000054The first entry of the permutation. St000451The length of the longest pattern of the form k 1 2. St000725The smallest label of a leaf of the increasing binary tree associated to a permutation. St000842The breadth of a permutation. St000891The number of distinct diagonal sums of a permutation matrix. St000028The number of stack-sorts needed to sort a permutation. St000141The maximum drop size of a permutation. St000352The Elizalde-Pak rank of a permutation. St000402Half the size of the symmetry class of a permutation. St000651The maximal size of a rise in a permutation. St000669The number of permutations obtained by switching ascents or descents of size 2. St000696The number of cycles in the breakpoint graph of a permutation. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St000022The number of fixed points of a permutation. St000119The number of occurrences of the pattern 321 in a permutation. St000123The difference in Coxeter length of a permutation and its image under the Simion-Schmidt map. St000223The number of nestings in the permutation. St000359The number of occurrences of the pattern 23-1. St000366The number of double descents of a permutation. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000441The number of successions of a permutation. St000534The number of 2-rises of a permutation. St000546The number of global descents of a permutation. St000648The number of 2-excedences of a permutation. St000665The number of rafts of a permutation. St000731The number of double exceedences of a permutation. St001115The number of even descents of a permutation. St001394The genus of a permutation. St000058The order of a permutation. St001616The number of neutral elements in a lattice. St001720The minimal length of a chain of small intervals in a lattice. St001613The binary logarithm of the size of the center of a lattice. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001881The number of factors of a lattice as a Cartesian product of lattices. St001615The number of join prime elements of a lattice. St001846The number of elements which do not have a complement in the lattice. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001875The number of simple modules with projective dimension at most 1. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2.
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