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Your data matches 11 different statistics following compositions of up to 3 maps.
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Matching statistic: St001875
Mp00002: Alternating sign matrices —to left key permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00206: Posets —antichains of maximal size⟶ Lattices
St001875: Lattices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00065: Permutations —permutation poset⟶ Posets
Mp00206: Posets —antichains of maximal size⟶ Lattices
St001875: Lattices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,0,0],[0,1,0],[0,0,1]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 3
[[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
=> [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> 3
[[0,0,1,0],[0,1,-1,1],[1,0,0,0],[0,0,1,0]]
=> [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> 3
[[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
=> [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> 3
[[0,0,1,0],[0,1,0,0],[1,-1,0,1],[0,1,0,0]]
=> [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> 3
[[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
[[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> [2,3,4,1] => ([(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 3
[[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [4,1,2,3,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 3
[[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [2,4,1,3,5] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 3
[[0,0,1,0,0],[0,1,-1,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [2,4,1,3,5] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 3
[[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [3,1,4,2,5] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 3
[[0,0,1,0,0],[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [3,1,4,2,5] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 3
[[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [3,4,1,2,5] => ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
[[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,0,0,1]]
=> [2,3,4,1,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 3
[[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [3,1,2,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> 3
[[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [2,3,1,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> 3
[[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [3,1,2,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> 3
[[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [2,3,1,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> 3
[[0,0,1,0,0],[0,1,-1,1,0],[1,0,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [2,3,1,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> 3
[[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [3,1,2,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> 3
[[0,0,1,0,0],[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [3,1,2,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> 3
[[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [3,1,2,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> 3
[[0,0,0,1,0],[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [3,1,2,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> 3
[[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[1,0,0,-1,1],[0,0,0,1,0]]
=> [2,3,1,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> 3
[[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[1,0,0,-1,1],[0,0,0,1,0]]
=> [2,3,1,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> 3
[[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[1,0,0,-1,1],[0,0,0,1,0]]
=> [2,3,1,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> 3
[[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [2,1,5,3,4] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,2),(2,1)],3)
=> 3
[[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [2,1,5,3,4] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,2),(2,1)],3)
=> 3
[[0,1,0,0,0],[0,0,1,0,0],[1,0,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [2,1,5,3,4] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,2),(2,1)],3)
=> 3
[[0,0,1,0,0],[0,1,0,0,0],[1,0,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [2,1,5,3,4] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,2),(2,1)],3)
=> 3
[[0,0,0,1,0],[1,0,0,0,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [2,1,5,3,4] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,2),(2,1)],3)
=> 3
[[0,1,0,0,0],[0,0,0,1,0],[1,0,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [2,1,5,3,4] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,2),(2,1)],3)
=> 3
[[0,0,1,0,0],[0,1,-1,1,0],[1,0,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [2,1,5,3,4] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,2),(2,1)],3)
=> 3
[[0,0,0,1,0],[0,1,0,0,0],[1,0,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [2,1,5,3,4] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,2),(2,1)],3)
=> 3
[[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,5,2,3,4] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> 3
[[0,1,0,0,0],[1,-1,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,5,2,3,4] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> 3
[[0,0,1,0,0],[1,0,-1,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,5,2,3,4] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> 3
[[0,0,0,1,0],[1,0,0,-1,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,5,2,3,4] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> 3
[[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] => ([(1,4),(3,2),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[[0,1,0,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [2,5,1,3,4] => ([(0,4),(1,2),(1,4),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[[0,0,1,0,0],[0,1,-1,0,1],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [2,5,1,3,4] => ([(0,4),(1,2),(1,4),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[[0,0,0,1,0],[0,1,0,-1,1],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [2,5,1,3,4] => ([(0,4),(1,2),(1,4),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 3
[[0,1,0,0,0],[1,-1,1,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 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] => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[[0,1,0,0,0],[0,0,1,0,0],[1,-1,0,0,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 3
[[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] => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 4
Description
The number of simple modules with projective dimension at most 1.
Matching statistic: St001615
Mp00002: Alternating sign matrices —to left key permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00206: Posets —antichains of maximal size⟶ Lattices
St001615: Lattices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00065: Permutations —permutation poset⟶ Posets
Mp00206: Posets —antichains of maximal size⟶ Lattices
St001615: Lattices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,0,0],[0,1,0],[0,0,1]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
=> [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[0,0,1,0],[0,1,-1,1],[1,0,0,0],[0,0,1,0]]
=> [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
=> [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[0,0,1,0],[0,1,0,0],[1,-1,0,1],[0,1,0,0]]
=> [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> [2,3,4,1] => ([(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [4,1,2,3,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [2,4,1,3,5] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[0,0,1,0,0],[0,1,-1,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [2,4,1,3,5] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [3,1,4,2,5] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[0,0,1,0,0],[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [3,1,4,2,5] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [3,4,1,2,5] => ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,0,0,1]]
=> [2,3,4,1,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [3,1,2,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [2,3,1,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [3,1,2,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [2,3,1,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[0,0,1,0,0],[0,1,-1,1,0],[1,0,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [2,3,1,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [3,1,2,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[0,0,1,0,0],[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [3,1,2,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [3,1,2,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[0,0,0,1,0],[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [3,1,2,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[1,0,0,-1,1],[0,0,0,1,0]]
=> [2,3,1,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[1,0,0,-1,1],[0,0,0,1,0]]
=> [2,3,1,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[1,0,0,-1,1],[0,0,0,1,0]]
=> [2,3,1,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [2,1,5,3,4] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [2,1,5,3,4] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[0,1,0,0,0],[0,0,1,0,0],[1,0,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [2,1,5,3,4] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[0,0,1,0,0],[0,1,0,0,0],[1,0,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [2,1,5,3,4] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[0,0,0,1,0],[1,0,0,0,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [2,1,5,3,4] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[0,1,0,0,0],[0,0,0,1,0],[1,0,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [2,1,5,3,4] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[0,0,1,0,0],[0,1,-1,1,0],[1,0,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [2,1,5,3,4] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[0,0,0,1,0],[0,1,0,0,0],[1,0,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [2,1,5,3,4] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,5,2,3,4] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[0,1,0,0,0],[1,-1,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,5,2,3,4] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[0,0,1,0,0],[1,0,-1,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,5,2,3,4] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[0,0,0,1,0],[1,0,0,-1,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,5,2,3,4] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 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]]
=> [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[[0,1,0,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [2,5,1,3,4] => ([(0,4),(1,2),(1,4),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[[0,0,1,0,0],[0,1,-1,0,1],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [2,5,1,3,4] => ([(0,4),(1,2),(1,4),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[[0,0,0,1,0],[0,1,0,-1,1],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [2,5,1,3,4] => ([(0,4),(1,2),(1,4),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[0,1,0,0,0],[1,-1,1,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 2 = 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]]
=> [3,1,5,2,4] => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[[0,1,0,0,0],[0,0,1,0,0],[1,-1,0,0,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 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]]
=> [3,1,5,2,4] => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
Description
The number of join prime elements of a lattice.
An element $x$ of a lattice $L$ is join-prime (or coprime) if $x \leq a \vee b$ implies $x \leq a$ or $x \leq b$ for every $a, b \in L$.
Matching statistic: St001617
Mp00002: Alternating sign matrices —to left key permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00206: Posets —antichains of maximal size⟶ Lattices
St001617: Lattices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00065: Permutations —permutation poset⟶ Posets
Mp00206: Posets —antichains of maximal size⟶ Lattices
St001617: Lattices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,0,0],[0,1,0],[0,0,1]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
=> [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[0,0,1,0],[0,1,-1,1],[1,0,0,0],[0,0,1,0]]
=> [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
=> [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[0,0,1,0],[0,1,0,0],[1,-1,0,1],[0,1,0,0]]
=> [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> [2,3,4,1] => ([(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [4,1,2,3,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [2,4,1,3,5] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[0,0,1,0,0],[0,1,-1,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [2,4,1,3,5] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [3,1,4,2,5] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[0,0,1,0,0],[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [3,1,4,2,5] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [3,4,1,2,5] => ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,0,0,1]]
=> [2,3,4,1,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [3,1,2,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [2,3,1,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [3,1,2,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [2,3,1,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[0,0,1,0,0],[0,1,-1,1,0],[1,0,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [2,3,1,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [3,1,2,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[0,0,1,0,0],[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [3,1,2,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [3,1,2,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[0,0,0,1,0],[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [3,1,2,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[1,0,0,-1,1],[0,0,0,1,0]]
=> [2,3,1,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[1,0,0,-1,1],[0,0,0,1,0]]
=> [2,3,1,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[1,0,0,-1,1],[0,0,0,1,0]]
=> [2,3,1,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [2,1,5,3,4] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [2,1,5,3,4] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[0,1,0,0,0],[0,0,1,0,0],[1,0,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [2,1,5,3,4] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[0,0,1,0,0],[0,1,0,0,0],[1,0,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [2,1,5,3,4] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[0,0,0,1,0],[1,0,0,0,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [2,1,5,3,4] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[0,1,0,0,0],[0,0,0,1,0],[1,0,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [2,1,5,3,4] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[0,0,1,0,0],[0,1,-1,1,0],[1,0,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [2,1,5,3,4] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[0,0,0,1,0],[0,1,0,0,0],[1,0,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [2,1,5,3,4] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,5,2,3,4] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[0,1,0,0,0],[1,-1,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,5,2,3,4] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[0,0,1,0,0],[1,0,-1,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,5,2,3,4] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[0,0,0,1,0],[1,0,0,-1,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,5,2,3,4] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 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]]
=> [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[[0,1,0,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [2,5,1,3,4] => ([(0,4),(1,2),(1,4),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[[0,0,1,0,0],[0,1,-1,0,1],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [2,5,1,3,4] => ([(0,4),(1,2),(1,4),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[[0,0,0,1,0],[0,1,0,-1,1],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [2,5,1,3,4] => ([(0,4),(1,2),(1,4),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[0,1,0,0,0],[1,-1,1,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 2 = 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]]
=> [3,1,5,2,4] => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[[0,1,0,0,0],[0,0,1,0,0],[1,-1,0,0,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 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]]
=> [3,1,5,2,4] => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
Description
The dimension of the space of valuations of a lattice.
A valuation, or modular function, on a lattice $L$ is a function $v:L\mapsto\mathbb R$ satisfying
$$
v(a\vee b) + v(a\wedge b) = v(a) + v(b).
$$
It was shown by Birkhoff [1, thm. X.2], that a lattice with a positive valuation must be modular. This was sharpened by Fleischer and Traynor [2, thm. 1], which states that the modular functions on an arbitrary lattice are in bijection with the modular functions on its modular quotient [[Mp00196]].
Moreover, Birkhoff [1, thm. X.2] showed that the dimension of the space of modular functions equals the number of subsets of projective prime intervals.
Matching statistic: St001622
Mp00002: Alternating sign matrices —to left key permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00206: Posets —antichains of maximal size⟶ Lattices
St001622: Lattices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00065: Permutations —permutation poset⟶ Posets
Mp00206: Posets —antichains of maximal size⟶ Lattices
St001622: Lattices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,0,0],[0,1,0],[0,0,1]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
=> [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[0,0,1,0],[0,1,-1,1],[1,0,0,0],[0,0,1,0]]
=> [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
=> [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[0,0,1,0],[0,1,0,0],[1,-1,0,1],[0,1,0,0]]
=> [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> [2,3,4,1] => ([(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [4,1,2,3,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [2,4,1,3,5] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[0,0,1,0,0],[0,1,-1,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [2,4,1,3,5] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [3,1,4,2,5] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[0,0,1,0,0],[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [3,1,4,2,5] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [3,4,1,2,5] => ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,0,0,1]]
=> [2,3,4,1,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [3,1,2,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [2,3,1,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [3,1,2,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [2,3,1,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[0,0,1,0,0],[0,1,-1,1,0],[1,0,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [2,3,1,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [3,1,2,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[0,0,1,0,0],[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [3,1,2,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [3,1,2,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[0,0,0,1,0],[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [3,1,2,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[1,0,0,-1,1],[0,0,0,1,0]]
=> [2,3,1,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[1,0,0,-1,1],[0,0,0,1,0]]
=> [2,3,1,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[1,0,0,-1,1],[0,0,0,1,0]]
=> [2,3,1,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [2,1,5,3,4] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [2,1,5,3,4] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[0,1,0,0,0],[0,0,1,0,0],[1,0,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [2,1,5,3,4] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[0,0,1,0,0],[0,1,0,0,0],[1,0,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [2,1,5,3,4] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[0,0,0,1,0],[1,0,0,0,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [2,1,5,3,4] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[0,1,0,0,0],[0,0,0,1,0],[1,0,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [2,1,5,3,4] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[0,0,1,0,0],[0,1,-1,1,0],[1,0,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [2,1,5,3,4] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[0,0,0,1,0],[0,1,0,0,0],[1,0,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [2,1,5,3,4] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,5,2,3,4] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[0,1,0,0,0],[1,-1,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,5,2,3,4] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[0,0,1,0,0],[1,0,-1,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,5,2,3,4] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[0,0,0,1,0],[1,0,0,-1,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,5,2,3,4] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 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]]
=> [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[[0,1,0,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [2,5,1,3,4] => ([(0,4),(1,2),(1,4),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[[0,0,1,0,0],[0,1,-1,0,1],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [2,5,1,3,4] => ([(0,4),(1,2),(1,4),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[[0,0,0,1,0],[0,1,0,-1,1],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [2,5,1,3,4] => ([(0,4),(1,2),(1,4),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[0,1,0,0,0],[1,-1,1,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 2 = 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]]
=> [3,1,5,2,4] => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[[0,1,0,0,0],[0,0,1,0,0],[1,-1,0,0,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 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]]
=> [3,1,5,2,4] => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
Description
The number of join-irreducible elements of a lattice.
An element $j$ of a lattice $L$ is '''join irreducible''' if it is not the least element and if $j=x\vee y$, then $j\in\{x,y\}$ for all $x,y\in L$.
Matching statistic: St000993
Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000993: Integer partitions ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 20%
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000993: Integer partitions ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 20%
Values
[[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ([],1)
=> [1]
=> ? = 3 - 2
[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,2],[3,3],[4]]
=> ([],1)
=> [1]
=> ? = 4 - 2
[[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [[1,1,1,4],[2,2,4],[3,4],[4]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> [7,1]
=> 1 = 3 - 2
[[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
=> [[1,1,2,2],[2,2,4],[3,4],[4]]
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> [6,3,1]
=> 1 = 3 - 2
[[0,0,1,0],[0,1,-1,1],[1,0,0,0],[0,0,1,0]]
=> [[1,1,2,3],[2,2,4],[3,4],[4]]
=> ([(0,6),(1,9),(1,10),(2,8),(3,7),(4,3),(4,12),(5,2),(5,12),(6,4),(6,5),(7,9),(7,11),(8,10),(8,11),(9,13),(10,13),(11,13),(12,1),(12,7),(12,8)],14)
=> [7,4,3]
=> ? = 3 - 2
[[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
=> [[1,1,1,3],[2,3,3],[3,4],[4]]
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> [6,1]
=> 1 = 3 - 2
[[0,0,1,0],[0,1,0,0],[1,-1,0,1],[0,1,0,0]]
=> [[1,1,2,3],[2,3,3],[3,4],[4]]
=> ([(0,3),(0,8),(1,10),(2,9),(3,11),(4,2),(5,4),(6,7),(7,1),(7,9),(8,5),(8,11),(9,10),(11,6)],12)
=> [7,5]
=> 1 = 3 - 2
[[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> [[1,1,3,3],[2,3,4],[3,4],[4]]
=> ([(0,8),(2,11),(2,12),(3,10),(4,9),(5,4),(5,14),(6,3),(6,14),(7,1),(8,5),(8,6),(9,11),(9,13),(10,12),(10,13),(11,15),(12,15),(13,15),(14,2),(14,9),(14,10),(15,7)],16)
=> [9,4,3]
=> ? = 3 - 2
[[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> [[1,2,2,2],[2,3,3],[3,4],[4]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> [7,1]
=> 1 = 3 - 2
[[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,2,2],[3,3,3],[4,4],[5]]
=> ([],1)
=> [1]
=> ? = 5 - 2
[[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,4],[2,2,2,4],[3,3,4],[4,4],[5]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> [7,1]
=> 1 = 3 - 2
[[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,2,2],[2,2,2,4],[3,3,4],[4,4],[5]]
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> [6,3,1]
=> 1 = 3 - 2
[[0,0,1,0,0],[0,1,-1,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,2,3],[2,2,2,4],[3,3,4],[4,4],[5]]
=> ?
=> ?
=> ? = 3 - 2
[[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,3],[2,2,3,3],[3,3,4],[4,4],[5]]
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> [6,1]
=> 1 = 3 - 2
[[0,0,1,0,0],[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,2,3],[2,2,3,3],[3,3,4],[4,4],[5]]
=> ?
=> ?
=> ? = 3 - 2
[[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,3,3],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ([(0,8),(2,11),(2,12),(3,10),(4,9),(5,4),(5,14),(6,3),(6,14),(7,1),(8,5),(8,6),(9,11),(9,13),(10,12),(10,13),(11,15),(12,15),(13,15),(14,2),(14,9),(14,10),(15,7)],16)
=> [9,4,3]
=> ? = 3 - 2
[[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,0,0,1]]
=> [[1,1,2,2,2],[2,2,3,3],[3,3,4],[4,4],[5]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> [7,1]
=> 1 = 3 - 2
[[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,1,3],[2,2,2,3],[3,3,3],[4,5],[5]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> [5,3]
=> 1 = 3 - 2
[[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,2,2],[2,2,2,3],[3,3,3],[4,5],[5]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> [5,3]
=> 1 = 3 - 2
[[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[1,1,1,1,4],[2,2,2,4],[3,3,4],[4,5],[5]]
=> ([(0,4),(0,7),(2,10),(3,9),(4,8),(5,3),(5,11),(6,1),(7,5),(7,8),(8,11),(9,10),(10,6),(11,2),(11,9)],12)
=> ?
=> ? = 3 - 2
[[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[1,1,1,2,2],[2,2,2,4],[3,3,4],[4,5],[5]]
=> ?
=> ?
=> ? = 3 - 2
[[0,0,1,0,0],[0,1,-1,1,0],[1,0,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[1,1,1,2,3],[2,2,2,4],[3,3,4],[4,5],[5]]
=> ?
=> ?
=> ? = 3 - 2
[[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [[1,1,1,1,3],[2,2,3,3],[3,3,4],[4,5],[5]]
=> ?
=> ?
=> ? = 3 - 2
[[0,0,1,0,0],[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [[1,1,1,2,3],[2,2,3,3],[3,3,4],[4,5],[5]]
=> ?
=> ?
=> ? = 3 - 2
[[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [[1,1,1,1,4],[2,2,3,4],[3,3,4],[4,5],[5]]
=> ?
=> ?
=> ? = 3 - 2
[[0,0,0,1,0],[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [[1,1,1,2,4],[2,2,3,4],[3,3,4],[4,5],[5]]
=> ?
=> ?
=> ? = 3 - 2
[[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[1,0,0,-1,1],[0,0,0,1,0]]
=> [[1,1,2,2,2],[2,2,3,3],[3,3,4],[4,5],[5]]
=> ?
=> ?
=> ? = 3 - 2
[[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[1,0,0,-1,1],[0,0,0,1,0]]
=> [[1,1,2,2,2],[2,2,3,4],[3,3,4],[4,5],[5]]
=> ?
=> ?
=> ? = 3 - 2
[[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[1,0,0,-1,1],[0,0,0,1,0]]
=> [[1,1,2,2,3],[2,2,3,4],[3,3,4],[4,5],[5]]
=> ?
=> ?
=> ? = 3 - 2
[[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [[1,1,1,1,2],[2,2,2,2],[3,3,5],[4,5],[5]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> [5,3]
=> 1 = 3 - 2
[[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [[1,1,1,1,3],[2,2,2,3],[3,3,5],[4,5],[5]]
=> ([(0,5),(0,6),(1,3),(1,13),(2,4),(2,12),(3,10),(4,8),(5,11),(6,2),(6,11),(8,9),(9,7),(10,7),(11,1),(11,12),(12,8),(12,13),(13,9),(13,10)],14)
=> ?
=> ? = 3 - 2
[[0,1,0,0,0],[0,0,1,0,0],[1,0,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [[1,1,1,2,2],[2,2,2,3],[3,3,5],[4,5],[5]]
=> ?
=> ?
=> ? = 3 - 2
[[0,0,1,0,0],[0,1,0,0,0],[1,0,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [[1,1,1,2,3],[2,2,2,3],[3,3,5],[4,5],[5]]
=> ?
=> ?
=> ? = 3 - 2
[[0,0,0,1,0],[1,0,0,0,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [[1,1,1,1,4],[2,2,2,4],[3,3,5],[4,5],[5]]
=> ([(0,6),(0,7),(2,5),(2,16),(3,4),(3,17),(4,14),(5,12),(6,13),(7,2),(7,13),(8,10),(8,11),(9,18),(10,18),(11,18),(12,15),(13,3),(13,16),(14,9),(14,10),(15,9),(15,11),(16,12),(16,17),(17,8),(17,14),(17,15),(18,1)],19)
=> ?
=> ? = 3 - 2
[[0,1,0,0,0],[0,0,0,1,0],[1,0,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [[1,1,1,2,2],[2,2,2,4],[3,3,5],[4,5],[5]]
=> ?
=> ?
=> ? = 3 - 2
[[0,0,1,0,0],[0,1,-1,1,0],[1,0,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [[1,1,1,2,3],[2,2,2,4],[3,3,5],[4,5],[5]]
=> ?
=> ?
=> ? = 3 - 2
[[0,0,0,1,0],[0,1,0,0,0],[1,0,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [[1,1,1,2,4],[2,2,2,4],[3,3,5],[4,5],[5]]
=> ?
=> ?
=> ? = 3 - 2
[[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [[1,1,1,1,1],[2,2,2,5],[3,3,5],[4,5],[5]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> [7,1]
=> 1 = 3 - 2
[[0,1,0,0,0],[1,-1,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [[1,1,1,1,2],[2,2,2,5],[3,3,5],[4,5],[5]]
=> ([(0,7),(1,11),(2,8),(3,9),(3,13),(4,9),(4,12),(5,1),(5,10),(6,3),(6,4),(6,8),(7,2),(7,6),(8,12),(8,13),(9,5),(9,14),(10,11),(12,14),(13,14),(14,10)],15)
=> ?
=> ? = 3 - 2
[[0,0,1,0,0],[1,0,-1,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [[1,1,1,1,3],[2,2,2,5],[3,3,5],[4,5],[5]]
=> ([(0,1),(1,4),(1,5),(2,16),(3,6),(3,17),(4,18),(5,7),(5,8),(5,18),(6,13),(7,11),(7,14),(8,10),(8,11),(10,19),(11,3),(11,19),(12,9),(13,9),(14,16),(14,19),(15,12),(16,15),(17,12),(17,13),(18,2),(18,10),(18,14),(19,15),(19,17)],20)
=> ?
=> ? = 3 - 2
[[0,0,0,1,0],[1,0,0,-1,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [[1,1,1,1,4],[2,2,2,5],[3,3,5],[4,5],[5]]
=> ([(0,2),(1,6),(2,4),(2,5),(3,7),(3,22),(4,23),(5,9),(5,10),(5,23),(6,20),(7,15),(8,12),(8,14),(9,13),(9,17),(10,13),(10,18),(12,1),(13,3),(13,24),(14,19),(15,16),(16,11),(17,14),(17,24),(18,12),(18,24),(19,21),(20,11),(21,16),(21,20),(22,15),(22,21),(23,8),(23,17),(23,18),(24,19),(24,22)],25)
=> ?
=> ? = 3 - 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]]
=> [[1,1,1,1,5],[2,2,2,5],[3,3,5],[4,5],[5]]
=> ([(0,9),(1,13),(2,14),(4,12),(5,11),(6,1),(6,12),(7,3),(8,5),(8,15),(9,10),(10,4),(10,6),(11,14),(12,8),(12,13),(13,15),(14,7),(15,2),(15,11)],16)
=> ?
=> ? = 4 - 2
[[0,1,0,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [[1,1,1,2,2],[2,2,2,5],[3,3,5],[4,5],[5]]
=> ([(0,1),(1,3),(1,4),(2,5),(2,18),(3,6),(3,19),(4,7),(4,8),(4,19),(5,13),(6,17),(7,12),(7,15),(8,12),(8,14),(10,21),(11,21),(12,2),(12,20),(13,9),(14,10),(14,20),(15,11),(15,20),(16,9),(17,10),(17,11),(18,13),(18,16),(19,14),(19,15),(19,17),(20,18),(20,21),(21,16)],22)
=> ?
=> ? = 4 - 2
[[0,0,1,0,0],[0,1,-1,0,1],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [[1,1,1,2,3],[2,2,2,5],[3,3,5],[4,5],[5]]
=> ?
=> ?
=> ? = 4 - 2
[[0,0,0,1,0],[0,1,0,-1,1],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [[1,1,1,2,4],[2,2,2,5],[3,3,5],[4,5],[5]]
=> ?
=> ?
=> ? = 4 - 2
[[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [[1,1,1,1,1],[2,2,3,3],[3,3,5],[4,5],[5]]
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> [6,3,1]
=> 1 = 3 - 2
[[0,1,0,0,0],[1,-1,1,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [[1,1,1,1,2],[2,2,3,3],[3,3,5],[4,5],[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]]
=> [[1,1,1,1,3],[2,2,3,3],[3,3,5],[4,5],[5]]
=> ([(0,6),(0,7),(1,5),(1,15),(2,4),(2,14),(3,13),(4,12),(5,3),(5,16),(6,9),(7,2),(7,9),(9,1),(9,14),(10,11),(11,8),(12,10),(13,8),(14,12),(14,15),(15,10),(15,16),(16,11),(16,13)],17)
=> ?
=> ? = 4 - 2
[[0,1,0,0,0],[0,0,1,0,0],[1,-1,0,0,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [[1,1,1,2,2],[2,2,3,3],[3,3,5],[4,5],[5]]
=> ?
=> ?
=> ? = 3 - 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]]
=> [[1,1,1,2,3],[2,2,3,3],[3,3,5],[4,5],[5]]
=> ?
=> ?
=> ? = 4 - 2
[[1,0,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [[1,1,1,1,1],[2,2,3,4],[3,3,5],[4,5],[5]]
=> ?
=> ?
=> ? = 3 - 2
[[0,1,0,0,0],[1,-1,0,1,0],[0,0,1,-1,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [[1,1,1,1,2],[2,2,3,4],[3,3,5],[4,5],[5]]
=> ?
=> ?
=> ? = 3 - 2
[[0,0,1,0,0],[1,0,-1,1,0],[0,0,1,-1,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [[1,1,1,1,3],[2,2,3,4],[3,3,5],[4,5],[5]]
=> ?
=> ?
=> ? = 3 - 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]]
=> [[1,1,1,1,4],[2,2,3,4],[3,3,5],[4,5],[5]]
=> ?
=> ?
=> ? = 4 - 2
[[0,1,0,0,0],[0,0,0,1,0],[1,-1,1,-1,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [[1,1,1,2,2],[2,2,3,4],[3,3,5],[4,5],[5]]
=> ?
=> ?
=> ? = 3 - 2
[[0,0,1,0,0],[0,1,-1,1,0],[1,-1,1,-1,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [[1,1,1,2,3],[2,2,3,4],[3,3,5],[4,5],[5]]
=> ?
=> ?
=> ? = 3 - 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]]
=> [[1,1,1,2,4],[2,2,3,4],[3,3,5],[4,5],[5]]
=> ?
=> ?
=> ? = 4 - 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]]
=> [[1,1,1,3,3],[2,2,3,4],[3,3,5],[4,5],[5]]
=> ?
=> ?
=> ? = 4 - 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]]
=> [[1,1,1,3,4],[2,2,3,4],[3,3,5],[4,5],[5]]
=> ?
=> ?
=> ? = 4 - 2
[[0,0,1,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [[1,1,1,3,3],[2,2,3,5],[3,3,5],[4,5],[5]]
=> ([(0,1),(1,4),(1,5),(2,8),(2,32),(3,6),(3,30),(4,7),(4,33),(5,10),(5,11),(5,33),(6,20),(7,31),(8,27),(9,15),(9,19),(10,18),(10,28),(11,18),(11,25),(13,39),(14,36),(14,39),(15,36),(16,38),(17,38),(18,2),(18,37),(19,26),(19,36),(20,21),(21,12),(22,12),(23,17),(23,35),(24,16),(24,35),(25,13),(25,37),(26,24),(26,34),(27,16),(27,17),(28,14),(28,19),(28,37),(29,21),(29,22),(30,20),(30,29),(31,13),(31,14),(31,15),(32,23),(32,24),(32,27),(33,9),(33,25),(33,28),(33,31),(34,30),(34,35),(35,29),(35,38),(36,3),(36,34),(37,26),(37,32),(37,39),(38,22),(39,23),(39,34)],40)
=> ?
=> ? = 4 - 2
[[0,0,0,1,0],[0,0,1,-1,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [[1,1,1,3,4],[2,2,3,5],[3,3,5],[4,5],[5]]
=> ?
=> ?
=> ? = 4 - 2
[[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,0,1,0]]
=> [[1,1,2,2,2],[2,2,3,3],[3,3,5],[4,5],[5]]
=> ([(0,1),(1,3),(1,4),(2,5),(2,18),(3,6),(3,19),(4,7),(4,8),(4,19),(5,13),(6,17),(7,12),(7,15),(8,12),(8,14),(10,21),(11,21),(12,2),(12,20),(13,9),(14,10),(14,20),(15,11),(15,20),(16,9),(17,10),(17,11),(18,13),(18,16),(19,14),(19,15),(19,17),(20,18),(20,21),(21,16)],22)
=> ?
=> ? = 4 - 2
[[0,1,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[1,0,0,0,0],[0,0,0,1,0]]
=> [[1,1,2,2,2],[2,2,3,4],[3,3,5],[4,5],[5]]
=> ?
=> ?
=> ? = 4 - 2
[[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[1,0,0,0,0],[0,0,0,1,0]]
=> [[1,1,2,2,3],[2,2,3,4],[3,3,5],[4,5],[5]]
=> ?
=> ?
=> ? = 4 - 2
[[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,0,1,0,0]]
=> [[1,1,1,1,2],[2,2,2,2],[3,4,4],[4,5],[5]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> [5,3]
=> 1 = 3 - 2
[[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0]]
=> [[1,1,1,1,1],[2,2,2,4],[3,4,4],[4,5],[5]]
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> [6,1]
=> 1 = 3 - 2
[[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0]]
=> [[1,1,1,1,1],[2,3,3,3],[3,4,4],[4,5],[5]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> [7,1]
=> 1 = 3 - 2
[[0,0,0,1,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,4],[2,2,2,2,4],[3,3,3,4],[4,4,4],[5,5],[6]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> [7,1]
=> 1 = 3 - 2
[[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,3],[2,2,2,2,3],[3,3,3,3],[4,4,5],[5,5],[6]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> [5,3]
=> 1 = 3 - 2
[[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,5],[4,4,5],[5,5],[6]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> [5,3]
=> 1 = 3 - 2
[[1,0,0,0,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,5],[3,3,3,5],[4,4,5],[5,5],[6]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> [7,1]
=> 1 = 3 - 2
[[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,3],[2,2,2,2,3],[3,3,3,3],[4,4,4],[5,6],[6]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> [5,3]
=> 1 = 3 - 2
[[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,4],[4,4,4],[5,6],[6]]
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> [4,2,2]
=> 1 = 3 - 2
[[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,-1,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,3],[2,2,2,2,3],[3,3,3,4],[4,4,4],[5,6],[6]]
=> ([(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(3,7),(3,8),(4,6),(4,8),(5,1),(5,9),(6,11),(7,11),(8,5),(8,11),(9,10),(11,9)],12)
=> [6,4,2]
=> 1 = 3 - 2
[[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,4],[3,3,3,4],[4,4,4],[5,6],[6]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> [5,3]
=> 1 = 3 - 2
[[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,3],[2,2,2,2,3],[3,3,3,3],[4,4,5],[5,6],[6]]
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> [6,4,2]
=> 1 = 3 - 2
[[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,5],[4,4,5],[5,6],[6]]
=> ([(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(3,7),(3,8),(4,6),(4,8),(5,1),(5,9),(6,11),(7,11),(8,5),(8,11),(9,10),(11,9)],12)
=> [6,4,2]
=> 1 = 3 - 2
[[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,3],[4,4,6],[5,6],[6]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> [5,3]
=> 1 = 3 - 2
[[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,3],[3,3,3,3],[4,4,6],[5,6],[6]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> [5,3]
=> 1 = 3 - 2
[[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,2],[2,2,2,2,3],[3,3,3,3],[4,4,6],[5,6],[6]]
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> [6,4,2]
=> 1 = 3 - 2
[[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,6],[4,4,6],[5,6],[6]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> [7,1]
=> 1 = 3 - 2
Description
The multiplicity of the largest part of an integer partition.
Matching statistic: St000510
Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000510: Integer partitions ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 20%
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000510: Integer partitions ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 20%
Values
[[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ([],1)
=> [1]
=> ? = 3 - 3
[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,2],[3,3],[4]]
=> ([],1)
=> [1]
=> ? = 4 - 3
[[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [[1,1,1,4],[2,2,4],[3,4],[4]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> [7,1]
=> 0 = 3 - 3
[[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
=> [[1,1,2,2],[2,2,4],[3,4],[4]]
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> [6,3,1]
=> 0 = 3 - 3
[[0,0,1,0],[0,1,-1,1],[1,0,0,0],[0,0,1,0]]
=> [[1,1,2,3],[2,2,4],[3,4],[4]]
=> ([(0,6),(1,9),(1,10),(2,8),(3,7),(4,3),(4,12),(5,2),(5,12),(6,4),(6,5),(7,9),(7,11),(8,10),(8,11),(9,13),(10,13),(11,13),(12,1),(12,7),(12,8)],14)
=> [7,4,3]
=> ? = 3 - 3
[[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
=> [[1,1,1,3],[2,3,3],[3,4],[4]]
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> [6,1]
=> 0 = 3 - 3
[[0,0,1,0],[0,1,0,0],[1,-1,0,1],[0,1,0,0]]
=> [[1,1,2,3],[2,3,3],[3,4],[4]]
=> ([(0,3),(0,8),(1,10),(2,9),(3,11),(4,2),(5,4),(6,7),(7,1),(7,9),(8,5),(8,11),(9,10),(11,6)],12)
=> [7,5]
=> 0 = 3 - 3
[[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> [[1,1,3,3],[2,3,4],[3,4],[4]]
=> ([(0,8),(2,11),(2,12),(3,10),(4,9),(5,4),(5,14),(6,3),(6,14),(7,1),(8,5),(8,6),(9,11),(9,13),(10,12),(10,13),(11,15),(12,15),(13,15),(14,2),(14,9),(14,10),(15,7)],16)
=> [9,4,3]
=> ? = 3 - 3
[[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> [[1,2,2,2],[2,3,3],[3,4],[4]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> [7,1]
=> 0 = 3 - 3
[[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,2,2],[3,3,3],[4,4],[5]]
=> ([],1)
=> [1]
=> ? = 5 - 3
[[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,4],[2,2,2,4],[3,3,4],[4,4],[5]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> [7,1]
=> 0 = 3 - 3
[[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,2,2],[2,2,2,4],[3,3,4],[4,4],[5]]
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> [6,3,1]
=> 0 = 3 - 3
[[0,0,1,0,0],[0,1,-1,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,2,3],[2,2,2,4],[3,3,4],[4,4],[5]]
=> ?
=> ?
=> ? = 3 - 3
[[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,3],[2,2,3,3],[3,3,4],[4,4],[5]]
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> [6,1]
=> 0 = 3 - 3
[[0,0,1,0,0],[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,2,3],[2,2,3,3],[3,3,4],[4,4],[5]]
=> ?
=> ?
=> ? = 3 - 3
[[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,3,3],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ([(0,8),(2,11),(2,12),(3,10),(4,9),(5,4),(5,14),(6,3),(6,14),(7,1),(8,5),(8,6),(9,11),(9,13),(10,12),(10,13),(11,15),(12,15),(13,15),(14,2),(14,9),(14,10),(15,7)],16)
=> [9,4,3]
=> ? = 3 - 3
[[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,0,0,1]]
=> [[1,1,2,2,2],[2,2,3,3],[3,3,4],[4,4],[5]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> [7,1]
=> 0 = 3 - 3
[[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,1,3],[2,2,2,3],[3,3,3],[4,5],[5]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> [5,3]
=> 0 = 3 - 3
[[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,2,2],[2,2,2,3],[3,3,3],[4,5],[5]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> [5,3]
=> 0 = 3 - 3
[[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[1,1,1,1,4],[2,2,2,4],[3,3,4],[4,5],[5]]
=> ([(0,4),(0,7),(2,10),(3,9),(4,8),(5,3),(5,11),(6,1),(7,5),(7,8),(8,11),(9,10),(10,6),(11,2),(11,9)],12)
=> ?
=> ? = 3 - 3
[[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[1,1,1,2,2],[2,2,2,4],[3,3,4],[4,5],[5]]
=> ?
=> ?
=> ? = 3 - 3
[[0,0,1,0,0],[0,1,-1,1,0],[1,0,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[1,1,1,2,3],[2,2,2,4],[3,3,4],[4,5],[5]]
=> ?
=> ?
=> ? = 3 - 3
[[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [[1,1,1,1,3],[2,2,3,3],[3,3,4],[4,5],[5]]
=> ?
=> ?
=> ? = 3 - 3
[[0,0,1,0,0],[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [[1,1,1,2,3],[2,2,3,3],[3,3,4],[4,5],[5]]
=> ?
=> ?
=> ? = 3 - 3
[[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [[1,1,1,1,4],[2,2,3,4],[3,3,4],[4,5],[5]]
=> ?
=> ?
=> ? = 3 - 3
[[0,0,0,1,0],[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [[1,1,1,2,4],[2,2,3,4],[3,3,4],[4,5],[5]]
=> ?
=> ?
=> ? = 3 - 3
[[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[1,0,0,-1,1],[0,0,0,1,0]]
=> [[1,1,2,2,2],[2,2,3,3],[3,3,4],[4,5],[5]]
=> ?
=> ?
=> ? = 3 - 3
[[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[1,0,0,-1,1],[0,0,0,1,0]]
=> [[1,1,2,2,2],[2,2,3,4],[3,3,4],[4,5],[5]]
=> ?
=> ?
=> ? = 3 - 3
[[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[1,0,0,-1,1],[0,0,0,1,0]]
=> [[1,1,2,2,3],[2,2,3,4],[3,3,4],[4,5],[5]]
=> ?
=> ?
=> ? = 3 - 3
[[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [[1,1,1,1,2],[2,2,2,2],[3,3,5],[4,5],[5]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> [5,3]
=> 0 = 3 - 3
[[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [[1,1,1,1,3],[2,2,2,3],[3,3,5],[4,5],[5]]
=> ([(0,5),(0,6),(1,3),(1,13),(2,4),(2,12),(3,10),(4,8),(5,11),(6,2),(6,11),(8,9),(9,7),(10,7),(11,1),(11,12),(12,8),(12,13),(13,9),(13,10)],14)
=> ?
=> ? = 3 - 3
[[0,1,0,0,0],[0,0,1,0,0],[1,0,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [[1,1,1,2,2],[2,2,2,3],[3,3,5],[4,5],[5]]
=> ?
=> ?
=> ? = 3 - 3
[[0,0,1,0,0],[0,1,0,0,0],[1,0,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [[1,1,1,2,3],[2,2,2,3],[3,3,5],[4,5],[5]]
=> ?
=> ?
=> ? = 3 - 3
[[0,0,0,1,0],[1,0,0,0,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [[1,1,1,1,4],[2,2,2,4],[3,3,5],[4,5],[5]]
=> ([(0,6),(0,7),(2,5),(2,16),(3,4),(3,17),(4,14),(5,12),(6,13),(7,2),(7,13),(8,10),(8,11),(9,18),(10,18),(11,18),(12,15),(13,3),(13,16),(14,9),(14,10),(15,9),(15,11),(16,12),(16,17),(17,8),(17,14),(17,15),(18,1)],19)
=> ?
=> ? = 3 - 3
[[0,1,0,0,0],[0,0,0,1,0],[1,0,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [[1,1,1,2,2],[2,2,2,4],[3,3,5],[4,5],[5]]
=> ?
=> ?
=> ? = 3 - 3
[[0,0,1,0,0],[0,1,-1,1,0],[1,0,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [[1,1,1,2,3],[2,2,2,4],[3,3,5],[4,5],[5]]
=> ?
=> ?
=> ? = 3 - 3
[[0,0,0,1,0],[0,1,0,0,0],[1,0,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [[1,1,1,2,4],[2,2,2,4],[3,3,5],[4,5],[5]]
=> ?
=> ?
=> ? = 3 - 3
[[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [[1,1,1,1,1],[2,2,2,5],[3,3,5],[4,5],[5]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> [7,1]
=> 0 = 3 - 3
[[0,1,0,0,0],[1,-1,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [[1,1,1,1,2],[2,2,2,5],[3,3,5],[4,5],[5]]
=> ([(0,7),(1,11),(2,8),(3,9),(3,13),(4,9),(4,12),(5,1),(5,10),(6,3),(6,4),(6,8),(7,2),(7,6),(8,12),(8,13),(9,5),(9,14),(10,11),(12,14),(13,14),(14,10)],15)
=> ?
=> ? = 3 - 3
[[0,0,1,0,0],[1,0,-1,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [[1,1,1,1,3],[2,2,2,5],[3,3,5],[4,5],[5]]
=> ([(0,1),(1,4),(1,5),(2,16),(3,6),(3,17),(4,18),(5,7),(5,8),(5,18),(6,13),(7,11),(7,14),(8,10),(8,11),(10,19),(11,3),(11,19),(12,9),(13,9),(14,16),(14,19),(15,12),(16,15),(17,12),(17,13),(18,2),(18,10),(18,14),(19,15),(19,17)],20)
=> ?
=> ? = 3 - 3
[[0,0,0,1,0],[1,0,0,-1,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [[1,1,1,1,4],[2,2,2,5],[3,3,5],[4,5],[5]]
=> ([(0,2),(1,6),(2,4),(2,5),(3,7),(3,22),(4,23),(5,9),(5,10),(5,23),(6,20),(7,15),(8,12),(8,14),(9,13),(9,17),(10,13),(10,18),(12,1),(13,3),(13,24),(14,19),(15,16),(16,11),(17,14),(17,24),(18,12),(18,24),(19,21),(20,11),(21,16),(21,20),(22,15),(22,21),(23,8),(23,17),(23,18),(24,19),(24,22)],25)
=> ?
=> ? = 3 - 3
[[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [[1,1,1,1,5],[2,2,2,5],[3,3,5],[4,5],[5]]
=> ([(0,9),(1,13),(2,14),(4,12),(5,11),(6,1),(6,12),(7,3),(8,5),(8,15),(9,10),(10,4),(10,6),(11,14),(12,8),(12,13),(13,15),(14,7),(15,2),(15,11)],16)
=> ?
=> ? = 4 - 3
[[0,1,0,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [[1,1,1,2,2],[2,2,2,5],[3,3,5],[4,5],[5]]
=> ([(0,1),(1,3),(1,4),(2,5),(2,18),(3,6),(3,19),(4,7),(4,8),(4,19),(5,13),(6,17),(7,12),(7,15),(8,12),(8,14),(10,21),(11,21),(12,2),(12,20),(13,9),(14,10),(14,20),(15,11),(15,20),(16,9),(17,10),(17,11),(18,13),(18,16),(19,14),(19,15),(19,17),(20,18),(20,21),(21,16)],22)
=> ?
=> ? = 4 - 3
[[0,0,1,0,0],[0,1,-1,0,1],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [[1,1,1,2,3],[2,2,2,5],[3,3,5],[4,5],[5]]
=> ?
=> ?
=> ? = 4 - 3
[[0,0,0,1,0],[0,1,0,-1,1],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [[1,1,1,2,4],[2,2,2,5],[3,3,5],[4,5],[5]]
=> ?
=> ?
=> ? = 4 - 3
[[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [[1,1,1,1,1],[2,2,3,3],[3,3,5],[4,5],[5]]
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> [6,3,1]
=> 0 = 3 - 3
[[0,1,0,0,0],[1,-1,1,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [[1,1,1,1,2],[2,2,3,3],[3,3,5],[4,5],[5]]
=> ?
=> ?
=> ? = 3 - 3
[[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [[1,1,1,1,3],[2,2,3,3],[3,3,5],[4,5],[5]]
=> ([(0,6),(0,7),(1,5),(1,15),(2,4),(2,14),(3,13),(4,12),(5,3),(5,16),(6,9),(7,2),(7,9),(9,1),(9,14),(10,11),(11,8),(12,10),(13,8),(14,12),(14,15),(15,10),(15,16),(16,11),(16,13)],17)
=> ?
=> ? = 4 - 3
[[0,1,0,0,0],[0,0,1,0,0],[1,-1,0,0,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [[1,1,1,2,2],[2,2,3,3],[3,3,5],[4,5],[5]]
=> ?
=> ?
=> ? = 3 - 3
[[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]]
=> [[1,1,1,2,3],[2,2,3,3],[3,3,5],[4,5],[5]]
=> ?
=> ?
=> ? = 4 - 3
[[1,0,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [[1,1,1,1,1],[2,2,3,4],[3,3,5],[4,5],[5]]
=> ?
=> ?
=> ? = 3 - 3
[[0,1,0,0,0],[1,-1,0,1,0],[0,0,1,-1,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [[1,1,1,1,2],[2,2,3,4],[3,3,5],[4,5],[5]]
=> ?
=> ?
=> ? = 3 - 3
[[0,0,1,0,0],[1,0,-1,1,0],[0,0,1,-1,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [[1,1,1,1,3],[2,2,3,4],[3,3,5],[4,5],[5]]
=> ?
=> ?
=> ? = 3 - 3
[[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]]
=> [[1,1,1,1,4],[2,2,3,4],[3,3,5],[4,5],[5]]
=> ?
=> ?
=> ? = 4 - 3
[[0,1,0,0,0],[0,0,0,1,0],[1,-1,1,-1,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [[1,1,1,2,2],[2,2,3,4],[3,3,5],[4,5],[5]]
=> ?
=> ?
=> ? = 3 - 3
[[0,0,1,0,0],[0,1,-1,1,0],[1,-1,1,-1,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [[1,1,1,2,3],[2,2,3,4],[3,3,5],[4,5],[5]]
=> ?
=> ?
=> ? = 3 - 3
[[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]]
=> [[1,1,1,2,4],[2,2,3,4],[3,3,5],[4,5],[5]]
=> ?
=> ?
=> ? = 4 - 3
[[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]]
=> [[1,1,1,3,3],[2,2,3,4],[3,3,5],[4,5],[5]]
=> ?
=> ?
=> ? = 4 - 3
[[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]]
=> [[1,1,1,3,4],[2,2,3,4],[3,3,5],[4,5],[5]]
=> ?
=> ?
=> ? = 4 - 3
[[0,0,1,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [[1,1,1,3,3],[2,2,3,5],[3,3,5],[4,5],[5]]
=> ([(0,1),(1,4),(1,5),(2,8),(2,32),(3,6),(3,30),(4,7),(4,33),(5,10),(5,11),(5,33),(6,20),(7,31),(8,27),(9,15),(9,19),(10,18),(10,28),(11,18),(11,25),(13,39),(14,36),(14,39),(15,36),(16,38),(17,38),(18,2),(18,37),(19,26),(19,36),(20,21),(21,12),(22,12),(23,17),(23,35),(24,16),(24,35),(25,13),(25,37),(26,24),(26,34),(27,16),(27,17),(28,14),(28,19),(28,37),(29,21),(29,22),(30,20),(30,29),(31,13),(31,14),(31,15),(32,23),(32,24),(32,27),(33,9),(33,25),(33,28),(33,31),(34,30),(34,35),(35,29),(35,38),(36,3),(36,34),(37,26),(37,32),(37,39),(38,22),(39,23),(39,34)],40)
=> ?
=> ? = 4 - 3
[[0,0,0,1,0],[0,0,1,-1,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [[1,1,1,3,4],[2,2,3,5],[3,3,5],[4,5],[5]]
=> ?
=> ?
=> ? = 4 - 3
[[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,0,1,0]]
=> [[1,1,2,2,2],[2,2,3,3],[3,3,5],[4,5],[5]]
=> ([(0,1),(1,3),(1,4),(2,5),(2,18),(3,6),(3,19),(4,7),(4,8),(4,19),(5,13),(6,17),(7,12),(7,15),(8,12),(8,14),(10,21),(11,21),(12,2),(12,20),(13,9),(14,10),(14,20),(15,11),(15,20),(16,9),(17,10),(17,11),(18,13),(18,16),(19,14),(19,15),(19,17),(20,18),(20,21),(21,16)],22)
=> ?
=> ? = 4 - 3
[[0,1,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[1,0,0,0,0],[0,0,0,1,0]]
=> [[1,1,2,2,2],[2,2,3,4],[3,3,5],[4,5],[5]]
=> ?
=> ?
=> ? = 4 - 3
[[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[1,0,0,0,0],[0,0,0,1,0]]
=> [[1,1,2,2,3],[2,2,3,4],[3,3,5],[4,5],[5]]
=> ?
=> ?
=> ? = 4 - 3
[[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,0,1,0,0]]
=> [[1,1,1,1,2],[2,2,2,2],[3,4,4],[4,5],[5]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> [5,3]
=> 0 = 3 - 3
[[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0]]
=> [[1,1,1,1,1],[2,2,2,4],[3,4,4],[4,5],[5]]
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> [6,1]
=> 0 = 3 - 3
[[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0]]
=> [[1,1,1,1,1],[2,3,3,3],[3,4,4],[4,5],[5]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> [7,1]
=> 0 = 3 - 3
[[0,0,0,1,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,4],[2,2,2,2,4],[3,3,3,4],[4,4,4],[5,5],[6]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> [7,1]
=> 0 = 3 - 3
[[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,3],[2,2,2,2,3],[3,3,3,3],[4,4,5],[5,5],[6]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> [5,3]
=> 0 = 3 - 3
[[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,5],[4,4,5],[5,5],[6]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> [5,3]
=> 0 = 3 - 3
[[1,0,0,0,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,5],[3,3,3,5],[4,4,5],[5,5],[6]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> [7,1]
=> 0 = 3 - 3
[[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,3],[2,2,2,2,3],[3,3,3,3],[4,4,4],[5,6],[6]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> [5,3]
=> 0 = 3 - 3
[[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,4],[4,4,4],[5,6],[6]]
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> [4,2,2]
=> 0 = 3 - 3
[[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,-1,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,3],[2,2,2,2,3],[3,3,3,4],[4,4,4],[5,6],[6]]
=> ([(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(3,7),(3,8),(4,6),(4,8),(5,1),(5,9),(6,11),(7,11),(8,5),(8,11),(9,10),(11,9)],12)
=> [6,4,2]
=> 0 = 3 - 3
[[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,4],[3,3,3,4],[4,4,4],[5,6],[6]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> [5,3]
=> 0 = 3 - 3
[[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,3],[2,2,2,2,3],[3,3,3,3],[4,4,5],[5,6],[6]]
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> [6,4,2]
=> 0 = 3 - 3
[[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,5],[4,4,5],[5,6],[6]]
=> ([(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(3,7),(3,8),(4,6),(4,8),(5,1),(5,9),(6,11),(7,11),(8,5),(8,11),(9,10),(11,9)],12)
=> [6,4,2]
=> 0 = 3 - 3
[[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,3],[4,4,6],[5,6],[6]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> [5,3]
=> 0 = 3 - 3
[[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,3],[3,3,3,3],[4,4,6],[5,6],[6]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> [5,3]
=> 0 = 3 - 3
[[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,2],[2,2,2,2,3],[3,3,3,3],[4,4,6],[5,6],[6]]
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> [6,4,2]
=> 0 = 3 - 3
[[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,6],[4,4,6],[5,6],[6]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> [7,1]
=> 0 = 3 - 3
Description
The number of invariant oriented cycles when acting with a permutation of given cycle type.
Matching statistic: St000929
Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000929: Integer partitions ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 20%
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000929: Integer partitions ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 20%
Values
[[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ([],1)
=> [1]
=> ? = 3 - 3
[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,2],[3,3],[4]]
=> ([],1)
=> [1]
=> ? = 4 - 3
[[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [[1,1,1,4],[2,2,4],[3,4],[4]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> [7,1]
=> 0 = 3 - 3
[[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
=> [[1,1,2,2],[2,2,4],[3,4],[4]]
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> [6,3,1]
=> 0 = 3 - 3
[[0,0,1,0],[0,1,-1,1],[1,0,0,0],[0,0,1,0]]
=> [[1,1,2,3],[2,2,4],[3,4],[4]]
=> ([(0,6),(1,9),(1,10),(2,8),(3,7),(4,3),(4,12),(5,2),(5,12),(6,4),(6,5),(7,9),(7,11),(8,10),(8,11),(9,13),(10,13),(11,13),(12,1),(12,7),(12,8)],14)
=> [7,4,3]
=> ? = 3 - 3
[[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
=> [[1,1,1,3],[2,3,3],[3,4],[4]]
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> [6,1]
=> 0 = 3 - 3
[[0,0,1,0],[0,1,0,0],[1,-1,0,1],[0,1,0,0]]
=> [[1,1,2,3],[2,3,3],[3,4],[4]]
=> ([(0,3),(0,8),(1,10),(2,9),(3,11),(4,2),(5,4),(6,7),(7,1),(7,9),(8,5),(8,11),(9,10),(11,6)],12)
=> [7,5]
=> 0 = 3 - 3
[[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> [[1,1,3,3],[2,3,4],[3,4],[4]]
=> ([(0,8),(2,11),(2,12),(3,10),(4,9),(5,4),(5,14),(6,3),(6,14),(7,1),(8,5),(8,6),(9,11),(9,13),(10,12),(10,13),(11,15),(12,15),(13,15),(14,2),(14,9),(14,10),(15,7)],16)
=> [9,4,3]
=> ? = 3 - 3
[[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> [[1,2,2,2],[2,3,3],[3,4],[4]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> [7,1]
=> 0 = 3 - 3
[[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,2,2],[3,3,3],[4,4],[5]]
=> ([],1)
=> [1]
=> ? = 5 - 3
[[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,4],[2,2,2,4],[3,3,4],[4,4],[5]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> [7,1]
=> 0 = 3 - 3
[[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,2,2],[2,2,2,4],[3,3,4],[4,4],[5]]
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> [6,3,1]
=> 0 = 3 - 3
[[0,0,1,0,0],[0,1,-1,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,2,3],[2,2,2,4],[3,3,4],[4,4],[5]]
=> ?
=> ?
=> ? = 3 - 3
[[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,3],[2,2,3,3],[3,3,4],[4,4],[5]]
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> [6,1]
=> 0 = 3 - 3
[[0,0,1,0,0],[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,2,3],[2,2,3,3],[3,3,4],[4,4],[5]]
=> ?
=> ?
=> ? = 3 - 3
[[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,3,3],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ([(0,8),(2,11),(2,12),(3,10),(4,9),(5,4),(5,14),(6,3),(6,14),(7,1),(8,5),(8,6),(9,11),(9,13),(10,12),(10,13),(11,15),(12,15),(13,15),(14,2),(14,9),(14,10),(15,7)],16)
=> [9,4,3]
=> ? = 3 - 3
[[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,0,0,1]]
=> [[1,1,2,2,2],[2,2,3,3],[3,3,4],[4,4],[5]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> [7,1]
=> 0 = 3 - 3
[[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,1,3],[2,2,2,3],[3,3,3],[4,5],[5]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> [5,3]
=> 0 = 3 - 3
[[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,2,2],[2,2,2,3],[3,3,3],[4,5],[5]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> [5,3]
=> 0 = 3 - 3
[[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[1,1,1,1,4],[2,2,2,4],[3,3,4],[4,5],[5]]
=> ([(0,4),(0,7),(2,10),(3,9),(4,8),(5,3),(5,11),(6,1),(7,5),(7,8),(8,11),(9,10),(10,6),(11,2),(11,9)],12)
=> ?
=> ? = 3 - 3
[[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[1,1,1,2,2],[2,2,2,4],[3,3,4],[4,5],[5]]
=> ?
=> ?
=> ? = 3 - 3
[[0,0,1,0,0],[0,1,-1,1,0],[1,0,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[1,1,1,2,3],[2,2,2,4],[3,3,4],[4,5],[5]]
=> ?
=> ?
=> ? = 3 - 3
[[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [[1,1,1,1,3],[2,2,3,3],[3,3,4],[4,5],[5]]
=> ?
=> ?
=> ? = 3 - 3
[[0,0,1,0,0],[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [[1,1,1,2,3],[2,2,3,3],[3,3,4],[4,5],[5]]
=> ?
=> ?
=> ? = 3 - 3
[[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [[1,1,1,1,4],[2,2,3,4],[3,3,4],[4,5],[5]]
=> ?
=> ?
=> ? = 3 - 3
[[0,0,0,1,0],[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [[1,1,1,2,4],[2,2,3,4],[3,3,4],[4,5],[5]]
=> ?
=> ?
=> ? = 3 - 3
[[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[1,0,0,-1,1],[0,0,0,1,0]]
=> [[1,1,2,2,2],[2,2,3,3],[3,3,4],[4,5],[5]]
=> ?
=> ?
=> ? = 3 - 3
[[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[1,0,0,-1,1],[0,0,0,1,0]]
=> [[1,1,2,2,2],[2,2,3,4],[3,3,4],[4,5],[5]]
=> ?
=> ?
=> ? = 3 - 3
[[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[1,0,0,-1,1],[0,0,0,1,0]]
=> [[1,1,2,2,3],[2,2,3,4],[3,3,4],[4,5],[5]]
=> ?
=> ?
=> ? = 3 - 3
[[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [[1,1,1,1,2],[2,2,2,2],[3,3,5],[4,5],[5]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> [5,3]
=> 0 = 3 - 3
[[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [[1,1,1,1,3],[2,2,2,3],[3,3,5],[4,5],[5]]
=> ([(0,5),(0,6),(1,3),(1,13),(2,4),(2,12),(3,10),(4,8),(5,11),(6,2),(6,11),(8,9),(9,7),(10,7),(11,1),(11,12),(12,8),(12,13),(13,9),(13,10)],14)
=> ?
=> ? = 3 - 3
[[0,1,0,0,0],[0,0,1,0,0],[1,0,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [[1,1,1,2,2],[2,2,2,3],[3,3,5],[4,5],[5]]
=> ?
=> ?
=> ? = 3 - 3
[[0,0,1,0,0],[0,1,0,0,0],[1,0,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [[1,1,1,2,3],[2,2,2,3],[3,3,5],[4,5],[5]]
=> ?
=> ?
=> ? = 3 - 3
[[0,0,0,1,0],[1,0,0,0,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [[1,1,1,1,4],[2,2,2,4],[3,3,5],[4,5],[5]]
=> ([(0,6),(0,7),(2,5),(2,16),(3,4),(3,17),(4,14),(5,12),(6,13),(7,2),(7,13),(8,10),(8,11),(9,18),(10,18),(11,18),(12,15),(13,3),(13,16),(14,9),(14,10),(15,9),(15,11),(16,12),(16,17),(17,8),(17,14),(17,15),(18,1)],19)
=> ?
=> ? = 3 - 3
[[0,1,0,0,0],[0,0,0,1,0],[1,0,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [[1,1,1,2,2],[2,2,2,4],[3,3,5],[4,5],[5]]
=> ?
=> ?
=> ? = 3 - 3
[[0,0,1,0,0],[0,1,-1,1,0],[1,0,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [[1,1,1,2,3],[2,2,2,4],[3,3,5],[4,5],[5]]
=> ?
=> ?
=> ? = 3 - 3
[[0,0,0,1,0],[0,1,0,0,0],[1,0,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [[1,1,1,2,4],[2,2,2,4],[3,3,5],[4,5],[5]]
=> ?
=> ?
=> ? = 3 - 3
[[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [[1,1,1,1,1],[2,2,2,5],[3,3,5],[4,5],[5]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> [7,1]
=> 0 = 3 - 3
[[0,1,0,0,0],[1,-1,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [[1,1,1,1,2],[2,2,2,5],[3,3,5],[4,5],[5]]
=> ([(0,7),(1,11),(2,8),(3,9),(3,13),(4,9),(4,12),(5,1),(5,10),(6,3),(6,4),(6,8),(7,2),(7,6),(8,12),(8,13),(9,5),(9,14),(10,11),(12,14),(13,14),(14,10)],15)
=> ?
=> ? = 3 - 3
[[0,0,1,0,0],[1,0,-1,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [[1,1,1,1,3],[2,2,2,5],[3,3,5],[4,5],[5]]
=> ([(0,1),(1,4),(1,5),(2,16),(3,6),(3,17),(4,18),(5,7),(5,8),(5,18),(6,13),(7,11),(7,14),(8,10),(8,11),(10,19),(11,3),(11,19),(12,9),(13,9),(14,16),(14,19),(15,12),(16,15),(17,12),(17,13),(18,2),(18,10),(18,14),(19,15),(19,17)],20)
=> ?
=> ? = 3 - 3
[[0,0,0,1,0],[1,0,0,-1,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [[1,1,1,1,4],[2,2,2,5],[3,3,5],[4,5],[5]]
=> ([(0,2),(1,6),(2,4),(2,5),(3,7),(3,22),(4,23),(5,9),(5,10),(5,23),(6,20),(7,15),(8,12),(8,14),(9,13),(9,17),(10,13),(10,18),(12,1),(13,3),(13,24),(14,19),(15,16),(16,11),(17,14),(17,24),(18,12),(18,24),(19,21),(20,11),(21,16),(21,20),(22,15),(22,21),(23,8),(23,17),(23,18),(24,19),(24,22)],25)
=> ?
=> ? = 3 - 3
[[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [[1,1,1,1,5],[2,2,2,5],[3,3,5],[4,5],[5]]
=> ([(0,9),(1,13),(2,14),(4,12),(5,11),(6,1),(6,12),(7,3),(8,5),(8,15),(9,10),(10,4),(10,6),(11,14),(12,8),(12,13),(13,15),(14,7),(15,2),(15,11)],16)
=> ?
=> ? = 4 - 3
[[0,1,0,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [[1,1,1,2,2],[2,2,2,5],[3,3,5],[4,5],[5]]
=> ([(0,1),(1,3),(1,4),(2,5),(2,18),(3,6),(3,19),(4,7),(4,8),(4,19),(5,13),(6,17),(7,12),(7,15),(8,12),(8,14),(10,21),(11,21),(12,2),(12,20),(13,9),(14,10),(14,20),(15,11),(15,20),(16,9),(17,10),(17,11),(18,13),(18,16),(19,14),(19,15),(19,17),(20,18),(20,21),(21,16)],22)
=> ?
=> ? = 4 - 3
[[0,0,1,0,0],[0,1,-1,0,1],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [[1,1,1,2,3],[2,2,2,5],[3,3,5],[4,5],[5]]
=> ?
=> ?
=> ? = 4 - 3
[[0,0,0,1,0],[0,1,0,-1,1],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [[1,1,1,2,4],[2,2,2,5],[3,3,5],[4,5],[5]]
=> ?
=> ?
=> ? = 4 - 3
[[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [[1,1,1,1,1],[2,2,3,3],[3,3,5],[4,5],[5]]
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> [6,3,1]
=> 0 = 3 - 3
[[0,1,0,0,0],[1,-1,1,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [[1,1,1,1,2],[2,2,3,3],[3,3,5],[4,5],[5]]
=> ?
=> ?
=> ? = 3 - 3
[[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [[1,1,1,1,3],[2,2,3,3],[3,3,5],[4,5],[5]]
=> ([(0,6),(0,7),(1,5),(1,15),(2,4),(2,14),(3,13),(4,12),(5,3),(5,16),(6,9),(7,2),(7,9),(9,1),(9,14),(10,11),(11,8),(12,10),(13,8),(14,12),(14,15),(15,10),(15,16),(16,11),(16,13)],17)
=> ?
=> ? = 4 - 3
[[0,1,0,0,0],[0,0,1,0,0],[1,-1,0,0,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [[1,1,1,2,2],[2,2,3,3],[3,3,5],[4,5],[5]]
=> ?
=> ?
=> ? = 3 - 3
[[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]]
=> [[1,1,1,2,3],[2,2,3,3],[3,3,5],[4,5],[5]]
=> ?
=> ?
=> ? = 4 - 3
[[1,0,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [[1,1,1,1,1],[2,2,3,4],[3,3,5],[4,5],[5]]
=> ?
=> ?
=> ? = 3 - 3
[[0,1,0,0,0],[1,-1,0,1,0],[0,0,1,-1,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [[1,1,1,1,2],[2,2,3,4],[3,3,5],[4,5],[5]]
=> ?
=> ?
=> ? = 3 - 3
[[0,0,1,0,0],[1,0,-1,1,0],[0,0,1,-1,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [[1,1,1,1,3],[2,2,3,4],[3,3,5],[4,5],[5]]
=> ?
=> ?
=> ? = 3 - 3
[[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]]
=> [[1,1,1,1,4],[2,2,3,4],[3,3,5],[4,5],[5]]
=> ?
=> ?
=> ? = 4 - 3
[[0,1,0,0,0],[0,0,0,1,0],[1,-1,1,-1,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [[1,1,1,2,2],[2,2,3,4],[3,3,5],[4,5],[5]]
=> ?
=> ?
=> ? = 3 - 3
[[0,0,1,0,0],[0,1,-1,1,0],[1,-1,1,-1,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [[1,1,1,2,3],[2,2,3,4],[3,3,5],[4,5],[5]]
=> ?
=> ?
=> ? = 3 - 3
[[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]]
=> [[1,1,1,2,4],[2,2,3,4],[3,3,5],[4,5],[5]]
=> ?
=> ?
=> ? = 4 - 3
[[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]]
=> [[1,1,1,3,3],[2,2,3,4],[3,3,5],[4,5],[5]]
=> ?
=> ?
=> ? = 4 - 3
[[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]]
=> [[1,1,1,3,4],[2,2,3,4],[3,3,5],[4,5],[5]]
=> ?
=> ?
=> ? = 4 - 3
[[0,0,1,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [[1,1,1,3,3],[2,2,3,5],[3,3,5],[4,5],[5]]
=> ([(0,1),(1,4),(1,5),(2,8),(2,32),(3,6),(3,30),(4,7),(4,33),(5,10),(5,11),(5,33),(6,20),(7,31),(8,27),(9,15),(9,19),(10,18),(10,28),(11,18),(11,25),(13,39),(14,36),(14,39),(15,36),(16,38),(17,38),(18,2),(18,37),(19,26),(19,36),(20,21),(21,12),(22,12),(23,17),(23,35),(24,16),(24,35),(25,13),(25,37),(26,24),(26,34),(27,16),(27,17),(28,14),(28,19),(28,37),(29,21),(29,22),(30,20),(30,29),(31,13),(31,14),(31,15),(32,23),(32,24),(32,27),(33,9),(33,25),(33,28),(33,31),(34,30),(34,35),(35,29),(35,38),(36,3),(36,34),(37,26),(37,32),(37,39),(38,22),(39,23),(39,34)],40)
=> ?
=> ? = 4 - 3
[[0,0,0,1,0],[0,0,1,-1,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [[1,1,1,3,4],[2,2,3,5],[3,3,5],[4,5],[5]]
=> ?
=> ?
=> ? = 4 - 3
[[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,0,1,0]]
=> [[1,1,2,2,2],[2,2,3,3],[3,3,5],[4,5],[5]]
=> ([(0,1),(1,3),(1,4),(2,5),(2,18),(3,6),(3,19),(4,7),(4,8),(4,19),(5,13),(6,17),(7,12),(7,15),(8,12),(8,14),(10,21),(11,21),(12,2),(12,20),(13,9),(14,10),(14,20),(15,11),(15,20),(16,9),(17,10),(17,11),(18,13),(18,16),(19,14),(19,15),(19,17),(20,18),(20,21),(21,16)],22)
=> ?
=> ? = 4 - 3
[[0,1,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[1,0,0,0,0],[0,0,0,1,0]]
=> [[1,1,2,2,2],[2,2,3,4],[3,3,5],[4,5],[5]]
=> ?
=> ?
=> ? = 4 - 3
[[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[1,0,0,0,0],[0,0,0,1,0]]
=> [[1,1,2,2,3],[2,2,3,4],[3,3,5],[4,5],[5]]
=> ?
=> ?
=> ? = 4 - 3
[[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,0,1,0,0]]
=> [[1,1,1,1,2],[2,2,2,2],[3,4,4],[4,5],[5]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> [5,3]
=> 0 = 3 - 3
[[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0]]
=> [[1,1,1,1,1],[2,2,2,4],[3,4,4],[4,5],[5]]
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> [6,1]
=> 0 = 3 - 3
[[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0]]
=> [[1,1,1,1,1],[2,3,3,3],[3,4,4],[4,5],[5]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> [7,1]
=> 0 = 3 - 3
[[0,0,0,1,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,4],[2,2,2,2,4],[3,3,3,4],[4,4,4],[5,5],[6]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> [7,1]
=> 0 = 3 - 3
[[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,3],[2,2,2,2,3],[3,3,3,3],[4,4,5],[5,5],[6]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> [5,3]
=> 0 = 3 - 3
[[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,5],[4,4,5],[5,5],[6]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> [5,3]
=> 0 = 3 - 3
[[1,0,0,0,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,5],[3,3,3,5],[4,4,5],[5,5],[6]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> [7,1]
=> 0 = 3 - 3
[[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,3],[2,2,2,2,3],[3,3,3,3],[4,4,4],[5,6],[6]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> [5,3]
=> 0 = 3 - 3
[[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,4],[4,4,4],[5,6],[6]]
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> [4,2,2]
=> 0 = 3 - 3
[[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,-1,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,3],[2,2,2,2,3],[3,3,3,4],[4,4,4],[5,6],[6]]
=> ([(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(3,7),(3,8),(4,6),(4,8),(5,1),(5,9),(6,11),(7,11),(8,5),(8,11),(9,10),(11,9)],12)
=> [6,4,2]
=> 0 = 3 - 3
[[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,4],[3,3,3,4],[4,4,4],[5,6],[6]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> [5,3]
=> 0 = 3 - 3
[[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,3],[2,2,2,2,3],[3,3,3,3],[4,4,5],[5,6],[6]]
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> [6,4,2]
=> 0 = 3 - 3
[[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,5],[4,4,5],[5,6],[6]]
=> ([(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(3,7),(3,8),(4,6),(4,8),(5,1),(5,9),(6,11),(7,11),(8,5),(8,11),(9,10),(11,9)],12)
=> [6,4,2]
=> 0 = 3 - 3
[[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,3],[4,4,6],[5,6],[6]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> [5,3]
=> 0 = 3 - 3
[[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,3],[3,3,3,3],[4,4,6],[5,6],[6]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> [5,3]
=> 0 = 3 - 3
[[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,2],[2,2,2,2,3],[3,3,3,3],[4,4,6],[5,6],[6]]
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> [6,4,2]
=> 0 = 3 - 3
[[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,6],[4,4,6],[5,6],[6]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> [7,1]
=> 0 = 3 - 3
Description
The constant term of the character polynomial of an integer partition.
The definition of the character polynomial can be found in [1]. Indeed, this constant term is $0$ for partitions $\lambda \neq 1^n$ and $1$ for $\lambda = 1^n$.
Matching statistic: St001123
Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St001123: Integer partitions ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 20%
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St001123: Integer partitions ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 20%
Values
[[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ([],1)
=> [1]
=> ? = 3 - 3
[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,2],[3,3],[4]]
=> ([],1)
=> [1]
=> ? = 4 - 3
[[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [[1,1,1,4],[2,2,4],[3,4],[4]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> [7,1]
=> 0 = 3 - 3
[[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
=> [[1,1,2,2],[2,2,4],[3,4],[4]]
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> [6,3,1]
=> 0 = 3 - 3
[[0,0,1,0],[0,1,-1,1],[1,0,0,0],[0,0,1,0]]
=> [[1,1,2,3],[2,2,4],[3,4],[4]]
=> ([(0,6),(1,9),(1,10),(2,8),(3,7),(4,3),(4,12),(5,2),(5,12),(6,4),(6,5),(7,9),(7,11),(8,10),(8,11),(9,13),(10,13),(11,13),(12,1),(12,7),(12,8)],14)
=> [7,4,3]
=> ? = 3 - 3
[[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
=> [[1,1,1,3],[2,3,3],[3,4],[4]]
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> [6,1]
=> 0 = 3 - 3
[[0,0,1,0],[0,1,0,0],[1,-1,0,1],[0,1,0,0]]
=> [[1,1,2,3],[2,3,3],[3,4],[4]]
=> ([(0,3),(0,8),(1,10),(2,9),(3,11),(4,2),(5,4),(6,7),(7,1),(7,9),(8,5),(8,11),(9,10),(11,6)],12)
=> [7,5]
=> 0 = 3 - 3
[[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> [[1,1,3,3],[2,3,4],[3,4],[4]]
=> ([(0,8),(2,11),(2,12),(3,10),(4,9),(5,4),(5,14),(6,3),(6,14),(7,1),(8,5),(8,6),(9,11),(9,13),(10,12),(10,13),(11,15),(12,15),(13,15),(14,2),(14,9),(14,10),(15,7)],16)
=> [9,4,3]
=> ? = 3 - 3
[[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> [[1,2,2,2],[2,3,3],[3,4],[4]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> [7,1]
=> 0 = 3 - 3
[[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,2,2],[3,3,3],[4,4],[5]]
=> ([],1)
=> [1]
=> ? = 5 - 3
[[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,4],[2,2,2,4],[3,3,4],[4,4],[5]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> [7,1]
=> 0 = 3 - 3
[[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,2,2],[2,2,2,4],[3,3,4],[4,4],[5]]
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> [6,3,1]
=> 0 = 3 - 3
[[0,0,1,0,0],[0,1,-1,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,2,3],[2,2,2,4],[3,3,4],[4,4],[5]]
=> ?
=> ?
=> ? = 3 - 3
[[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,3],[2,2,3,3],[3,3,4],[4,4],[5]]
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> [6,1]
=> 0 = 3 - 3
[[0,0,1,0,0],[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,2,3],[2,2,3,3],[3,3,4],[4,4],[5]]
=> ?
=> ?
=> ? = 3 - 3
[[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,3,3],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ([(0,8),(2,11),(2,12),(3,10),(4,9),(5,4),(5,14),(6,3),(6,14),(7,1),(8,5),(8,6),(9,11),(9,13),(10,12),(10,13),(11,15),(12,15),(13,15),(14,2),(14,9),(14,10),(15,7)],16)
=> [9,4,3]
=> ? = 3 - 3
[[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,0,0,1]]
=> [[1,1,2,2,2],[2,2,3,3],[3,3,4],[4,4],[5]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> [7,1]
=> 0 = 3 - 3
[[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,1,3],[2,2,2,3],[3,3,3],[4,5],[5]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> [5,3]
=> 0 = 3 - 3
[[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,2,2],[2,2,2,3],[3,3,3],[4,5],[5]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> [5,3]
=> 0 = 3 - 3
[[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[1,1,1,1,4],[2,2,2,4],[3,3,4],[4,5],[5]]
=> ([(0,4),(0,7),(2,10),(3,9),(4,8),(5,3),(5,11),(6,1),(7,5),(7,8),(8,11),(9,10),(10,6),(11,2),(11,9)],12)
=> ?
=> ? = 3 - 3
[[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[1,1,1,2,2],[2,2,2,4],[3,3,4],[4,5],[5]]
=> ?
=> ?
=> ? = 3 - 3
[[0,0,1,0,0],[0,1,-1,1,0],[1,0,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[1,1,1,2,3],[2,2,2,4],[3,3,4],[4,5],[5]]
=> ?
=> ?
=> ? = 3 - 3
[[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [[1,1,1,1,3],[2,2,3,3],[3,3,4],[4,5],[5]]
=> ?
=> ?
=> ? = 3 - 3
[[0,0,1,0,0],[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [[1,1,1,2,3],[2,2,3,3],[3,3,4],[4,5],[5]]
=> ?
=> ?
=> ? = 3 - 3
[[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [[1,1,1,1,4],[2,2,3,4],[3,3,4],[4,5],[5]]
=> ?
=> ?
=> ? = 3 - 3
[[0,0,0,1,0],[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [[1,1,1,2,4],[2,2,3,4],[3,3,4],[4,5],[5]]
=> ?
=> ?
=> ? = 3 - 3
[[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[1,0,0,-1,1],[0,0,0,1,0]]
=> [[1,1,2,2,2],[2,2,3,3],[3,3,4],[4,5],[5]]
=> ?
=> ?
=> ? = 3 - 3
[[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[1,0,0,-1,1],[0,0,0,1,0]]
=> [[1,1,2,2,2],[2,2,3,4],[3,3,4],[4,5],[5]]
=> ?
=> ?
=> ? = 3 - 3
[[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[1,0,0,-1,1],[0,0,0,1,0]]
=> [[1,1,2,2,3],[2,2,3,4],[3,3,4],[4,5],[5]]
=> ?
=> ?
=> ? = 3 - 3
[[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [[1,1,1,1,2],[2,2,2,2],[3,3,5],[4,5],[5]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> [5,3]
=> 0 = 3 - 3
[[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [[1,1,1,1,3],[2,2,2,3],[3,3,5],[4,5],[5]]
=> ([(0,5),(0,6),(1,3),(1,13),(2,4),(2,12),(3,10),(4,8),(5,11),(6,2),(6,11),(8,9),(9,7),(10,7),(11,1),(11,12),(12,8),(12,13),(13,9),(13,10)],14)
=> ?
=> ? = 3 - 3
[[0,1,0,0,0],[0,0,1,0,0],[1,0,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [[1,1,1,2,2],[2,2,2,3],[3,3,5],[4,5],[5]]
=> ?
=> ?
=> ? = 3 - 3
[[0,0,1,0,0],[0,1,0,0,0],[1,0,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [[1,1,1,2,3],[2,2,2,3],[3,3,5],[4,5],[5]]
=> ?
=> ?
=> ? = 3 - 3
[[0,0,0,1,0],[1,0,0,0,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [[1,1,1,1,4],[2,2,2,4],[3,3,5],[4,5],[5]]
=> ([(0,6),(0,7),(2,5),(2,16),(3,4),(3,17),(4,14),(5,12),(6,13),(7,2),(7,13),(8,10),(8,11),(9,18),(10,18),(11,18),(12,15),(13,3),(13,16),(14,9),(14,10),(15,9),(15,11),(16,12),(16,17),(17,8),(17,14),(17,15),(18,1)],19)
=> ?
=> ? = 3 - 3
[[0,1,0,0,0],[0,0,0,1,0],[1,0,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [[1,1,1,2,2],[2,2,2,4],[3,3,5],[4,5],[5]]
=> ?
=> ?
=> ? = 3 - 3
[[0,0,1,0,0],[0,1,-1,1,0],[1,0,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [[1,1,1,2,3],[2,2,2,4],[3,3,5],[4,5],[5]]
=> ?
=> ?
=> ? = 3 - 3
[[0,0,0,1,0],[0,1,0,0,0],[1,0,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [[1,1,1,2,4],[2,2,2,4],[3,3,5],[4,5],[5]]
=> ?
=> ?
=> ? = 3 - 3
[[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [[1,1,1,1,1],[2,2,2,5],[3,3,5],[4,5],[5]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> [7,1]
=> 0 = 3 - 3
[[0,1,0,0,0],[1,-1,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [[1,1,1,1,2],[2,2,2,5],[3,3,5],[4,5],[5]]
=> ([(0,7),(1,11),(2,8),(3,9),(3,13),(4,9),(4,12),(5,1),(5,10),(6,3),(6,4),(6,8),(7,2),(7,6),(8,12),(8,13),(9,5),(9,14),(10,11),(12,14),(13,14),(14,10)],15)
=> ?
=> ? = 3 - 3
[[0,0,1,0,0],[1,0,-1,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [[1,1,1,1,3],[2,2,2,5],[3,3,5],[4,5],[5]]
=> ([(0,1),(1,4),(1,5),(2,16),(3,6),(3,17),(4,18),(5,7),(5,8),(5,18),(6,13),(7,11),(7,14),(8,10),(8,11),(10,19),(11,3),(11,19),(12,9),(13,9),(14,16),(14,19),(15,12),(16,15),(17,12),(17,13),(18,2),(18,10),(18,14),(19,15),(19,17)],20)
=> ?
=> ? = 3 - 3
[[0,0,0,1,0],[1,0,0,-1,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [[1,1,1,1,4],[2,2,2,5],[3,3,5],[4,5],[5]]
=> ([(0,2),(1,6),(2,4),(2,5),(3,7),(3,22),(4,23),(5,9),(5,10),(5,23),(6,20),(7,15),(8,12),(8,14),(9,13),(9,17),(10,13),(10,18),(12,1),(13,3),(13,24),(14,19),(15,16),(16,11),(17,14),(17,24),(18,12),(18,24),(19,21),(20,11),(21,16),(21,20),(22,15),(22,21),(23,8),(23,17),(23,18),(24,19),(24,22)],25)
=> ?
=> ? = 3 - 3
[[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [[1,1,1,1,5],[2,2,2,5],[3,3,5],[4,5],[5]]
=> ([(0,9),(1,13),(2,14),(4,12),(5,11),(6,1),(6,12),(7,3),(8,5),(8,15),(9,10),(10,4),(10,6),(11,14),(12,8),(12,13),(13,15),(14,7),(15,2),(15,11)],16)
=> ?
=> ? = 4 - 3
[[0,1,0,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [[1,1,1,2,2],[2,2,2,5],[3,3,5],[4,5],[5]]
=> ([(0,1),(1,3),(1,4),(2,5),(2,18),(3,6),(3,19),(4,7),(4,8),(4,19),(5,13),(6,17),(7,12),(7,15),(8,12),(8,14),(10,21),(11,21),(12,2),(12,20),(13,9),(14,10),(14,20),(15,11),(15,20),(16,9),(17,10),(17,11),(18,13),(18,16),(19,14),(19,15),(19,17),(20,18),(20,21),(21,16)],22)
=> ?
=> ? = 4 - 3
[[0,0,1,0,0],[0,1,-1,0,1],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [[1,1,1,2,3],[2,2,2,5],[3,3,5],[4,5],[5]]
=> ?
=> ?
=> ? = 4 - 3
[[0,0,0,1,0],[0,1,0,-1,1],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [[1,1,1,2,4],[2,2,2,5],[3,3,5],[4,5],[5]]
=> ?
=> ?
=> ? = 4 - 3
[[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [[1,1,1,1,1],[2,2,3,3],[3,3,5],[4,5],[5]]
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> [6,3,1]
=> 0 = 3 - 3
[[0,1,0,0,0],[1,-1,1,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [[1,1,1,1,2],[2,2,3,3],[3,3,5],[4,5],[5]]
=> ?
=> ?
=> ? = 3 - 3
[[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [[1,1,1,1,3],[2,2,3,3],[3,3,5],[4,5],[5]]
=> ([(0,6),(0,7),(1,5),(1,15),(2,4),(2,14),(3,13),(4,12),(5,3),(5,16),(6,9),(7,2),(7,9),(9,1),(9,14),(10,11),(11,8),(12,10),(13,8),(14,12),(14,15),(15,10),(15,16),(16,11),(16,13)],17)
=> ?
=> ? = 4 - 3
[[0,1,0,0,0],[0,0,1,0,0],[1,-1,0,0,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [[1,1,1,2,2],[2,2,3,3],[3,3,5],[4,5],[5]]
=> ?
=> ?
=> ? = 3 - 3
[[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]]
=> [[1,1,1,2,3],[2,2,3,3],[3,3,5],[4,5],[5]]
=> ?
=> ?
=> ? = 4 - 3
[[1,0,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [[1,1,1,1,1],[2,2,3,4],[3,3,5],[4,5],[5]]
=> ?
=> ?
=> ? = 3 - 3
[[0,1,0,0,0],[1,-1,0,1,0],[0,0,1,-1,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [[1,1,1,1,2],[2,2,3,4],[3,3,5],[4,5],[5]]
=> ?
=> ?
=> ? = 3 - 3
[[0,0,1,0,0],[1,0,-1,1,0],[0,0,1,-1,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [[1,1,1,1,3],[2,2,3,4],[3,3,5],[4,5],[5]]
=> ?
=> ?
=> ? = 3 - 3
[[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]]
=> [[1,1,1,1,4],[2,2,3,4],[3,3,5],[4,5],[5]]
=> ?
=> ?
=> ? = 4 - 3
[[0,1,0,0,0],[0,0,0,1,0],[1,-1,1,-1,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [[1,1,1,2,2],[2,2,3,4],[3,3,5],[4,5],[5]]
=> ?
=> ?
=> ? = 3 - 3
[[0,0,1,0,0],[0,1,-1,1,0],[1,-1,1,-1,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [[1,1,1,2,3],[2,2,3,4],[3,3,5],[4,5],[5]]
=> ?
=> ?
=> ? = 3 - 3
[[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]]
=> [[1,1,1,2,4],[2,2,3,4],[3,3,5],[4,5],[5]]
=> ?
=> ?
=> ? = 4 - 3
[[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]]
=> [[1,1,1,3,3],[2,2,3,4],[3,3,5],[4,5],[5]]
=> ?
=> ?
=> ? = 4 - 3
[[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]]
=> [[1,1,1,3,4],[2,2,3,4],[3,3,5],[4,5],[5]]
=> ?
=> ?
=> ? = 4 - 3
[[0,0,1,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [[1,1,1,3,3],[2,2,3,5],[3,3,5],[4,5],[5]]
=> ([(0,1),(1,4),(1,5),(2,8),(2,32),(3,6),(3,30),(4,7),(4,33),(5,10),(5,11),(5,33),(6,20),(7,31),(8,27),(9,15),(9,19),(10,18),(10,28),(11,18),(11,25),(13,39),(14,36),(14,39),(15,36),(16,38),(17,38),(18,2),(18,37),(19,26),(19,36),(20,21),(21,12),(22,12),(23,17),(23,35),(24,16),(24,35),(25,13),(25,37),(26,24),(26,34),(27,16),(27,17),(28,14),(28,19),(28,37),(29,21),(29,22),(30,20),(30,29),(31,13),(31,14),(31,15),(32,23),(32,24),(32,27),(33,9),(33,25),(33,28),(33,31),(34,30),(34,35),(35,29),(35,38),(36,3),(36,34),(37,26),(37,32),(37,39),(38,22),(39,23),(39,34)],40)
=> ?
=> ? = 4 - 3
[[0,0,0,1,0],[0,0,1,-1,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [[1,1,1,3,4],[2,2,3,5],[3,3,5],[4,5],[5]]
=> ?
=> ?
=> ? = 4 - 3
[[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,0,1,0]]
=> [[1,1,2,2,2],[2,2,3,3],[3,3,5],[4,5],[5]]
=> ([(0,1),(1,3),(1,4),(2,5),(2,18),(3,6),(3,19),(4,7),(4,8),(4,19),(5,13),(6,17),(7,12),(7,15),(8,12),(8,14),(10,21),(11,21),(12,2),(12,20),(13,9),(14,10),(14,20),(15,11),(15,20),(16,9),(17,10),(17,11),(18,13),(18,16),(19,14),(19,15),(19,17),(20,18),(20,21),(21,16)],22)
=> ?
=> ? = 4 - 3
[[0,1,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[1,0,0,0,0],[0,0,0,1,0]]
=> [[1,1,2,2,2],[2,2,3,4],[3,3,5],[4,5],[5]]
=> ?
=> ?
=> ? = 4 - 3
[[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[1,0,0,0,0],[0,0,0,1,0]]
=> [[1,1,2,2,3],[2,2,3,4],[3,3,5],[4,5],[5]]
=> ?
=> ?
=> ? = 4 - 3
[[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,0,1,0,0]]
=> [[1,1,1,1,2],[2,2,2,2],[3,4,4],[4,5],[5]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> [5,3]
=> 0 = 3 - 3
[[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0]]
=> [[1,1,1,1,1],[2,2,2,4],[3,4,4],[4,5],[5]]
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> [6,1]
=> 0 = 3 - 3
[[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0]]
=> [[1,1,1,1,1],[2,3,3,3],[3,4,4],[4,5],[5]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> [7,1]
=> 0 = 3 - 3
[[0,0,0,1,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,4],[2,2,2,2,4],[3,3,3,4],[4,4,4],[5,5],[6]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> [7,1]
=> 0 = 3 - 3
[[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,3],[2,2,2,2,3],[3,3,3,3],[4,4,5],[5,5],[6]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> [5,3]
=> 0 = 3 - 3
[[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,5],[4,4,5],[5,5],[6]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> [5,3]
=> 0 = 3 - 3
[[1,0,0,0,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,5],[3,3,3,5],[4,4,5],[5,5],[6]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> [7,1]
=> 0 = 3 - 3
[[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,3],[2,2,2,2,3],[3,3,3,3],[4,4,4],[5,6],[6]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> [5,3]
=> 0 = 3 - 3
[[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,4],[4,4,4],[5,6],[6]]
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> [4,2,2]
=> 0 = 3 - 3
[[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,-1,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,3],[2,2,2,2,3],[3,3,3,4],[4,4,4],[5,6],[6]]
=> ([(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(3,7),(3,8),(4,6),(4,8),(5,1),(5,9),(6,11),(7,11),(8,5),(8,11),(9,10),(11,9)],12)
=> [6,4,2]
=> 0 = 3 - 3
[[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,4],[3,3,3,4],[4,4,4],[5,6],[6]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> [5,3]
=> 0 = 3 - 3
[[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,3],[2,2,2,2,3],[3,3,3,3],[4,4,5],[5,6],[6]]
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> [6,4,2]
=> 0 = 3 - 3
[[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,5],[4,4,5],[5,6],[6]]
=> ([(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(3,7),(3,8),(4,6),(4,8),(5,1),(5,9),(6,11),(7,11),(8,5),(8,11),(9,10),(11,9)],12)
=> [6,4,2]
=> 0 = 3 - 3
[[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,3],[4,4,6],[5,6],[6]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> [5,3]
=> 0 = 3 - 3
[[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,3],[3,3,3,3],[4,4,6],[5,6],[6]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> [5,3]
=> 0 = 3 - 3
[[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,2],[2,2,2,2,3],[3,3,3,3],[4,4,6],[5,6],[6]]
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> [6,4,2]
=> 0 = 3 - 3
[[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,6],[4,4,6],[5,6],[6]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> [7,1]
=> 0 = 3 - 3
Description
The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition.
The Kronecker coefficient is the multiplicity $g_{\mu,\nu}^\lambda$ of the Specht module $S^\lambda$ in $S^\mu\otimes S^\nu$:
$$ S^\mu\otimes S^\nu = \bigoplus_\lambda g_{\mu,\nu}^\lambda S^\lambda $$
This statistic records the Kronecker coefficient $g_{\lambda,\lambda}^{21^{n-2}}$, for $\lambda\vdash n$.
Matching statistic: St000068
Mp00004: Alternating sign matrices —rotate clockwise⟶ Alternating sign matrices
Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
St000068: Posets ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 20%
Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
St000068: Posets ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 20%
Values
[[1,0,0],[0,1,0],[0,0,1]]
=> [[0,0,1],[0,1,0],[1,0,0]]
=> [[1,2,3],[2,3],[3]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> 1 = 3 - 2
[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> [[1,2,3,4],[2,3,4],[3,4],[4]]
=> ([(0,5),(0,16),(0,17),(1,23),(2,19),(3,11),(3,21),(4,10),(4,20),(5,12),(5,13),(6,50),(7,51),(8,24),(8,58),(9,25),(9,59),(10,14),(10,52),(11,15),(11,53),(12,26),(12,60),(13,27),(13,60),(14,54),(15,55),(16,4),(16,29),(17,3),(17,29),(18,48),(18,49),(19,32),(19,33),(20,46),(20,52),(21,47),(21,53),(22,34),(22,35),(23,18),(23,54),(23,55),(24,40),(24,42),(25,41),(25,43),(26,46),(26,56),(27,47),(27,57),(28,63),(29,1),(30,62),(31,61),(32,61),(33,61),(34,6),(34,62),(35,7),(35,62),(36,58),(37,59),(38,32),(39,33),(40,44),(41,45),(42,38),(43,39),(44,31),(45,31),(46,36),(47,37),(48,40),(48,63),(49,41),(49,63),(50,38),(51,39),(52,8),(52,36),(53,9),(53,37),(54,28),(54,48),(55,28),(55,49),(56,30),(56,34),(57,30),(57,35),(58,42),(58,50),(59,43),(59,51),(60,22),(60,56),(60,57),(62,2),(63,44),(63,45)],64)
=> ? = 4 - 2
[[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> [[1,1,2,3],[2,2,3],[3,3],[4]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> 1 = 3 - 2
[[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
=> [[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
=> [[1,1,2,2],[2,2,4],[3,4],[4]]
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> 1 = 3 - 2
[[0,0,1,0],[0,1,-1,1],[1,0,0,0],[0,0,1,0]]
=> [[0,1,0,0],[0,0,1,0],[1,0,-1,1],[0,0,1,0]]
=> [[1,1,2,2],[2,2,3],[3,4],[4]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 3 - 2
[[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
=> [[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
=> [[1,1,1,3],[2,3,3],[3,4],[4]]
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> 1 = 3 - 2
[[0,0,1,0],[0,1,0,0],[1,-1,0,1],[0,1,0,0]]
=> [[0,1,0,0],[1,-1,1,0],[0,0,0,1],[0,1,0,0]]
=> [[1,1,1,2],[2,3,3],[3,4],[4]]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 1 = 3 - 2
[[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> [[1,1,1,2],[2,2,2],[3,4],[4]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 3 - 2
[[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> [[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> [[1,1,1,1],[2,3,4],[3,4],[4]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> 1 = 3 - 2
[[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0]]
=> [[1,2,3,4,5],[2,3,4,5],[3,4,5],[4,5],[5]]
=> 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=> ? = 5 - 2
[[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[1,0,0,0,0]]
=> [[1,2,2,3,4],[2,3,3,4],[3,4,4],[4,5],[5]]
=> ([(0,21),(0,22),(0,44),(1,103),(2,102),(3,19),(3,105),(4,20),(4,104),(5,40),(5,60),(6,37),(6,101),(7,42),(7,129),(8,36),(8,132),(9,34),(9,126),(10,41),(10,127),(11,38),(11,108),(12,32),(12,107),(13,43),(13,59),(14,35),(14,100),(15,18),(15,33),(15,106),(16,39),(16,128),(17,119),(18,26),(18,27),(18,99),(19,17),(19,130),(20,113),(21,16),(21,97),(22,11),(22,116),(23,78),(23,120),(24,58),(24,74),(25,57),(25,70),(26,98),(26,124),(27,23),(27,98),(27,115),(28,63),(28,117),(29,64),(29,73),(30,93),(30,95),(31,111),(31,118),(32,50),(32,89),(33,88),(33,99),(34,85),(34,87),(35,30),(35,86),(35,122),(36,84),(36,125),(37,55),(37,96),(38,51),(38,90),(39,51),(39,91),(40,49),(40,92),(41,28),(41,121),(41,123),(42,31),(42,124),(42,131),(43,29),(43,120),(43,130),(44,15),(44,97),(44,116),(45,145),(46,135),(47,133),(48,146),(49,134),(50,141),(51,142),(52,139),(53,136),(54,136),(55,138),(56,147),(57,2),(57,143),(58,1),(58,140),(59,14),(60,6),(61,77),(62,49),(62,145),(63,75),(64,84),(64,144),(65,54),(66,57),(66,133),(67,65),(68,53),(69,53),(70,62),(70,143),(71,126),(72,127),(73,121),(73,144),(74,25),(74,66),(74,140),(75,94),(76,52),(76,146),(77,52),(77,134),(78,100),(79,85),(79,147),(80,81),(80,141),(81,45),(81,143),(82,46),(82,144),(83,47),(83,140),(84,61),(85,68),(86,95),(86,135),(87,65),(88,59),(89,104),(89,141),(90,105),(90,142),(91,110),(91,142),(92,94),(92,134),(93,79),(93,137),(94,55),(94,139),(95,48),(95,137),(96,54),(96,138),(97,7),(97,128),(98,8),(98,114),(99,12),(99,115),(100,122),(101,96),(102,67),(103,109),(104,9),(104,71),(105,10),(105,72),(106,13),(106,88),(107,4),(107,89),(108,3),(108,90),(109,75),(109,92),(110,58),(110,83),(111,50),(111,80),(112,76),(112,77),(113,56),(113,79),(114,80),(114,132),(115,78),(115,107),(116,106),(116,108),(117,48),(117,76),(118,47),(118,66),(119,46),(119,86),(120,64),(120,82),(121,112),(121,117),(122,93),(122,113),(122,135),(123,63),(123,109),(124,111),(124,114),(125,45),(125,62),(126,67),(126,87),(127,103),(127,123),(128,91),(128,129),(129,24),(129,110),(129,131),(130,73),(130,82),(130,119),(131,74),(131,83),(131,118),(132,70),(132,81),(132,125),(133,60),(134,139),(135,56),(135,137),(137,146),(137,147),(138,136),(139,138),(140,5),(140,133),(141,71),(142,72),(143,102),(143,145),(144,61),(144,112),(145,101),(146,69),(147,68),(147,69)],148)
=> ? = 3 - 2
[[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[0,0,1,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0]]
=> [[1,2,2,3,3],[2,3,3,5],[3,4,5],[4,5],[5]]
=> ([(0,14),(0,15),(1,10),(1,11),(2,87),(3,13),(3,113),(4,89),(5,31),(5,107),(6,25),(6,108),(7,27),(7,29),(7,114),(8,24),(8,88),(9,12),(9,115),(10,90),(11,26),(11,28),(11,90),(12,112),(13,93),(14,1),(14,86),(15,7),(15,20),(15,86),(16,70),(16,71),(17,83),(17,84),(18,69),(18,99),(19,22),(19,82),(19,104),(20,23),(20,98),(20,114),(21,42),(21,100),(22,74),(22,75),(23,62),(23,95),(24,55),(24,58),(25,54),(25,56),(26,57),(26,85),(27,47),(27,66),(28,85),(28,109),(29,18),(29,47),(29,92),(30,81),(30,97),(30,103),(31,30),(31,99),(31,109),(31,110),(32,138),(33,147),(34,146),(35,117),(36,117),(36,147),(37,140),(37,148),(38,139),(39,137),(40,136),(41,150),(42,142),(43,125),(44,129),(45,135),(46,127),(46,151),(47,6),(47,116),(48,71),(48,148),(48,151),(49,64),(50,77),(50,152),(51,73),(51,150),(52,43),(53,65),(54,96),(54,153),(55,76),(55,118),(56,43),(56,153),(57,88),(58,89),(58,118),(59,67),(59,149),(60,68),(60,118),(60,149),(61,59),(61,144),(62,94),(63,44),(63,146),(64,32),(64,133),(65,38),(65,132),(66,34),(66,116),(67,35),(67,145),(68,36),(68,143),(68,145),(69,44),(69,119),(70,33),(70,143),(71,17),(71,91),(71,143),(72,78),(72,130),(73,45),(73,123),(74,32),(74,121),(75,40),(75,121),(76,41),(76,126),(77,39),(77,123),(78,40),(78,124),(79,38),(79,124),(80,37),(80,129),(80,142),(81,54),(81,122),(81,144),(82,75),(82,131),(83,72),(83,128),(84,39),(84,128),(85,9),(85,111),(86,5),(86,98),(87,19),(87,101),(88,4),(88,58),(89,3),(89,106),(90,8),(90,57),(91,77),(91,84),(91,120),(92,63),(92,69),(92,116),(93,78),(93,79),(94,42),(94,80),(95,34),(95,63),(96,41),(96,51),(97,55),(97,60),(97,144),(98,62),(98,107),(99,61),(99,81),(99,119),(100,16),(100,46),(100,48),(100,142),(101,49),(101,104),(102,65),(102,79),(102,130),(103,37),(103,48),(103,122),(104,64),(104,74),(104,131),(105,53),(105,102),(106,105),(106,113),(107,21),(107,94),(107,110),(108,52),(108,56),(109,61),(109,97),(109,111),(110,80),(110,100),(110,103),(110,119),(111,59),(111,60),(111,115),(112,33),(112,35),(112,36),(113,72),(113,93),(113,102),(114,66),(114,92),(114,95),(115,67),(115,68),(115,70),(115,112),(116,108),(116,146),(117,45),(117,134),(118,106),(118,126),(119,46),(119,122),(119,129),(120,123),(120,128),(120,134),(121,136),(121,138),(122,140),(122,151),(122,153),(123,135),(123,137),(124,136),(124,139),(125,152),(126,105),(126,150),(127,154),(128,130),(128,137),(129,127),(129,140),(130,124),(130,132),(131,121),(131,133),(132,139),(133,138),(134,131),(134,135),(135,133),(136,141),(137,132),(138,141),(139,141),(140,125),(140,154),(142,2),(142,127),(142,148),(143,83),(143,120),(143,147),(144,76),(144,96),(144,149),(145,73),(145,117),(145,120),(146,52),(147,82),(147,134),(148,87),(148,154),(149,51),(149,126),(149,145),(150,53),(151,50),(151,91),(151,154),(152,49),(153,50),(153,125),(154,101),(154,152)],155)
=> ? = 3 - 2
[[0,0,1,0,0],[0,1,-1,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[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]]
=> [[1,2,2,3,3],[2,3,3,4],[3,4,5],[4,5],[5]]
=> ?
=> ? = 3 - 2
[[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[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]]
=> [[1,2,2,2,4],[2,3,4,4],[3,4,5],[4,5],[5]]
=> ([(0,16),(0,17),(0,28),(1,9),(2,70),(3,13),(3,85),(4,27),(4,69),(5,83),(6,15),(6,43),(7,26),(7,84),(8,12),(8,29),(8,71),(9,68),(10,77),(11,76),(12,14),(12,78),(13,10),(13,86),(14,11),(14,87),(15,75),(16,8),(16,48),(17,7),(17,67),(18,42),(18,61),(19,25),(19,62),(19,63),(20,22),(20,59),(20,79),(21,24),(21,41),(21,60),(22,23),(22,80),(22,81),(23,64),(23,66),(24,37),(24,65),(25,36),(25,46),(26,50),(26,58),(27,31),(27,32),(28,48),(28,67),(29,57),(29,58),(29,78),(31,103),(32,103),(33,89),(34,89),(35,90),(36,88),(37,92),(38,102),(39,93),(40,101),(41,4),(41,100),(42,5),(42,97),(43,1),(44,53),(45,51),(45,95),(46,41),(46,88),(47,54),(48,71),(49,79),(50,73),(50,98),(51,68),(51,102),(52,31),(52,96),(53,30),(54,30),(55,33),(55,97),(56,35),(56,99),(57,70),(57,98),(58,74),(58,98),(59,81),(59,90),(60,37),(60,100),(61,34),(61,97),(62,36),(62,99),(63,21),(63,46),(63,99),(64,52),(64,91),(65,52),(65,92),(66,45),(66,91),(67,3),(67,84),(68,47),(69,32),(70,20),(70,49),(71,2),(71,57),(72,53),(72,54),(73,42),(73,55),(74,56),(74,62),(75,38),(75,51),(76,35),(76,59),(77,33),(77,34),(78,19),(78,74),(78,87),(79,80),(79,83),(79,90),(80,66),(80,82),(80,94),(81,64),(81,65),(81,94),(82,45),(82,75),(82,93),(83,39),(83,82),(84,50),(84,85),(85,18),(85,73),(85,86),(86,55),(86,61),(86,77),(87,56),(87,63),(87,76),(88,100),(89,43),(90,39),(90,94),(91,95),(91,96),(92,40),(92,96),(93,38),(93,40),(93,95),(94,91),(94,92),(94,93),(95,101),(95,102),(96,101),(96,103),(97,6),(97,89),(98,49),(99,60),(99,88),(100,69),(101,44),(101,72),(102,47),(102,72),(103,44)],104)
=> ? = 3 - 2
[[0,0,1,0,0],[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[0,0,1,0,0],[0,1,-1,1,0],[0,0,0,0,1],[0,0,1,0,0],[1,0,0,0,0]]
=> [[1,2,2,2,3],[2,3,4,4],[3,4,5],[4,5],[5]]
=> ?
=> ? = 3 - 2
[[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[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]]
=> [[1,2,2,2,3],[2,3,3,3],[3,4,5],[4,5],[5]]
=> ([(0,8),(0,9),(0,19),(1,42),(2,4),(2,20),(2,29),(3,50),(4,5),(4,52),(5,3),(5,53),(6,11),(6,35),(7,12),(7,31),(8,7),(8,34),(9,2),(9,33),(10,18),(10,46),(10,48),(11,27),(11,28),(12,25),(12,47),(13,15),(13,43),(13,44),(14,16),(14,26),(14,45),(15,17),(15,41),(15,51),(16,24),(16,40),(17,23),(17,39),(18,22),(18,32),(19,33),(19,34),(20,38),(20,47),(20,52),(21,55),(22,54),(23,57),(24,56),(25,62),(26,6),(26,63),(27,58),(28,58),(29,1),(29,38),(30,27),(30,60),(31,25),(32,26),(32,54),(33,29),(34,31),(35,28),(36,43),(37,21),(37,61),(38,42),(38,62),(39,30),(39,57),(40,30),(40,56),(41,23),(41,59),(42,13),(42,36),(43,41),(43,55),(44,51),(44,55),(45,24),(45,63),(46,22),(46,61),(47,49),(47,62),(48,14),(48,32),(48,61),(49,37),(49,46),(50,21),(50,44),(51,39),(51,40),(51,59),(52,10),(52,49),(52,53),(53,37),(53,48),(53,50),(54,63),(55,59),(56,60),(57,60),(59,56),(59,57),(60,58),(61,45),(61,54),(62,36),(63,35)],64)
=> ? = 3 - 2
[[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,0,0,1]]
=> [[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0]]
=> [[1,2,2,2,2],[2,3,4,5],[3,4,5],[4,5],[5]]
=> ([(0,5),(0,14),(0,15),(1,28),(2,9),(2,27),(3,13),(3,26),(4,21),(5,10),(5,11),(6,91),(7,81),(8,57),(9,6),(9,67),(10,18),(10,85),(11,19),(11,85),(12,20),(12,74),(13,17),(13,24),(13,93),(14,2),(14,37),(15,3),(15,37),(16,45),(16,70),(17,72),(17,88),(18,71),(18,89),(19,73),(19,90),(20,66),(20,68),(21,42),(21,65),(22,43),(22,44),(23,29),(23,87),(23,92),(24,72),(24,94),(25,31),(25,69),(25,86),(26,73),(26,93),(27,67),(27,71),(28,30),(28,91),(28,94),(29,48),(29,49),(30,50),(30,79),(31,56),(31,63),(31,64),(32,97),(33,99),(33,108),(34,98),(35,95),(35,97),(36,96),(37,1),(38,100),(39,100),(40,108),(41,109),(42,99),(43,7),(43,96),(44,8),(44,96),(45,101),(46,42),(47,65),(47,109),(48,53),(49,52),(50,66),(50,107),(51,74),(52,38),(53,39),(54,33),(54,106),(55,32),(55,107),(56,54),(56,95),(57,46),(58,62),(58,104),(59,61),(59,104),(60,58),(60,102),(61,39),(61,103),(62,38),(62,103),(63,47),(63,105),(64,70),(64,95),(64,105),(65,60),(65,99),(66,78),(67,12),(67,51),(68,46),(69,16),(69,64),(69,98),(70,23),(70,82),(70,101),(71,51),(72,80),(73,77),(74,57),(74,68),(75,53),(75,61),(76,52),(76,62),(77,34),(77,86),(78,33),(78,40),(79,35),(79,56),(79,107),(80,32),(80,35),(81,41),(81,47),(82,84),(82,87),(83,58),(83,76),(84,59),(84,75),(85,22),(85,89),(85,90),(86,63),(86,81),(86,98),(87,48),(87,75),(88,34),(88,69),(89,36),(89,44),(90,36),(90,43),(91,50),(91,55),(92,49),(92,76),(93,25),(93,77),(93,88),(94,55),(94,79),(94,80),(95,82),(95,106),(96,4),(97,40),(97,106),(98,41),(98,45),(98,105),(99,102),(101,83),(101,92),(102,104),(103,100),(104,103),(105,101),(105,109),(106,84),(106,108),(107,54),(107,78),(107,97),(108,59),(108,102),(109,60),(109,83)],110)
=> ? = 3 - 2
[[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0]]
=> [[1,1,3,3,4],[2,3,4,4],[3,4,5],[4,5],[5]]
=> ([(0,12),(0,13),(1,6),(2,61),(3,5),(3,94),(4,81),(5,80),(6,32),(7,4),(7,95),(8,22),(8,87),(9,20),(9,85),(10,19),(10,92),(11,17),(11,18),(11,98),(12,51),(13,11),(13,21),(13,51),(14,46),(14,47),(15,42),(15,89),(16,73),(16,75),(17,27),(17,48),(17,97),(18,48),(18,90),(19,43),(19,91),(20,44),(20,76),(21,25),(21,88),(21,98),(22,26),(22,74),(22,96),(23,24),(23,45),(23,77),(24,66),(24,67),(25,53),(25,93),(26,68),(26,84),(27,74),(27,78),(28,37),(28,60),(28,72),(29,124),(30,124),(31,101),(31,123),(33,117),(33,125),(34,118),(35,99),(36,116),(36,128),(37,101),(37,121),(38,109),(38,125),(39,100),(39,103),(40,109),(40,129),(41,126),(42,119),(43,107),(44,113),(45,9),(45,106),(46,108),(47,2),(47,108),(48,10),(48,122),(49,65),(49,128),(50,64),(51,8),(51,88),(52,39),(52,121),(52,123),(53,79),(54,45),(54,100),(55,32),(56,29),(56,127),(57,30),(57,127),(58,35),(58,120),(59,34),(59,99),(60,63),(60,101),(61,55),(62,56),(62,104),(63,83),(63,111),(64,57),(64,104),(65,44),(65,114),(66,58),(66,110),(67,65),(67,110),(68,43),(68,115),(69,34),(69,105),(70,36),(70,112),(71,33),(71,103),(71,119),(72,23),(72,54),(72,121),(73,50),(74,68),(74,102),(75,3),(75,82),(76,64),(76,113),(77,66),(77,106),(78,39),(78,54),(78,102),(79,42),(79,71),(80,29),(80,30),(81,35),(81,59),(82,62),(82,94),(83,49),(83,67),(83,112),(84,33),(84,38),(84,115),(85,50),(85,76),(86,59),(86,69),(86,120),(87,15),(87,79),(87,96),(88,53),(88,87),(89,38),(89,40),(89,119),(90,31),(90,60),(90,122),(91,36),(91,49),(91,107),(92,70),(92,83),(92,91),(93,31),(93,37),(93,52),(94,56),(94,57),(94,80),(95,58),(95,81),(95,86),(96,71),(96,84),(96,89),(96,102),(97,52),(97,72),(97,78),(97,122),(98,28),(98,90),(98,93),(98,97),(99,46),(99,118),(100,106),(101,7),(101,111),(102,40),(102,103),(102,115),(103,117),(103,129),(104,127),(105,47),(105,118),(106,85),(107,41),(107,116),(108,1),(109,41),(109,130),(110,114),(110,120),(111,95),(111,112),(112,86),(112,110),(112,128),(113,104),(114,105),(114,113),(115,107),(115,109),(115,117),(116,126),(117,116),(117,130),(118,108),(119,16),(119,125),(119,129),(120,14),(120,99),(120,105),(121,77),(121,100),(122,63),(122,92),(122,123),(123,70),(123,111),(124,55),(125,75),(125,130),(126,62),(127,61),(127,124),(128,69),(128,114),(129,73),(129,130),(130,82),(130,126)],131)
=> ? = 3 - 2
[[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0]]
=> [[1,1,3,3,3],[2,3,4,5],[3,4,5],[4,5],[5]]
=> ([(0,2),(0,3),(1,8),(1,16),(2,1),(3,7),(3,10),(4,13),(4,59),(5,12),(5,81),(6,20),(6,82),(7,11),(7,83),(8,60),(9,23),(9,58),(10,19),(10,22),(10,83),(11,78),(12,64),(13,24),(14,52),(14,72),(15,17),(15,73),(15,80),(16,60),(16,74),(17,39),(17,70),(18,36),(18,44),(19,51),(19,69),(20,33),(20,75),(21,28),(21,38),(21,50),(22,51),(22,71),(22,74),(23,49),(23,56),(23,57),(25,101),(26,113),(27,107),(27,110),(28,107),(28,111),(29,110),(29,111),(30,112),(31,90),(31,98),(32,94),(32,114),(33,96),(34,91),(34,92),(35,89),(35,92),(36,108),(37,94),(37,103),(38,9),(38,111),(39,45),(40,31),(40,102),(40,114),(41,31),(41,104),(42,37),(42,99),(43,34),(43,85),(43,87),(44,30),(44,108),(45,55),(46,33),(46,86),(47,48),(47,86),(48,25),(48,100),(49,42),(49,85),(50,14),(50,76),(50,107),(51,15),(51,61),(51,84),(52,46),(52,109),(53,32),(53,93),(54,26),(54,95),(55,25),(55,91),(56,35),(56,85),(56,97),(57,53),(57,97),(58,57),(59,24),(60,5),(60,63),(61,67),(61,73),(62,34),(62,35),(62,113),(63,67),(63,81),(64,46),(64,47),(65,45),(65,68),(66,47),(66,68),(66,109),(67,65),(67,66),(68,48),(68,55),(68,87),(69,29),(69,38),(69,84),(70,26),(70,62),(71,27),(71,50),(71,84),(72,43),(72,49),(72,109),(73,39),(73,65),(74,61),(74,63),(75,32),(75,40),(75,96),(76,72),(76,79),(76,88),(77,40),(77,41),(77,93),(78,27),(78,28),(78,29),(79,43),(79,56),(79,62),(79,95),(80,54),(80,70),(80,79),(81,52),(81,64),(81,66),(82,53),(82,75),(82,77),(83,21),(83,69),(83,71),(83,78),(84,76),(84,80),(84,110),(85,92),(85,99),(86,37),(86,96),(86,100),(87,91),(87,99),(87,100),(88,82),(88,95),(89,104),(90,108),(90,116),(91,101),(91,106),(92,106),(93,18),(93,104),(93,114),(94,115),(95,77),(95,97),(95,113),(96,94),(96,102),(97,89),(97,93),(98,116),(99,103),(99,106),(100,101),(100,102),(100,103),(101,98),(101,105),(102,98),(102,115),(103,105),(103,115),(104,36),(104,90),(105,116),(106,105),(107,6),(107,88),(108,4),(108,112),(109,42),(109,86),(109,87),(110,54),(110,88),(111,58),(112,59),(113,41),(113,89),(114,44),(114,90),(114,115),(115,30),(115,116),(116,112)],117)
=> ? = 3 - 2
[[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[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]]
=> [[1,1,2,3,4],[2,3,3,4],[3,4,4],[4,5],[5]]
=> ?
=> ? = 3 - 2
[[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[0,0,1,0,0],[0,0,0,0,1],[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0]]
=> [[1,1,2,3,3],[2,3,3,5],[3,4,5],[4,5],[5]]
=> ?
=> ? = 3 - 2
[[0,0,1,0,0],[0,1,-1,1,0],[1,0,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[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]]
=> [[1,1,2,3,3],[2,3,3,4],[3,4,5],[4,5],[5]]
=> ?
=> ? = 3 - 2
[[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [[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]]
=> [[1,1,2,2,4],[2,3,4,4],[3,4,5],[4,5],[5]]
=> ?
=> ? = 3 - 2
[[0,0,1,0,0],[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [[0,0,1,0,0],[0,1,-1,1,0],[0,0,0,0,1],[1,-1,1,0,0],[0,1,0,0,0]]
=> [[1,1,2,2,3],[2,3,4,4],[3,4,5],[4,5],[5]]
=> ?
=> ? = 3 - 2
[[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [[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]]
=> [[1,1,2,2,4],[2,3,3,4],[3,4,4],[4,5],[5]]
=> ?
=> ? = 3 - 2
[[0,0,0,1,0],[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[1,-1,0,0,1],[0,1,0,0,0]]
=> [[1,1,2,2,3],[2,3,3,4],[3,4,4],[4,5],[5]]
=> ?
=> ? = 3 - 2
[[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[1,0,0,-1,1],[0,0,0,1,0]]
=> [[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[1,-1,1,0,0],[0,1,0,0,0]]
=> [[1,1,2,2,2],[2,3,4,5],[3,4,5],[4,5],[5]]
=> ?
=> ? = 3 - 2
[[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[1,0,0,-1,1],[0,0,0,1,0]]
=> [[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[1,-1,0,1,0],[0,1,0,0,0]]
=> [[1,1,2,2,2],[2,3,3,5],[3,4,5],[4,5],[5]]
=> ?
=> ? = 3 - 2
[[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[1,0,0,-1,1],[0,0,0,1,0]]
=> [[0,1,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[1,-1,0,1,0],[0,1,0,0,0]]
=> [[1,1,2,2,2],[2,3,3,4],[3,4,5],[4,5],[5]]
=> ?
=> ? = 3 - 2
[[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [[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]]
=> [[1,1,2,4,4],[2,2,4,5],[3,4,5],[4,5],[5]]
=> ([(0,2),(0,3),(1,8),(1,16),(2,1),(3,7),(3,10),(4,13),(4,59),(5,12),(5,81),(6,20),(6,82),(7,11),(7,83),(8,60),(9,23),(9,58),(10,19),(10,22),(10,83),(11,78),(12,64),(13,24),(14,52),(14,72),(15,17),(15,73),(15,80),(16,60),(16,74),(17,39),(17,70),(18,36),(18,44),(19,51),(19,69),(20,33),(20,75),(21,28),(21,38),(21,50),(22,51),(22,71),(22,74),(23,49),(23,56),(23,57),(25,101),(26,113),(27,107),(27,110),(28,107),(28,111),(29,110),(29,111),(30,112),(31,90),(31,98),(32,94),(32,114),(33,96),(34,91),(34,92),(35,89),(35,92),(36,108),(37,94),(37,103),(38,9),(38,111),(39,45),(40,31),(40,102),(40,114),(41,31),(41,104),(42,37),(42,99),(43,34),(43,85),(43,87),(44,30),(44,108),(45,55),(46,33),(46,86),(47,48),(47,86),(48,25),(48,100),(49,42),(49,85),(50,14),(50,76),(50,107),(51,15),(51,61),(51,84),(52,46),(52,109),(53,32),(53,93),(54,26),(54,95),(55,25),(55,91),(56,35),(56,85),(56,97),(57,53),(57,97),(58,57),(59,24),(60,5),(60,63),(61,67),(61,73),(62,34),(62,35),(62,113),(63,67),(63,81),(64,46),(64,47),(65,45),(65,68),(66,47),(66,68),(66,109),(67,65),(67,66),(68,48),(68,55),(68,87),(69,29),(69,38),(69,84),(70,26),(70,62),(71,27),(71,50),(71,84),(72,43),(72,49),(72,109),(73,39),(73,65),(74,61),(74,63),(75,32),(75,40),(75,96),(76,72),(76,79),(76,88),(77,40),(77,41),(77,93),(78,27),(78,28),(78,29),(79,43),(79,56),(79,62),(79,95),(80,54),(80,70),(80,79),(81,52),(81,64),(81,66),(82,53),(82,75),(82,77),(83,21),(83,69),(83,71),(83,78),(84,76),(84,80),(84,110),(85,92),(85,99),(86,37),(86,96),(86,100),(87,91),(87,99),(87,100),(88,82),(88,95),(89,104),(90,108),(90,116),(91,101),(91,106),(92,106),(93,18),(93,104),(93,114),(94,115),(95,77),(95,97),(95,113),(96,94),(96,102),(97,89),(97,93),(98,116),(99,103),(99,106),(100,101),(100,102),(100,103),(101,98),(101,105),(102,98),(102,115),(103,105),(103,115),(104,36),(104,90),(105,116),(106,105),(107,6),(107,88),(108,4),(108,112),(109,42),(109,86),(109,87),(110,54),(110,88),(111,58),(112,59),(113,41),(113,89),(114,44),(114,90),(114,115),(115,30),(115,116),(116,112)],117)
=> ? = 3 - 2
[[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [[0,0,0,1,0],[0,0,1,0,0],[0,1,-1,0,1],[1,0,0,0,0],[0,0,1,0,0]]
=> [[1,1,2,3,4],[2,2,4,4],[3,4,5],[4,5],[5]]
=> ?
=> ? = 3 - 2
[[0,1,0,0,0],[0,0,1,0,0],[1,0,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [[0,0,1,0,0],[0,0,0,0,1],[0,1,-1,1,0],[1,0,0,0,0],[0,0,1,0,0]]
=> [[1,1,2,3,3],[2,2,4,5],[3,4,5],[4,5],[5]]
=> ?
=> ? = 3 - 2
[[0,0,1,0,0],[0,1,0,0,0],[1,0,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [[0,0,1,0,0],[0,0,0,1,0],[0,1,-1,0,1],[1,0,0,0,0],[0,0,1,0,0]]
=> [[1,1,2,3,3],[2,2,4,4],[3,4,5],[4,5],[5]]
=> ?
=> ? = 3 - 2
[[0,0,0,1,0],[1,0,0,0,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [[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]]
=> [[1,1,2,3,4],[2,2,3,4],[3,4,4],[4,5],[5]]
=> ?
=> ? = 3 - 2
[[0,1,0,0,0],[0,0,0,1,0],[1,0,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [[0,0,1,0,0],[0,0,0,0,1],[0,1,0,0,0],[1,0,-1,1,0],[0,0,1,0,0]]
=> [[1,1,2,3,3],[2,2,3,5],[3,4,5],[4,5],[5]]
=> ?
=> ? = 3 - 2
[[0,0,1,0,0],[0,1,-1,1,0],[1,0,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [[0,0,1,0,0],[0,0,0,1,0],[0,1,0,-1,1],[1,0,-1,1,0],[0,0,1,0,0]]
=> [[1,1,2,3,3],[2,2,3,4],[3,4,5],[4,5],[5]]
=> ?
=> ? = 3 - 2
[[0,0,0,1,0],[0,1,0,0,0],[1,0,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[1,0,-1,0,1],[0,0,1,0,0]]
=> [[1,1,2,3,3],[2,2,3,4],[3,4,4],[4,5],[5]]
=> ?
=> ? = 3 - 2
[[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [[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]]
=> [[1,1,2,3,5],[2,2,3,5],[3,3,5],[4,5],[5]]
=> ([(0,5),(0,14),(0,15),(1,28),(2,9),(2,27),(3,13),(3,26),(4,21),(5,10),(5,11),(6,91),(7,81),(8,57),(9,6),(9,67),(10,18),(10,85),(11,19),(11,85),(12,20),(12,74),(13,17),(13,24),(13,93),(14,2),(14,37),(15,3),(15,37),(16,45),(16,70),(17,72),(17,88),(18,71),(18,89),(19,73),(19,90),(20,66),(20,68),(21,42),(21,65),(22,43),(22,44),(23,29),(23,87),(23,92),(24,72),(24,94),(25,31),(25,69),(25,86),(26,73),(26,93),(27,67),(27,71),(28,30),(28,91),(28,94),(29,48),(29,49),(30,50),(30,79),(31,56),(31,63),(31,64),(32,97),(33,99),(33,108),(34,98),(35,95),(35,97),(36,96),(37,1),(38,100),(39,100),(40,108),(41,109),(42,99),(43,7),(43,96),(44,8),(44,96),(45,101),(46,42),(47,65),(47,109),(48,53),(49,52),(50,66),(50,107),(51,74),(52,38),(53,39),(54,33),(54,106),(55,32),(55,107),(56,54),(56,95),(57,46),(58,62),(58,104),(59,61),(59,104),(60,58),(60,102),(61,39),(61,103),(62,38),(62,103),(63,47),(63,105),(64,70),(64,95),(64,105),(65,60),(65,99),(66,78),(67,12),(67,51),(68,46),(69,16),(69,64),(69,98),(70,23),(70,82),(70,101),(71,51),(72,80),(73,77),(74,57),(74,68),(75,53),(75,61),(76,52),(76,62),(77,34),(77,86),(78,33),(78,40),(79,35),(79,56),(79,107),(80,32),(80,35),(81,41),(81,47),(82,84),(82,87),(83,58),(83,76),(84,59),(84,75),(85,22),(85,89),(85,90),(86,63),(86,81),(86,98),(87,48),(87,75),(88,34),(88,69),(89,36),(89,44),(90,36),(90,43),(91,50),(91,55),(92,49),(92,76),(93,25),(93,77),(93,88),(94,55),(94,79),(94,80),(95,82),(95,106),(96,4),(97,40),(97,106),(98,41),(98,45),(98,105),(99,102),(101,83),(101,92),(102,104),(103,100),(104,103),(105,101),(105,109),(106,84),(106,108),(107,54),(107,78),(107,97),(108,59),(108,102),(109,60),(109,83)],110)
=> ? = 3 - 2
[[0,1,0,0,0],[1,-1,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [[0,0,0,1,0],[0,0,1,-1,1],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0]]
=> [[1,1,2,3,4],[2,2,3,5],[3,3,5],[4,5],[5]]
=> ?
=> ? = 3 - 2
[[0,0,1,0,0],[1,0,-1,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [[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]]
=> [[1,1,2,3,4],[2,2,3,4],[3,3,5],[4,5],[5]]
=> ?
=> ? = 3 - 2
[[0,0,0,1,0],[1,0,0,-1,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,-1,1],[0,0,0,1,0]]
=> [[1,1,2,3,4],[2,2,3,4],[3,3,4],[4,5],[5]]
=> ([(0,16),(0,17),(0,28),(1,22),(2,19),(3,11),(3,21),(4,54),(5,55),(6,26),(6,57),(7,25),(7,67),(8,23),(8,64),(9,24),(9,65),(10,14),(10,56),(11,15),(11,58),(12,3),(12,37),(13,27),(13,66),(14,59),(15,60),(16,13),(16,53),(17,6),(17,61),(18,51),(18,52),(19,33),(19,34),(20,35),(20,36),(21,50),(21,58),(22,18),(22,59),(22,60),(23,41),(23,43),(24,42),(24,44),(25,50),(25,63),(26,29),(26,46),(27,29),(27,45),(28,12),(28,53),(28,61),(29,70),(30,71),(31,69),(32,68),(33,68),(34,68),(35,4),(35,69),(36,5),(36,69),(37,1),(38,33),(39,34),(40,64),(41,48),(42,49),(43,38),(44,39),(45,62),(45,70),(46,56),(46,70),(47,65),(48,32),(49,32),(50,40),(51,42),(51,71),(52,41),(52,71),(53,7),(53,66),(54,38),(55,39),(56,9),(56,47),(57,10),(57,46),(58,8),(58,40),(59,30),(59,51),(60,30),(60,52),(61,37),(61,57),(62,31),(62,36),(63,31),(63,35),(64,43),(64,54),(65,44),(65,55),(66,45),(66,67),(67,20),(67,62),(67,63),(69,2),(70,47),(71,48),(71,49)],72)
=> ? = 3 - 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]]
=> [[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1]]
=> [[1,1,2,3,4],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ([(0,5),(0,16),(0,17),(1,23),(2,19),(3,11),(3,21),(4,10),(4,20),(5,12),(5,13),(6,50),(7,51),(8,24),(8,58),(9,25),(9,59),(10,14),(10,52),(11,15),(11,53),(12,26),(12,60),(13,27),(13,60),(14,54),(15,55),(16,4),(16,29),(17,3),(17,29),(18,48),(18,49),(19,32),(19,33),(20,46),(20,52),(21,47),(21,53),(22,34),(22,35),(23,18),(23,54),(23,55),(24,40),(24,42),(25,41),(25,43),(26,46),(26,56),(27,47),(27,57),(28,63),(29,1),(30,62),(31,61),(32,61),(33,61),(34,6),(34,62),(35,7),(35,62),(36,58),(37,59),(38,32),(39,33),(40,44),(41,45),(42,38),(43,39),(44,31),(45,31),(46,36),(47,37),(48,40),(48,63),(49,41),(49,63),(50,38),(51,39),(52,8),(52,36),(53,9),(53,37),(54,28),(54,48),(55,28),(55,49),(56,30),(56,34),(57,30),(57,35),(58,42),(58,50),(59,43),(59,51),(60,22),(60,56),(60,57),(62,2),(63,44),(63,45)],64)
=> ? = 4 - 2
[[0,1,0,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [[0,0,1,0,0],[0,0,0,0,1],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0]]
=> [[1,1,2,3,3],[2,2,3,5],[3,3,5],[4,5],[5]]
=> ([(0,2),(0,3),(1,7),(1,14),(2,1),(3,6),(3,8),(4,10),(4,62),(5,16),(5,59),(6,9),(6,63),(7,44),(8,17),(8,18),(8,63),(9,60),(10,46),(11,35),(11,36),(12,15),(12,53),(12,61),(13,24),(13,37),(14,44),(14,54),(15,27),(15,51),(16,26),(16,39),(17,33),(17,34),(18,34),(18,52),(18,54),(19,75),(20,74),(21,78),(22,77),(23,76),(23,77),(24,76),(25,68),(26,67),(27,29),(28,25),(28,78),(29,42),(30,20),(30,73),(31,20),(32,19),(32,72),(33,22),(33,64),(34,12),(34,45),(34,64),(35,38),(35,66),(36,40),(36,66),(37,11),(37,57),(37,76),(38,25),(38,71),(39,30),(39,67),(40,26),(40,70),(41,32),(41,70),(42,19),(42,68),(43,21),(43,69),(44,4),(44,47),(45,50),(45,53),(46,40),(46,41),(47,50),(47,62),(48,29),(48,55),(49,41),(49,55),(49,66),(50,48),(50,49),(51,21),(51,28),(52,23),(52,37),(52,64),(53,27),(53,48),(54,45),(54,47),(55,32),(55,42),(55,71),(56,28),(56,38),(56,69),(57,35),(57,56),(57,65),(58,30),(58,31),(59,39),(59,58),(60,22),(60,23),(60,24),(61,43),(61,51),(61,56),(62,36),(62,46),(62,49),(63,13),(63,33),(63,52),(63,60),(64,57),(64,61),(64,77),(65,59),(65,69),(66,70),(66,71),(67,73),(68,75),(69,58),(69,78),(70,67),(70,72),(71,68),(71,72),(72,73),(72,75),(73,74),(75,74),(76,5),(76,65),(77,43),(77,65),(78,31)],79)
=> ? = 4 - 2
[[0,0,1,0,0],[0,1,-1,0,1],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [[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]]
=> [[1,1,2,3,3],[2,2,3,4],[3,3,5],[4,5],[5]]
=> ?
=> ? = 4 - 2
[[0,0,0,1,0],[0,1,0,-1,1],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[1,0,0,-1,1],[0,0,0,1,0]]
=> [[1,1,2,3,3],[2,2,3,4],[3,3,4],[4,5],[5]]
=> ?
=> ? = 4 - 2
[[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [[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]]
=> [[1,1,2,2,5],[2,2,4,5],[3,4,5],[4,5],[5]]
=> ([(0,14),(0,15),(1,10),(1,11),(2,87),(3,13),(3,113),(4,89),(5,31),(5,107),(6,25),(6,108),(7,27),(7,29),(7,114),(8,24),(8,88),(9,12),(9,115),(10,90),(11,26),(11,28),(11,90),(12,112),(13,93),(14,1),(14,86),(15,7),(15,20),(15,86),(16,70),(16,71),(17,83),(17,84),(18,69),(18,99),(19,22),(19,82),(19,104),(20,23),(20,98),(20,114),(21,42),(21,100),(22,74),(22,75),(23,62),(23,95),(24,55),(24,58),(25,54),(25,56),(26,57),(26,85),(27,47),(27,66),(28,85),(28,109),(29,18),(29,47),(29,92),(30,81),(30,97),(30,103),(31,30),(31,99),(31,109),(31,110),(32,138),(33,147),(34,146),(35,117),(36,117),(36,147),(37,140),(37,148),(38,139),(39,137),(40,136),(41,150),(42,142),(43,125),(44,129),(45,135),(46,127),(46,151),(47,6),(47,116),(48,71),(48,148),(48,151),(49,64),(50,77),(50,152),(51,73),(51,150),(52,43),(53,65),(54,96),(54,153),(55,76),(55,118),(56,43),(56,153),(57,88),(58,89),(58,118),(59,67),(59,149),(60,68),(60,118),(60,149),(61,59),(61,144),(62,94),(63,44),(63,146),(64,32),(64,133),(65,38),(65,132),(66,34),(66,116),(67,35),(67,145),(68,36),(68,143),(68,145),(69,44),(69,119),(70,33),(70,143),(71,17),(71,91),(71,143),(72,78),(72,130),(73,45),(73,123),(74,32),(74,121),(75,40),(75,121),(76,41),(76,126),(77,39),(77,123),(78,40),(78,124),(79,38),(79,124),(80,37),(80,129),(80,142),(81,54),(81,122),(81,144),(82,75),(82,131),(83,72),(83,128),(84,39),(84,128),(85,9),(85,111),(86,5),(86,98),(87,19),(87,101),(88,4),(88,58),(89,3),(89,106),(90,8),(90,57),(91,77),(91,84),(91,120),(92,63),(92,69),(92,116),(93,78),(93,79),(94,42),(94,80),(95,34),(95,63),(96,41),(96,51),(97,55),(97,60),(97,144),(98,62),(98,107),(99,61),(99,81),(99,119),(100,16),(100,46),(100,48),(100,142),(101,49),(101,104),(102,65),(102,79),(102,130),(103,37),(103,48),(103,122),(104,64),(104,74),(104,131),(105,53),(105,102),(106,105),(106,113),(107,21),(107,94),(107,110),(108,52),(108,56),(109,61),(109,97),(109,111),(110,80),(110,100),(110,103),(110,119),(111,59),(111,60),(111,115),(112,33),(112,35),(112,36),(113,72),(113,93),(113,102),(114,66),(114,92),(114,95),(115,67),(115,68),(115,70),(115,112),(116,108),(116,146),(117,45),(117,134),(118,106),(118,126),(119,46),(119,122),(119,129),(120,123),(120,128),(120,134),(121,136),(121,138),(122,140),(122,151),(122,153),(123,135),(123,137),(124,136),(124,139),(125,152),(126,105),(126,150),(127,154),(128,130),(128,137),(129,127),(129,140),(130,124),(130,132),(131,121),(131,133),(132,139),(133,138),(134,131),(134,135),(135,133),(136,141),(137,132),(138,141),(139,141),(140,125),(140,154),(142,2),(142,127),(142,148),(143,83),(143,120),(143,147),(144,76),(144,96),(144,149),(145,73),(145,117),(145,120),(146,52),(147,82),(147,134),(148,87),(148,154),(149,51),(149,126),(149,145),(150,53),(151,50),(151,91),(151,154),(152,49),(153,50),(153,125),(154,101),(154,152)],155)
=> ? = 3 - 2
[[0,1,0,0,0],[1,-1,1,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [[0,0,0,1,0],[0,1,0,-1,1],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0]]
=> [[1,1,2,2,4],[2,2,4,5],[3,4,5],[4,5],[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]]
=> [[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0]]
=> [[1,1,2,2,4],[2,2,4,4],[3,4,5],[4,5],[5]]
=> ([(0,7),(0,8),(1,2),(1,45),(2,35),(3,37),(4,14),(4,49),(5,9),(5,48),(6,10),(6,12),(6,54),(7,31),(8,6),(8,13),(8,31),(9,27),(9,42),(10,28),(10,41),(11,26),(11,52),(12,17),(12,28),(12,51),(13,15),(13,50),(13,54),(14,16),(14,40),(14,53),(15,33),(15,44),(16,38),(16,47),(17,30),(17,40),(19,64),(19,66),(20,62),(21,65),(22,58),(23,60),(23,66),(24,60),(24,63),(25,67),(26,55),(27,59),(28,5),(28,56),(29,34),(30,22),(30,57),(31,4),(31,50),(32,22),(32,65),(33,43),(34,18),(35,18),(36,20),(37,1),(37,46),(38,27),(38,61),(39,19),(39,55),(39,58),(40,38),(40,57),(41,21),(41,56),(42,20),(42,59),(43,26),(43,39),(44,21),(44,32),(45,34),(45,35),(46,29),(46,45),(47,19),(47,23),(47,61),(48,36),(48,42),(49,11),(49,43),(49,53),(50,33),(50,49),(51,30),(51,32),(51,56),(52,23),(52,24),(52,55),(53,39),(53,47),(53,52),(53,57),(54,41),(54,44),(54,51),(55,3),(55,63),(55,66),(56,48),(56,65),(57,24),(57,58),(57,61),(58,63),(58,64),(59,25),(59,62),(60,25),(60,68),(61,59),(61,60),(61,64),(62,67),(63,68),(64,62),(64,68),(65,36),(66,37),(66,68),(67,29),(68,46),(68,67)],69)
=> ? = 4 - 2
[[0,1,0,0,0],[0,0,1,0,0],[1,-1,0,0,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [[0,0,1,0,0],[0,1,-1,0,1],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0]]
=> [[1,1,2,2,3],[2,2,4,5],[3,4,5],[4,5],[5]]
=> ?
=> ? = 3 - 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]]
=> [[0,0,1,0,0],[0,1,-1,1,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0]]
=> [[1,1,2,2,3],[2,2,4,4],[3,4,5],[4,5],[5]]
=> ?
=> ? = 4 - 2
[[1,0,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [[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]]
=> [[1,1,2,2,5],[2,2,3,5],[3,4,5],[4,5],[5]]
=> ?
=> ? = 3 - 2
[[0,1,0,0,0],[1,-1,0,1,0],[0,0,1,-1,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [[0,0,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[1,0,-1,1,0],[0,0,1,0,0]]
=> [[1,1,2,2,4],[2,2,3,5],[3,4,5],[4,5],[5]]
=> ?
=> ? = 3 - 2
[[0,0,1,0,0],[1,0,-1,1,0],[0,0,1,-1,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [[0,0,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[1,0,-1,1,0],[0,0,1,0,0]]
=> [[1,1,2,2,4],[2,2,3,4],[3,4,5],[4,5],[5]]
=> ?
=> ? = 3 - 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]]
=> [[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]]
=> [[1,1,2,2,4],[2,2,3,4],[3,4,4],[4,5],[5]]
=> ?
=> ? = 4 - 2
[[0,1,0,0,0],[0,0,0,1,0],[1,-1,1,-1,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [[0,0,1,0,0],[0,1,-1,0,1],[0,0,1,0,0],[1,0,-1,1,0],[0,0,1,0,0]]
=> [[1,1,2,2,3],[2,2,3,5],[3,4,5],[4,5],[5]]
=> ?
=> ? = 3 - 2
[[0,0,1,0,0],[0,1,-1,1,0],[1,-1,1,-1,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[1,0,-1,1,0],[0,0,1,0,0]]
=> [[1,1,2,2,3],[2,2,3,4],[3,4,5],[4,5],[5]]
=> ?
=> ? = 3 - 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]]
=> [[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[1,0,-1,0,1],[0,0,1,0,0]]
=> [[1,1,2,2,3],[2,2,3,4],[3,4,4],[4,5],[5]]
=> ?
=> ? = 4 - 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]]
=> [[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[1,0,-1,1,0],[0,0,1,0,0]]
=> [[1,1,2,2,3],[2,2,3,3],[3,4,5],[4,5],[5]]
=> ?
=> ? = 4 - 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]]
=> [[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,2,3],[2,2,2,3],[3,3,3],[4,4],[5]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> 1 = 3 - 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]]
=> [[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,2,2],[2,2,2,4],[3,3,4],[4,4],[5]]
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> 1 = 3 - 2
[[0,0,0,0,1],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0]]
=> [[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,3],[2,2,3,3],[3,3,4],[4,4],[5]]
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> 1 = 3 - 2
[[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0]]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,2],[2,2,2,2],[3,3,4],[4,4],[5]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 3 - 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]]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,1,2],[2,2,2,2],[3,3,3],[4,5],[5]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 3 - 2
[[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0]]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> 1 = 3 - 2
[[0,0,1,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0]]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [[1,1,1,1,1],[2,2,3,3],[3,3,5],[4,5],[5]]
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> 1 = 3 - 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]]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0]]
=> [[1,1,1,1,1],[2,2,2,4],[3,4,4],[4,5],[5]]
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> 1 = 3 - 2
[[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0]]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,1,1],[2,2,2,3],[3,3,3],[4,5],[5]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 3 - 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]]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [[1,1,1,1,1],[2,2,2,2],[3,4,5],[4,5],[5]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> 1 = 3 - 2
[[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0]]
=> [[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,2,3],[2,2,2,2,3],[3,3,3,3],[4,4,4],[5,5],[6]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> 1 = 3 - 2
[[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0]]
=> [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,4],[4,4,4],[5,5],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 3 - 2
[[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0]]
=> [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,3],[4,4,5],[5,5],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 3 - 2
[[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0]]
=> [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,3],[4,4,4],[5,6],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 3 - 2
[[0,0,0,0,0,1],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[1,0,0,0,0,0]]
=> [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,3,4],[3,3,3,4],[4,4,4],[5,5],[6]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> 1 = 3 - 2
[[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[1,0,0,0,0,0]]
=> [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,3],[3,3,3,3],[4,4,5],[5,5],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 3 - 2
[[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[1,0,0,0,0,0]]
=> [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,3],[3,3,3,3],[4,4,4],[5,6],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 3 - 2
[[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[1,0,0,0,0,0]]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,4,5],[4,4,5],[5,5],[6]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> 1 = 3 - 2
[[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0]]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,4],[4,4,4],[5,6],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 3 - 2
[[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0]]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,3],[4,5,6],[5,6],[6]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> 1 = 3 - 2
Description
The number of minimal elements in a poset.
Matching statistic: St000069
Mp00004: Alternating sign matrices —rotate clockwise⟶ Alternating sign matrices
Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
St000069: Posets ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 20%
Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
St000069: Posets ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 20%
Values
[[1,0,0],[0,1,0],[0,0,1]]
=> [[0,0,1],[0,1,0],[1,0,0]]
=> [[1,2,3],[2,3],[3]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> 1 = 3 - 2
[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> [[1,2,3,4],[2,3,4],[3,4],[4]]
=> ([(0,5),(0,16),(0,17),(1,23),(2,19),(3,11),(3,21),(4,10),(4,20),(5,12),(5,13),(6,50),(7,51),(8,24),(8,58),(9,25),(9,59),(10,14),(10,52),(11,15),(11,53),(12,26),(12,60),(13,27),(13,60),(14,54),(15,55),(16,4),(16,29),(17,3),(17,29),(18,48),(18,49),(19,32),(19,33),(20,46),(20,52),(21,47),(21,53),(22,34),(22,35),(23,18),(23,54),(23,55),(24,40),(24,42),(25,41),(25,43),(26,46),(26,56),(27,47),(27,57),(28,63),(29,1),(30,62),(31,61),(32,61),(33,61),(34,6),(34,62),(35,7),(35,62),(36,58),(37,59),(38,32),(39,33),(40,44),(41,45),(42,38),(43,39),(44,31),(45,31),(46,36),(47,37),(48,40),(48,63),(49,41),(49,63),(50,38),(51,39),(52,8),(52,36),(53,9),(53,37),(54,28),(54,48),(55,28),(55,49),(56,30),(56,34),(57,30),(57,35),(58,42),(58,50),(59,43),(59,51),(60,22),(60,56),(60,57),(62,2),(63,44),(63,45)],64)
=> ? = 4 - 2
[[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> [[1,1,2,3],[2,2,3],[3,3],[4]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> 1 = 3 - 2
[[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
=> [[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
=> [[1,1,2,2],[2,2,4],[3,4],[4]]
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> 1 = 3 - 2
[[0,0,1,0],[0,1,-1,1],[1,0,0,0],[0,0,1,0]]
=> [[0,1,0,0],[0,0,1,0],[1,0,-1,1],[0,0,1,0]]
=> [[1,1,2,2],[2,2,3],[3,4],[4]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 3 - 2
[[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
=> [[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
=> [[1,1,1,3],[2,3,3],[3,4],[4]]
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> 1 = 3 - 2
[[0,0,1,0],[0,1,0,0],[1,-1,0,1],[0,1,0,0]]
=> [[0,1,0,0],[1,-1,1,0],[0,0,0,1],[0,1,0,0]]
=> [[1,1,1,2],[2,3,3],[3,4],[4]]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 1 = 3 - 2
[[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> [[1,1,1,2],[2,2,2],[3,4],[4]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 3 - 2
[[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> [[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> [[1,1,1,1],[2,3,4],[3,4],[4]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> 1 = 3 - 2
[[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0]]
=> [[1,2,3,4,5],[2,3,4,5],[3,4,5],[4,5],[5]]
=> 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=> ? = 5 - 2
[[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[1,0,0,0,0]]
=> [[1,2,2,3,4],[2,3,3,4],[3,4,4],[4,5],[5]]
=> ([(0,21),(0,22),(0,44),(1,103),(2,102),(3,19),(3,105),(4,20),(4,104),(5,40),(5,60),(6,37),(6,101),(7,42),(7,129),(8,36),(8,132),(9,34),(9,126),(10,41),(10,127),(11,38),(11,108),(12,32),(12,107),(13,43),(13,59),(14,35),(14,100),(15,18),(15,33),(15,106),(16,39),(16,128),(17,119),(18,26),(18,27),(18,99),(19,17),(19,130),(20,113),(21,16),(21,97),(22,11),(22,116),(23,78),(23,120),(24,58),(24,74),(25,57),(25,70),(26,98),(26,124),(27,23),(27,98),(27,115),(28,63),(28,117),(29,64),(29,73),(30,93),(30,95),(31,111),(31,118),(32,50),(32,89),(33,88),(33,99),(34,85),(34,87),(35,30),(35,86),(35,122),(36,84),(36,125),(37,55),(37,96),(38,51),(38,90),(39,51),(39,91),(40,49),(40,92),(41,28),(41,121),(41,123),(42,31),(42,124),(42,131),(43,29),(43,120),(43,130),(44,15),(44,97),(44,116),(45,145),(46,135),(47,133),(48,146),(49,134),(50,141),(51,142),(52,139),(53,136),(54,136),(55,138),(56,147),(57,2),(57,143),(58,1),(58,140),(59,14),(60,6),(61,77),(62,49),(62,145),(63,75),(64,84),(64,144),(65,54),(66,57),(66,133),(67,65),(68,53),(69,53),(70,62),(70,143),(71,126),(72,127),(73,121),(73,144),(74,25),(74,66),(74,140),(75,94),(76,52),(76,146),(77,52),(77,134),(78,100),(79,85),(79,147),(80,81),(80,141),(81,45),(81,143),(82,46),(82,144),(83,47),(83,140),(84,61),(85,68),(86,95),(86,135),(87,65),(88,59),(89,104),(89,141),(90,105),(90,142),(91,110),(91,142),(92,94),(92,134),(93,79),(93,137),(94,55),(94,139),(95,48),(95,137),(96,54),(96,138),(97,7),(97,128),(98,8),(98,114),(99,12),(99,115),(100,122),(101,96),(102,67),(103,109),(104,9),(104,71),(105,10),(105,72),(106,13),(106,88),(107,4),(107,89),(108,3),(108,90),(109,75),(109,92),(110,58),(110,83),(111,50),(111,80),(112,76),(112,77),(113,56),(113,79),(114,80),(114,132),(115,78),(115,107),(116,106),(116,108),(117,48),(117,76),(118,47),(118,66),(119,46),(119,86),(120,64),(120,82),(121,112),(121,117),(122,93),(122,113),(122,135),(123,63),(123,109),(124,111),(124,114),(125,45),(125,62),(126,67),(126,87),(127,103),(127,123),(128,91),(128,129),(129,24),(129,110),(129,131),(130,73),(130,82),(130,119),(131,74),(131,83),(131,118),(132,70),(132,81),(132,125),(133,60),(134,139),(135,56),(135,137),(137,146),(137,147),(138,136),(139,138),(140,5),(140,133),(141,71),(142,72),(143,102),(143,145),(144,61),(144,112),(145,101),(146,69),(147,68),(147,69)],148)
=> ? = 3 - 2
[[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[0,0,1,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0]]
=> [[1,2,2,3,3],[2,3,3,5],[3,4,5],[4,5],[5]]
=> ([(0,14),(0,15),(1,10),(1,11),(2,87),(3,13),(3,113),(4,89),(5,31),(5,107),(6,25),(6,108),(7,27),(7,29),(7,114),(8,24),(8,88),(9,12),(9,115),(10,90),(11,26),(11,28),(11,90),(12,112),(13,93),(14,1),(14,86),(15,7),(15,20),(15,86),(16,70),(16,71),(17,83),(17,84),(18,69),(18,99),(19,22),(19,82),(19,104),(20,23),(20,98),(20,114),(21,42),(21,100),(22,74),(22,75),(23,62),(23,95),(24,55),(24,58),(25,54),(25,56),(26,57),(26,85),(27,47),(27,66),(28,85),(28,109),(29,18),(29,47),(29,92),(30,81),(30,97),(30,103),(31,30),(31,99),(31,109),(31,110),(32,138),(33,147),(34,146),(35,117),(36,117),(36,147),(37,140),(37,148),(38,139),(39,137),(40,136),(41,150),(42,142),(43,125),(44,129),(45,135),(46,127),(46,151),(47,6),(47,116),(48,71),(48,148),(48,151),(49,64),(50,77),(50,152),(51,73),(51,150),(52,43),(53,65),(54,96),(54,153),(55,76),(55,118),(56,43),(56,153),(57,88),(58,89),(58,118),(59,67),(59,149),(60,68),(60,118),(60,149),(61,59),(61,144),(62,94),(63,44),(63,146),(64,32),(64,133),(65,38),(65,132),(66,34),(66,116),(67,35),(67,145),(68,36),(68,143),(68,145),(69,44),(69,119),(70,33),(70,143),(71,17),(71,91),(71,143),(72,78),(72,130),(73,45),(73,123),(74,32),(74,121),(75,40),(75,121),(76,41),(76,126),(77,39),(77,123),(78,40),(78,124),(79,38),(79,124),(80,37),(80,129),(80,142),(81,54),(81,122),(81,144),(82,75),(82,131),(83,72),(83,128),(84,39),(84,128),(85,9),(85,111),(86,5),(86,98),(87,19),(87,101),(88,4),(88,58),(89,3),(89,106),(90,8),(90,57),(91,77),(91,84),(91,120),(92,63),(92,69),(92,116),(93,78),(93,79),(94,42),(94,80),(95,34),(95,63),(96,41),(96,51),(97,55),(97,60),(97,144),(98,62),(98,107),(99,61),(99,81),(99,119),(100,16),(100,46),(100,48),(100,142),(101,49),(101,104),(102,65),(102,79),(102,130),(103,37),(103,48),(103,122),(104,64),(104,74),(104,131),(105,53),(105,102),(106,105),(106,113),(107,21),(107,94),(107,110),(108,52),(108,56),(109,61),(109,97),(109,111),(110,80),(110,100),(110,103),(110,119),(111,59),(111,60),(111,115),(112,33),(112,35),(112,36),(113,72),(113,93),(113,102),(114,66),(114,92),(114,95),(115,67),(115,68),(115,70),(115,112),(116,108),(116,146),(117,45),(117,134),(118,106),(118,126),(119,46),(119,122),(119,129),(120,123),(120,128),(120,134),(121,136),(121,138),(122,140),(122,151),(122,153),(123,135),(123,137),(124,136),(124,139),(125,152),(126,105),(126,150),(127,154),(128,130),(128,137),(129,127),(129,140),(130,124),(130,132),(131,121),(131,133),(132,139),(133,138),(134,131),(134,135),(135,133),(136,141),(137,132),(138,141),(139,141),(140,125),(140,154),(142,2),(142,127),(142,148),(143,83),(143,120),(143,147),(144,76),(144,96),(144,149),(145,73),(145,117),(145,120),(146,52),(147,82),(147,134),(148,87),(148,154),(149,51),(149,126),(149,145),(150,53),(151,50),(151,91),(151,154),(152,49),(153,50),(153,125),(154,101),(154,152)],155)
=> ? = 3 - 2
[[0,0,1,0,0],[0,1,-1,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[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]]
=> [[1,2,2,3,3],[2,3,3,4],[3,4,5],[4,5],[5]]
=> ?
=> ? = 3 - 2
[[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[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]]
=> [[1,2,2,2,4],[2,3,4,4],[3,4,5],[4,5],[5]]
=> ([(0,16),(0,17),(0,28),(1,9),(2,70),(3,13),(3,85),(4,27),(4,69),(5,83),(6,15),(6,43),(7,26),(7,84),(8,12),(8,29),(8,71),(9,68),(10,77),(11,76),(12,14),(12,78),(13,10),(13,86),(14,11),(14,87),(15,75),(16,8),(16,48),(17,7),(17,67),(18,42),(18,61),(19,25),(19,62),(19,63),(20,22),(20,59),(20,79),(21,24),(21,41),(21,60),(22,23),(22,80),(22,81),(23,64),(23,66),(24,37),(24,65),(25,36),(25,46),(26,50),(26,58),(27,31),(27,32),(28,48),(28,67),(29,57),(29,58),(29,78),(31,103),(32,103),(33,89),(34,89),(35,90),(36,88),(37,92),(38,102),(39,93),(40,101),(41,4),(41,100),(42,5),(42,97),(43,1),(44,53),(45,51),(45,95),(46,41),(46,88),(47,54),(48,71),(49,79),(50,73),(50,98),(51,68),(51,102),(52,31),(52,96),(53,30),(54,30),(55,33),(55,97),(56,35),(56,99),(57,70),(57,98),(58,74),(58,98),(59,81),(59,90),(60,37),(60,100),(61,34),(61,97),(62,36),(62,99),(63,21),(63,46),(63,99),(64,52),(64,91),(65,52),(65,92),(66,45),(66,91),(67,3),(67,84),(68,47),(69,32),(70,20),(70,49),(71,2),(71,57),(72,53),(72,54),(73,42),(73,55),(74,56),(74,62),(75,38),(75,51),(76,35),(76,59),(77,33),(77,34),(78,19),(78,74),(78,87),(79,80),(79,83),(79,90),(80,66),(80,82),(80,94),(81,64),(81,65),(81,94),(82,45),(82,75),(82,93),(83,39),(83,82),(84,50),(84,85),(85,18),(85,73),(85,86),(86,55),(86,61),(86,77),(87,56),(87,63),(87,76),(88,100),(89,43),(90,39),(90,94),(91,95),(91,96),(92,40),(92,96),(93,38),(93,40),(93,95),(94,91),(94,92),(94,93),(95,101),(95,102),(96,101),(96,103),(97,6),(97,89),(98,49),(99,60),(99,88),(100,69),(101,44),(101,72),(102,47),(102,72),(103,44)],104)
=> ? = 3 - 2
[[0,0,1,0,0],[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[0,0,1,0,0],[0,1,-1,1,0],[0,0,0,0,1],[0,0,1,0,0],[1,0,0,0,0]]
=> [[1,2,2,2,3],[2,3,4,4],[3,4,5],[4,5],[5]]
=> ?
=> ? = 3 - 2
[[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[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]]
=> [[1,2,2,2,3],[2,3,3,3],[3,4,5],[4,5],[5]]
=> ([(0,8),(0,9),(0,19),(1,42),(2,4),(2,20),(2,29),(3,50),(4,5),(4,52),(5,3),(5,53),(6,11),(6,35),(7,12),(7,31),(8,7),(8,34),(9,2),(9,33),(10,18),(10,46),(10,48),(11,27),(11,28),(12,25),(12,47),(13,15),(13,43),(13,44),(14,16),(14,26),(14,45),(15,17),(15,41),(15,51),(16,24),(16,40),(17,23),(17,39),(18,22),(18,32),(19,33),(19,34),(20,38),(20,47),(20,52),(21,55),(22,54),(23,57),(24,56),(25,62),(26,6),(26,63),(27,58),(28,58),(29,1),(29,38),(30,27),(30,60),(31,25),(32,26),(32,54),(33,29),(34,31),(35,28),(36,43),(37,21),(37,61),(38,42),(38,62),(39,30),(39,57),(40,30),(40,56),(41,23),(41,59),(42,13),(42,36),(43,41),(43,55),(44,51),(44,55),(45,24),(45,63),(46,22),(46,61),(47,49),(47,62),(48,14),(48,32),(48,61),(49,37),(49,46),(50,21),(50,44),(51,39),(51,40),(51,59),(52,10),(52,49),(52,53),(53,37),(53,48),(53,50),(54,63),(55,59),(56,60),(57,60),(59,56),(59,57),(60,58),(61,45),(61,54),(62,36),(63,35)],64)
=> ? = 3 - 2
[[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,0,0,1]]
=> [[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0]]
=> [[1,2,2,2,2],[2,3,4,5],[3,4,5],[4,5],[5]]
=> ([(0,5),(0,14),(0,15),(1,28),(2,9),(2,27),(3,13),(3,26),(4,21),(5,10),(5,11),(6,91),(7,81),(8,57),(9,6),(9,67),(10,18),(10,85),(11,19),(11,85),(12,20),(12,74),(13,17),(13,24),(13,93),(14,2),(14,37),(15,3),(15,37),(16,45),(16,70),(17,72),(17,88),(18,71),(18,89),(19,73),(19,90),(20,66),(20,68),(21,42),(21,65),(22,43),(22,44),(23,29),(23,87),(23,92),(24,72),(24,94),(25,31),(25,69),(25,86),(26,73),(26,93),(27,67),(27,71),(28,30),(28,91),(28,94),(29,48),(29,49),(30,50),(30,79),(31,56),(31,63),(31,64),(32,97),(33,99),(33,108),(34,98),(35,95),(35,97),(36,96),(37,1),(38,100),(39,100),(40,108),(41,109),(42,99),(43,7),(43,96),(44,8),(44,96),(45,101),(46,42),(47,65),(47,109),(48,53),(49,52),(50,66),(50,107),(51,74),(52,38),(53,39),(54,33),(54,106),(55,32),(55,107),(56,54),(56,95),(57,46),(58,62),(58,104),(59,61),(59,104),(60,58),(60,102),(61,39),(61,103),(62,38),(62,103),(63,47),(63,105),(64,70),(64,95),(64,105),(65,60),(65,99),(66,78),(67,12),(67,51),(68,46),(69,16),(69,64),(69,98),(70,23),(70,82),(70,101),(71,51),(72,80),(73,77),(74,57),(74,68),(75,53),(75,61),(76,52),(76,62),(77,34),(77,86),(78,33),(78,40),(79,35),(79,56),(79,107),(80,32),(80,35),(81,41),(81,47),(82,84),(82,87),(83,58),(83,76),(84,59),(84,75),(85,22),(85,89),(85,90),(86,63),(86,81),(86,98),(87,48),(87,75),(88,34),(88,69),(89,36),(89,44),(90,36),(90,43),(91,50),(91,55),(92,49),(92,76),(93,25),(93,77),(93,88),(94,55),(94,79),(94,80),(95,82),(95,106),(96,4),(97,40),(97,106),(98,41),(98,45),(98,105),(99,102),(101,83),(101,92),(102,104),(103,100),(104,103),(105,101),(105,109),(106,84),(106,108),(107,54),(107,78),(107,97),(108,59),(108,102),(109,60),(109,83)],110)
=> ? = 3 - 2
[[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0]]
=> [[1,1,3,3,4],[2,3,4,4],[3,4,5],[4,5],[5]]
=> ([(0,12),(0,13),(1,6),(2,61),(3,5),(3,94),(4,81),(5,80),(6,32),(7,4),(7,95),(8,22),(8,87),(9,20),(9,85),(10,19),(10,92),(11,17),(11,18),(11,98),(12,51),(13,11),(13,21),(13,51),(14,46),(14,47),(15,42),(15,89),(16,73),(16,75),(17,27),(17,48),(17,97),(18,48),(18,90),(19,43),(19,91),(20,44),(20,76),(21,25),(21,88),(21,98),(22,26),(22,74),(22,96),(23,24),(23,45),(23,77),(24,66),(24,67),(25,53),(25,93),(26,68),(26,84),(27,74),(27,78),(28,37),(28,60),(28,72),(29,124),(30,124),(31,101),(31,123),(33,117),(33,125),(34,118),(35,99),(36,116),(36,128),(37,101),(37,121),(38,109),(38,125),(39,100),(39,103),(40,109),(40,129),(41,126),(42,119),(43,107),(44,113),(45,9),(45,106),(46,108),(47,2),(47,108),(48,10),(48,122),(49,65),(49,128),(50,64),(51,8),(51,88),(52,39),(52,121),(52,123),(53,79),(54,45),(54,100),(55,32),(56,29),(56,127),(57,30),(57,127),(58,35),(58,120),(59,34),(59,99),(60,63),(60,101),(61,55),(62,56),(62,104),(63,83),(63,111),(64,57),(64,104),(65,44),(65,114),(66,58),(66,110),(67,65),(67,110),(68,43),(68,115),(69,34),(69,105),(70,36),(70,112),(71,33),(71,103),(71,119),(72,23),(72,54),(72,121),(73,50),(74,68),(74,102),(75,3),(75,82),(76,64),(76,113),(77,66),(77,106),(78,39),(78,54),(78,102),(79,42),(79,71),(80,29),(80,30),(81,35),(81,59),(82,62),(82,94),(83,49),(83,67),(83,112),(84,33),(84,38),(84,115),(85,50),(85,76),(86,59),(86,69),(86,120),(87,15),(87,79),(87,96),(88,53),(88,87),(89,38),(89,40),(89,119),(90,31),(90,60),(90,122),(91,36),(91,49),(91,107),(92,70),(92,83),(92,91),(93,31),(93,37),(93,52),(94,56),(94,57),(94,80),(95,58),(95,81),(95,86),(96,71),(96,84),(96,89),(96,102),(97,52),(97,72),(97,78),(97,122),(98,28),(98,90),(98,93),(98,97),(99,46),(99,118),(100,106),(101,7),(101,111),(102,40),(102,103),(102,115),(103,117),(103,129),(104,127),(105,47),(105,118),(106,85),(107,41),(107,116),(108,1),(109,41),(109,130),(110,114),(110,120),(111,95),(111,112),(112,86),(112,110),(112,128),(113,104),(114,105),(114,113),(115,107),(115,109),(115,117),(116,126),(117,116),(117,130),(118,108),(119,16),(119,125),(119,129),(120,14),(120,99),(120,105),(121,77),(121,100),(122,63),(122,92),(122,123),(123,70),(123,111),(124,55),(125,75),(125,130),(126,62),(127,61),(127,124),(128,69),(128,114),(129,73),(129,130),(130,82),(130,126)],131)
=> ? = 3 - 2
[[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0]]
=> [[1,1,3,3,3],[2,3,4,5],[3,4,5],[4,5],[5]]
=> ([(0,2),(0,3),(1,8),(1,16),(2,1),(3,7),(3,10),(4,13),(4,59),(5,12),(5,81),(6,20),(6,82),(7,11),(7,83),(8,60),(9,23),(9,58),(10,19),(10,22),(10,83),(11,78),(12,64),(13,24),(14,52),(14,72),(15,17),(15,73),(15,80),(16,60),(16,74),(17,39),(17,70),(18,36),(18,44),(19,51),(19,69),(20,33),(20,75),(21,28),(21,38),(21,50),(22,51),(22,71),(22,74),(23,49),(23,56),(23,57),(25,101),(26,113),(27,107),(27,110),(28,107),(28,111),(29,110),(29,111),(30,112),(31,90),(31,98),(32,94),(32,114),(33,96),(34,91),(34,92),(35,89),(35,92),(36,108),(37,94),(37,103),(38,9),(38,111),(39,45),(40,31),(40,102),(40,114),(41,31),(41,104),(42,37),(42,99),(43,34),(43,85),(43,87),(44,30),(44,108),(45,55),(46,33),(46,86),(47,48),(47,86),(48,25),(48,100),(49,42),(49,85),(50,14),(50,76),(50,107),(51,15),(51,61),(51,84),(52,46),(52,109),(53,32),(53,93),(54,26),(54,95),(55,25),(55,91),(56,35),(56,85),(56,97),(57,53),(57,97),(58,57),(59,24),(60,5),(60,63),(61,67),(61,73),(62,34),(62,35),(62,113),(63,67),(63,81),(64,46),(64,47),(65,45),(65,68),(66,47),(66,68),(66,109),(67,65),(67,66),(68,48),(68,55),(68,87),(69,29),(69,38),(69,84),(70,26),(70,62),(71,27),(71,50),(71,84),(72,43),(72,49),(72,109),(73,39),(73,65),(74,61),(74,63),(75,32),(75,40),(75,96),(76,72),(76,79),(76,88),(77,40),(77,41),(77,93),(78,27),(78,28),(78,29),(79,43),(79,56),(79,62),(79,95),(80,54),(80,70),(80,79),(81,52),(81,64),(81,66),(82,53),(82,75),(82,77),(83,21),(83,69),(83,71),(83,78),(84,76),(84,80),(84,110),(85,92),(85,99),(86,37),(86,96),(86,100),(87,91),(87,99),(87,100),(88,82),(88,95),(89,104),(90,108),(90,116),(91,101),(91,106),(92,106),(93,18),(93,104),(93,114),(94,115),(95,77),(95,97),(95,113),(96,94),(96,102),(97,89),(97,93),(98,116),(99,103),(99,106),(100,101),(100,102),(100,103),(101,98),(101,105),(102,98),(102,115),(103,105),(103,115),(104,36),(104,90),(105,116),(106,105),(107,6),(107,88),(108,4),(108,112),(109,42),(109,86),(109,87),(110,54),(110,88),(111,58),(112,59),(113,41),(113,89),(114,44),(114,90),(114,115),(115,30),(115,116),(116,112)],117)
=> ? = 3 - 2
[[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[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]]
=> [[1,1,2,3,4],[2,3,3,4],[3,4,4],[4,5],[5]]
=> ?
=> ? = 3 - 2
[[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[0,0,1,0,0],[0,0,0,0,1],[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0]]
=> [[1,1,2,3,3],[2,3,3,5],[3,4,5],[4,5],[5]]
=> ?
=> ? = 3 - 2
[[0,0,1,0,0],[0,1,-1,1,0],[1,0,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[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]]
=> [[1,1,2,3,3],[2,3,3,4],[3,4,5],[4,5],[5]]
=> ?
=> ? = 3 - 2
[[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [[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]]
=> [[1,1,2,2,4],[2,3,4,4],[3,4,5],[4,5],[5]]
=> ?
=> ? = 3 - 2
[[0,0,1,0,0],[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [[0,0,1,0,0],[0,1,-1,1,0],[0,0,0,0,1],[1,-1,1,0,0],[0,1,0,0,0]]
=> [[1,1,2,2,3],[2,3,4,4],[3,4,5],[4,5],[5]]
=> ?
=> ? = 3 - 2
[[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [[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]]
=> [[1,1,2,2,4],[2,3,3,4],[3,4,4],[4,5],[5]]
=> ?
=> ? = 3 - 2
[[0,0,0,1,0],[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[1,-1,0,0,1],[0,1,0,0,0]]
=> [[1,1,2,2,3],[2,3,3,4],[3,4,4],[4,5],[5]]
=> ?
=> ? = 3 - 2
[[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[1,0,0,-1,1],[0,0,0,1,0]]
=> [[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[1,-1,1,0,0],[0,1,0,0,0]]
=> [[1,1,2,2,2],[2,3,4,5],[3,4,5],[4,5],[5]]
=> ?
=> ? = 3 - 2
[[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[1,0,0,-1,1],[0,0,0,1,0]]
=> [[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[1,-1,0,1,0],[0,1,0,0,0]]
=> [[1,1,2,2,2],[2,3,3,5],[3,4,5],[4,5],[5]]
=> ?
=> ? = 3 - 2
[[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[1,0,0,-1,1],[0,0,0,1,0]]
=> [[0,1,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[1,-1,0,1,0],[0,1,0,0,0]]
=> [[1,1,2,2,2],[2,3,3,4],[3,4,5],[4,5],[5]]
=> ?
=> ? = 3 - 2
[[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [[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]]
=> [[1,1,2,4,4],[2,2,4,5],[3,4,5],[4,5],[5]]
=> ([(0,2),(0,3),(1,8),(1,16),(2,1),(3,7),(3,10),(4,13),(4,59),(5,12),(5,81),(6,20),(6,82),(7,11),(7,83),(8,60),(9,23),(9,58),(10,19),(10,22),(10,83),(11,78),(12,64),(13,24),(14,52),(14,72),(15,17),(15,73),(15,80),(16,60),(16,74),(17,39),(17,70),(18,36),(18,44),(19,51),(19,69),(20,33),(20,75),(21,28),(21,38),(21,50),(22,51),(22,71),(22,74),(23,49),(23,56),(23,57),(25,101),(26,113),(27,107),(27,110),(28,107),(28,111),(29,110),(29,111),(30,112),(31,90),(31,98),(32,94),(32,114),(33,96),(34,91),(34,92),(35,89),(35,92),(36,108),(37,94),(37,103),(38,9),(38,111),(39,45),(40,31),(40,102),(40,114),(41,31),(41,104),(42,37),(42,99),(43,34),(43,85),(43,87),(44,30),(44,108),(45,55),(46,33),(46,86),(47,48),(47,86),(48,25),(48,100),(49,42),(49,85),(50,14),(50,76),(50,107),(51,15),(51,61),(51,84),(52,46),(52,109),(53,32),(53,93),(54,26),(54,95),(55,25),(55,91),(56,35),(56,85),(56,97),(57,53),(57,97),(58,57),(59,24),(60,5),(60,63),(61,67),(61,73),(62,34),(62,35),(62,113),(63,67),(63,81),(64,46),(64,47),(65,45),(65,68),(66,47),(66,68),(66,109),(67,65),(67,66),(68,48),(68,55),(68,87),(69,29),(69,38),(69,84),(70,26),(70,62),(71,27),(71,50),(71,84),(72,43),(72,49),(72,109),(73,39),(73,65),(74,61),(74,63),(75,32),(75,40),(75,96),(76,72),(76,79),(76,88),(77,40),(77,41),(77,93),(78,27),(78,28),(78,29),(79,43),(79,56),(79,62),(79,95),(80,54),(80,70),(80,79),(81,52),(81,64),(81,66),(82,53),(82,75),(82,77),(83,21),(83,69),(83,71),(83,78),(84,76),(84,80),(84,110),(85,92),(85,99),(86,37),(86,96),(86,100),(87,91),(87,99),(87,100),(88,82),(88,95),(89,104),(90,108),(90,116),(91,101),(91,106),(92,106),(93,18),(93,104),(93,114),(94,115),(95,77),(95,97),(95,113),(96,94),(96,102),(97,89),(97,93),(98,116),(99,103),(99,106),(100,101),(100,102),(100,103),(101,98),(101,105),(102,98),(102,115),(103,105),(103,115),(104,36),(104,90),(105,116),(106,105),(107,6),(107,88),(108,4),(108,112),(109,42),(109,86),(109,87),(110,54),(110,88),(111,58),(112,59),(113,41),(113,89),(114,44),(114,90),(114,115),(115,30),(115,116),(116,112)],117)
=> ? = 3 - 2
[[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [[0,0,0,1,0],[0,0,1,0,0],[0,1,-1,0,1],[1,0,0,0,0],[0,0,1,0,0]]
=> [[1,1,2,3,4],[2,2,4,4],[3,4,5],[4,5],[5]]
=> ?
=> ? = 3 - 2
[[0,1,0,0,0],[0,0,1,0,0],[1,0,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [[0,0,1,0,0],[0,0,0,0,1],[0,1,-1,1,0],[1,0,0,0,0],[0,0,1,0,0]]
=> [[1,1,2,3,3],[2,2,4,5],[3,4,5],[4,5],[5]]
=> ?
=> ? = 3 - 2
[[0,0,1,0,0],[0,1,0,0,0],[1,0,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [[0,0,1,0,0],[0,0,0,1,0],[0,1,-1,0,1],[1,0,0,0,0],[0,0,1,0,0]]
=> [[1,1,2,3,3],[2,2,4,4],[3,4,5],[4,5],[5]]
=> ?
=> ? = 3 - 2
[[0,0,0,1,0],[1,0,0,0,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [[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]]
=> [[1,1,2,3,4],[2,2,3,4],[3,4,4],[4,5],[5]]
=> ?
=> ? = 3 - 2
[[0,1,0,0,0],[0,0,0,1,0],[1,0,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [[0,0,1,0,0],[0,0,0,0,1],[0,1,0,0,0],[1,0,-1,1,0],[0,0,1,0,0]]
=> [[1,1,2,3,3],[2,2,3,5],[3,4,5],[4,5],[5]]
=> ?
=> ? = 3 - 2
[[0,0,1,0,0],[0,1,-1,1,0],[1,0,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [[0,0,1,0,0],[0,0,0,1,0],[0,1,0,-1,1],[1,0,-1,1,0],[0,0,1,0,0]]
=> [[1,1,2,3,3],[2,2,3,4],[3,4,5],[4,5],[5]]
=> ?
=> ? = 3 - 2
[[0,0,0,1,0],[0,1,0,0,0],[1,0,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[1,0,-1,0,1],[0,0,1,0,0]]
=> [[1,1,2,3,3],[2,2,3,4],[3,4,4],[4,5],[5]]
=> ?
=> ? = 3 - 2
[[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [[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]]
=> [[1,1,2,3,5],[2,2,3,5],[3,3,5],[4,5],[5]]
=> ([(0,5),(0,14),(0,15),(1,28),(2,9),(2,27),(3,13),(3,26),(4,21),(5,10),(5,11),(6,91),(7,81),(8,57),(9,6),(9,67),(10,18),(10,85),(11,19),(11,85),(12,20),(12,74),(13,17),(13,24),(13,93),(14,2),(14,37),(15,3),(15,37),(16,45),(16,70),(17,72),(17,88),(18,71),(18,89),(19,73),(19,90),(20,66),(20,68),(21,42),(21,65),(22,43),(22,44),(23,29),(23,87),(23,92),(24,72),(24,94),(25,31),(25,69),(25,86),(26,73),(26,93),(27,67),(27,71),(28,30),(28,91),(28,94),(29,48),(29,49),(30,50),(30,79),(31,56),(31,63),(31,64),(32,97),(33,99),(33,108),(34,98),(35,95),(35,97),(36,96),(37,1),(38,100),(39,100),(40,108),(41,109),(42,99),(43,7),(43,96),(44,8),(44,96),(45,101),(46,42),(47,65),(47,109),(48,53),(49,52),(50,66),(50,107),(51,74),(52,38),(53,39),(54,33),(54,106),(55,32),(55,107),(56,54),(56,95),(57,46),(58,62),(58,104),(59,61),(59,104),(60,58),(60,102),(61,39),(61,103),(62,38),(62,103),(63,47),(63,105),(64,70),(64,95),(64,105),(65,60),(65,99),(66,78),(67,12),(67,51),(68,46),(69,16),(69,64),(69,98),(70,23),(70,82),(70,101),(71,51),(72,80),(73,77),(74,57),(74,68),(75,53),(75,61),(76,52),(76,62),(77,34),(77,86),(78,33),(78,40),(79,35),(79,56),(79,107),(80,32),(80,35),(81,41),(81,47),(82,84),(82,87),(83,58),(83,76),(84,59),(84,75),(85,22),(85,89),(85,90),(86,63),(86,81),(86,98),(87,48),(87,75),(88,34),(88,69),(89,36),(89,44),(90,36),(90,43),(91,50),(91,55),(92,49),(92,76),(93,25),(93,77),(93,88),(94,55),(94,79),(94,80),(95,82),(95,106),(96,4),(97,40),(97,106),(98,41),(98,45),(98,105),(99,102),(101,83),(101,92),(102,104),(103,100),(104,103),(105,101),(105,109),(106,84),(106,108),(107,54),(107,78),(107,97),(108,59),(108,102),(109,60),(109,83)],110)
=> ? = 3 - 2
[[0,1,0,0,0],[1,-1,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [[0,0,0,1,0],[0,0,1,-1,1],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0]]
=> [[1,1,2,3,4],[2,2,3,5],[3,3,5],[4,5],[5]]
=> ?
=> ? = 3 - 2
[[0,0,1,0,0],[1,0,-1,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [[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]]
=> [[1,1,2,3,4],[2,2,3,4],[3,3,5],[4,5],[5]]
=> ?
=> ? = 3 - 2
[[0,0,0,1,0],[1,0,0,-1,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,-1,1],[0,0,0,1,0]]
=> [[1,1,2,3,4],[2,2,3,4],[3,3,4],[4,5],[5]]
=> ([(0,16),(0,17),(0,28),(1,22),(2,19),(3,11),(3,21),(4,54),(5,55),(6,26),(6,57),(7,25),(7,67),(8,23),(8,64),(9,24),(9,65),(10,14),(10,56),(11,15),(11,58),(12,3),(12,37),(13,27),(13,66),(14,59),(15,60),(16,13),(16,53),(17,6),(17,61),(18,51),(18,52),(19,33),(19,34),(20,35),(20,36),(21,50),(21,58),(22,18),(22,59),(22,60),(23,41),(23,43),(24,42),(24,44),(25,50),(25,63),(26,29),(26,46),(27,29),(27,45),(28,12),(28,53),(28,61),(29,70),(30,71),(31,69),(32,68),(33,68),(34,68),(35,4),(35,69),(36,5),(36,69),(37,1),(38,33),(39,34),(40,64),(41,48),(42,49),(43,38),(44,39),(45,62),(45,70),(46,56),(46,70),(47,65),(48,32),(49,32),(50,40),(51,42),(51,71),(52,41),(52,71),(53,7),(53,66),(54,38),(55,39),(56,9),(56,47),(57,10),(57,46),(58,8),(58,40),(59,30),(59,51),(60,30),(60,52),(61,37),(61,57),(62,31),(62,36),(63,31),(63,35),(64,43),(64,54),(65,44),(65,55),(66,45),(66,67),(67,20),(67,62),(67,63),(69,2),(70,47),(71,48),(71,49)],72)
=> ? = 3 - 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]]
=> [[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1]]
=> [[1,1,2,3,4],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ([(0,5),(0,16),(0,17),(1,23),(2,19),(3,11),(3,21),(4,10),(4,20),(5,12),(5,13),(6,50),(7,51),(8,24),(8,58),(9,25),(9,59),(10,14),(10,52),(11,15),(11,53),(12,26),(12,60),(13,27),(13,60),(14,54),(15,55),(16,4),(16,29),(17,3),(17,29),(18,48),(18,49),(19,32),(19,33),(20,46),(20,52),(21,47),(21,53),(22,34),(22,35),(23,18),(23,54),(23,55),(24,40),(24,42),(25,41),(25,43),(26,46),(26,56),(27,47),(27,57),(28,63),(29,1),(30,62),(31,61),(32,61),(33,61),(34,6),(34,62),(35,7),(35,62),(36,58),(37,59),(38,32),(39,33),(40,44),(41,45),(42,38),(43,39),(44,31),(45,31),(46,36),(47,37),(48,40),(48,63),(49,41),(49,63),(50,38),(51,39),(52,8),(52,36),(53,9),(53,37),(54,28),(54,48),(55,28),(55,49),(56,30),(56,34),(57,30),(57,35),(58,42),(58,50),(59,43),(59,51),(60,22),(60,56),(60,57),(62,2),(63,44),(63,45)],64)
=> ? = 4 - 2
[[0,1,0,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [[0,0,1,0,0],[0,0,0,0,1],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0]]
=> [[1,1,2,3,3],[2,2,3,5],[3,3,5],[4,5],[5]]
=> ([(0,2),(0,3),(1,7),(1,14),(2,1),(3,6),(3,8),(4,10),(4,62),(5,16),(5,59),(6,9),(6,63),(7,44),(8,17),(8,18),(8,63),(9,60),(10,46),(11,35),(11,36),(12,15),(12,53),(12,61),(13,24),(13,37),(14,44),(14,54),(15,27),(15,51),(16,26),(16,39),(17,33),(17,34),(18,34),(18,52),(18,54),(19,75),(20,74),(21,78),(22,77),(23,76),(23,77),(24,76),(25,68),(26,67),(27,29),(28,25),(28,78),(29,42),(30,20),(30,73),(31,20),(32,19),(32,72),(33,22),(33,64),(34,12),(34,45),(34,64),(35,38),(35,66),(36,40),(36,66),(37,11),(37,57),(37,76),(38,25),(38,71),(39,30),(39,67),(40,26),(40,70),(41,32),(41,70),(42,19),(42,68),(43,21),(43,69),(44,4),(44,47),(45,50),(45,53),(46,40),(46,41),(47,50),(47,62),(48,29),(48,55),(49,41),(49,55),(49,66),(50,48),(50,49),(51,21),(51,28),(52,23),(52,37),(52,64),(53,27),(53,48),(54,45),(54,47),(55,32),(55,42),(55,71),(56,28),(56,38),(56,69),(57,35),(57,56),(57,65),(58,30),(58,31),(59,39),(59,58),(60,22),(60,23),(60,24),(61,43),(61,51),(61,56),(62,36),(62,46),(62,49),(63,13),(63,33),(63,52),(63,60),(64,57),(64,61),(64,77),(65,59),(65,69),(66,70),(66,71),(67,73),(68,75),(69,58),(69,78),(70,67),(70,72),(71,68),(71,72),(72,73),(72,75),(73,74),(75,74),(76,5),(76,65),(77,43),(77,65),(78,31)],79)
=> ? = 4 - 2
[[0,0,1,0,0],[0,1,-1,0,1],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [[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]]
=> [[1,1,2,3,3],[2,2,3,4],[3,3,5],[4,5],[5]]
=> ?
=> ? = 4 - 2
[[0,0,0,1,0],[0,1,0,-1,1],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[1,0,0,-1,1],[0,0,0,1,0]]
=> [[1,1,2,3,3],[2,2,3,4],[3,3,4],[4,5],[5]]
=> ?
=> ? = 4 - 2
[[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [[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]]
=> [[1,1,2,2,5],[2,2,4,5],[3,4,5],[4,5],[5]]
=> ([(0,14),(0,15),(1,10),(1,11),(2,87),(3,13),(3,113),(4,89),(5,31),(5,107),(6,25),(6,108),(7,27),(7,29),(7,114),(8,24),(8,88),(9,12),(9,115),(10,90),(11,26),(11,28),(11,90),(12,112),(13,93),(14,1),(14,86),(15,7),(15,20),(15,86),(16,70),(16,71),(17,83),(17,84),(18,69),(18,99),(19,22),(19,82),(19,104),(20,23),(20,98),(20,114),(21,42),(21,100),(22,74),(22,75),(23,62),(23,95),(24,55),(24,58),(25,54),(25,56),(26,57),(26,85),(27,47),(27,66),(28,85),(28,109),(29,18),(29,47),(29,92),(30,81),(30,97),(30,103),(31,30),(31,99),(31,109),(31,110),(32,138),(33,147),(34,146),(35,117),(36,117),(36,147),(37,140),(37,148),(38,139),(39,137),(40,136),(41,150),(42,142),(43,125),(44,129),(45,135),(46,127),(46,151),(47,6),(47,116),(48,71),(48,148),(48,151),(49,64),(50,77),(50,152),(51,73),(51,150),(52,43),(53,65),(54,96),(54,153),(55,76),(55,118),(56,43),(56,153),(57,88),(58,89),(58,118),(59,67),(59,149),(60,68),(60,118),(60,149),(61,59),(61,144),(62,94),(63,44),(63,146),(64,32),(64,133),(65,38),(65,132),(66,34),(66,116),(67,35),(67,145),(68,36),(68,143),(68,145),(69,44),(69,119),(70,33),(70,143),(71,17),(71,91),(71,143),(72,78),(72,130),(73,45),(73,123),(74,32),(74,121),(75,40),(75,121),(76,41),(76,126),(77,39),(77,123),(78,40),(78,124),(79,38),(79,124),(80,37),(80,129),(80,142),(81,54),(81,122),(81,144),(82,75),(82,131),(83,72),(83,128),(84,39),(84,128),(85,9),(85,111),(86,5),(86,98),(87,19),(87,101),(88,4),(88,58),(89,3),(89,106),(90,8),(90,57),(91,77),(91,84),(91,120),(92,63),(92,69),(92,116),(93,78),(93,79),(94,42),(94,80),(95,34),(95,63),(96,41),(96,51),(97,55),(97,60),(97,144),(98,62),(98,107),(99,61),(99,81),(99,119),(100,16),(100,46),(100,48),(100,142),(101,49),(101,104),(102,65),(102,79),(102,130),(103,37),(103,48),(103,122),(104,64),(104,74),(104,131),(105,53),(105,102),(106,105),(106,113),(107,21),(107,94),(107,110),(108,52),(108,56),(109,61),(109,97),(109,111),(110,80),(110,100),(110,103),(110,119),(111,59),(111,60),(111,115),(112,33),(112,35),(112,36),(113,72),(113,93),(113,102),(114,66),(114,92),(114,95),(115,67),(115,68),(115,70),(115,112),(116,108),(116,146),(117,45),(117,134),(118,106),(118,126),(119,46),(119,122),(119,129),(120,123),(120,128),(120,134),(121,136),(121,138),(122,140),(122,151),(122,153),(123,135),(123,137),(124,136),(124,139),(125,152),(126,105),(126,150),(127,154),(128,130),(128,137),(129,127),(129,140),(130,124),(130,132),(131,121),(131,133),(132,139),(133,138),(134,131),(134,135),(135,133),(136,141),(137,132),(138,141),(139,141),(140,125),(140,154),(142,2),(142,127),(142,148),(143,83),(143,120),(143,147),(144,76),(144,96),(144,149),(145,73),(145,117),(145,120),(146,52),(147,82),(147,134),(148,87),(148,154),(149,51),(149,126),(149,145),(150,53),(151,50),(151,91),(151,154),(152,49),(153,50),(153,125),(154,101),(154,152)],155)
=> ? = 3 - 2
[[0,1,0,0,0],[1,-1,1,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [[0,0,0,1,0],[0,1,0,-1,1],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0]]
=> [[1,1,2,2,4],[2,2,4,5],[3,4,5],[4,5],[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]]
=> [[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0]]
=> [[1,1,2,2,4],[2,2,4,4],[3,4,5],[4,5],[5]]
=> ([(0,7),(0,8),(1,2),(1,45),(2,35),(3,37),(4,14),(4,49),(5,9),(5,48),(6,10),(6,12),(6,54),(7,31),(8,6),(8,13),(8,31),(9,27),(9,42),(10,28),(10,41),(11,26),(11,52),(12,17),(12,28),(12,51),(13,15),(13,50),(13,54),(14,16),(14,40),(14,53),(15,33),(15,44),(16,38),(16,47),(17,30),(17,40),(19,64),(19,66),(20,62),(21,65),(22,58),(23,60),(23,66),(24,60),(24,63),(25,67),(26,55),(27,59),(28,5),(28,56),(29,34),(30,22),(30,57),(31,4),(31,50),(32,22),(32,65),(33,43),(34,18),(35,18),(36,20),(37,1),(37,46),(38,27),(38,61),(39,19),(39,55),(39,58),(40,38),(40,57),(41,21),(41,56),(42,20),(42,59),(43,26),(43,39),(44,21),(44,32),(45,34),(45,35),(46,29),(46,45),(47,19),(47,23),(47,61),(48,36),(48,42),(49,11),(49,43),(49,53),(50,33),(50,49),(51,30),(51,32),(51,56),(52,23),(52,24),(52,55),(53,39),(53,47),(53,52),(53,57),(54,41),(54,44),(54,51),(55,3),(55,63),(55,66),(56,48),(56,65),(57,24),(57,58),(57,61),(58,63),(58,64),(59,25),(59,62),(60,25),(60,68),(61,59),(61,60),(61,64),(62,67),(63,68),(64,62),(64,68),(65,36),(66,37),(66,68),(67,29),(68,46),(68,67)],69)
=> ? = 4 - 2
[[0,1,0,0,0],[0,0,1,0,0],[1,-1,0,0,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [[0,0,1,0,0],[0,1,-1,0,1],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0]]
=> [[1,1,2,2,3],[2,2,4,5],[3,4,5],[4,5],[5]]
=> ?
=> ? = 3 - 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]]
=> [[0,0,1,0,0],[0,1,-1,1,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0]]
=> [[1,1,2,2,3],[2,2,4,4],[3,4,5],[4,5],[5]]
=> ?
=> ? = 4 - 2
[[1,0,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [[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]]
=> [[1,1,2,2,5],[2,2,3,5],[3,4,5],[4,5],[5]]
=> ?
=> ? = 3 - 2
[[0,1,0,0,0],[1,-1,0,1,0],[0,0,1,-1,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [[0,0,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[1,0,-1,1,0],[0,0,1,0,0]]
=> [[1,1,2,2,4],[2,2,3,5],[3,4,5],[4,5],[5]]
=> ?
=> ? = 3 - 2
[[0,0,1,0,0],[1,0,-1,1,0],[0,0,1,-1,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [[0,0,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[1,0,-1,1,0],[0,0,1,0,0]]
=> [[1,1,2,2,4],[2,2,3,4],[3,4,5],[4,5],[5]]
=> ?
=> ? = 3 - 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]]
=> [[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]]
=> [[1,1,2,2,4],[2,2,3,4],[3,4,4],[4,5],[5]]
=> ?
=> ? = 4 - 2
[[0,1,0,0,0],[0,0,0,1,0],[1,-1,1,-1,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [[0,0,1,0,0],[0,1,-1,0,1],[0,0,1,0,0],[1,0,-1,1,0],[0,0,1,0,0]]
=> [[1,1,2,2,3],[2,2,3,5],[3,4,5],[4,5],[5]]
=> ?
=> ? = 3 - 2
[[0,0,1,0,0],[0,1,-1,1,0],[1,-1,1,-1,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[1,0,-1,1,0],[0,0,1,0,0]]
=> [[1,1,2,2,3],[2,2,3,4],[3,4,5],[4,5],[5]]
=> ?
=> ? = 3 - 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]]
=> [[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[1,0,-1,0,1],[0,0,1,0,0]]
=> [[1,1,2,2,3],[2,2,3,4],[3,4,4],[4,5],[5]]
=> ?
=> ? = 4 - 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]]
=> [[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[1,0,-1,1,0],[0,0,1,0,0]]
=> [[1,1,2,2,3],[2,2,3,3],[3,4,5],[4,5],[5]]
=> ?
=> ? = 4 - 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]]
=> [[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,2,3],[2,2,2,3],[3,3,3],[4,4],[5]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> 1 = 3 - 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]]
=> [[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,2,2],[2,2,2,4],[3,3,4],[4,4],[5]]
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> 1 = 3 - 2
[[0,0,0,0,1],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0]]
=> [[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,3],[2,2,3,3],[3,3,4],[4,4],[5]]
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> 1 = 3 - 2
[[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0]]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,2],[2,2,2,2],[3,3,4],[4,4],[5]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 3 - 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]]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,1,2],[2,2,2,2],[3,3,3],[4,5],[5]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 3 - 2
[[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0]]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> 1 = 3 - 2
[[0,0,1,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0]]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0]]
=> [[1,1,1,1,1],[2,2,3,3],[3,3,5],[4,5],[5]]
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> 1 = 3 - 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]]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0]]
=> [[1,1,1,1,1],[2,2,2,4],[3,4,4],[4,5],[5]]
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> 1 = 3 - 2
[[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0]]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,1,1],[2,2,2,3],[3,3,3],[4,5],[5]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 3 - 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]]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [[1,1,1,1,1],[2,2,2,2],[3,4,5],[4,5],[5]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> 1 = 3 - 2
[[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0]]
=> [[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,2,3],[2,2,2,2,3],[3,3,3,3],[4,4,4],[5,5],[6]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> 1 = 3 - 2
[[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0]]
=> [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,4],[4,4,4],[5,5],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 3 - 2
[[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0]]
=> [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,3],[4,4,5],[5,5],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 3 - 2
[[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0]]
=> [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,3],[4,4,4],[5,6],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 3 - 2
[[0,0,0,0,0,1],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[1,0,0,0,0,0]]
=> [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,3,4],[3,3,3,4],[4,4,4],[5,5],[6]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> 1 = 3 - 2
[[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[1,0,0,0,0,0]]
=> [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,3],[3,3,3,3],[4,4,5],[5,5],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 3 - 2
[[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[1,0,0,0,0,0]]
=> [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,3],[3,3,3,3],[4,4,4],[5,6],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 3 - 2
[[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[1,0,0,0,0,0]]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,4,5],[4,4,5],[5,5],[6]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> 1 = 3 - 2
[[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0]]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,4],[4,4,4],[5,6],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 3 - 2
[[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0]]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,3],[4,5,6],[5,6],[6]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> 1 = 3 - 2
Description
The number of maximal elements of a poset.
The following 1 statistic also match your data. Click on any of them to see the details.
St001568The smallest positive integer that does not appear twice in the partition.
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