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Your data matches 92 different statistics following compositions of up to 3 maps.
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Matching statistic: St001880
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00002: Alternating sign matrices —to left key permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001880: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00252: Permutations —restriction⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001880: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [1,2,3,4] => [1,2,3] => ([(0,2),(2,1)],3)
=> 3
[[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [1,2,4,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 3
[[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,2,4,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 3
[[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,2,4,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 3
[[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> [1,4,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 3
[[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
=> [1,4,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 3
[[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
=> [1,4,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 3
[[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [4,1,2,3] => [1,2,3] => ([(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] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 4
[[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,3,2,4,5] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,3,2,4,5] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,2,3,5,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 4
[[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,2,3,5,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 4
[[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,2,3,5,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 4
[[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,2,3,5,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 4
[[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[[0,0,1,0,0],[1,0,-1,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[[0,1,0,0,0],[1,-1,1,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[[0,1,0,0,0],[0,0,1,0,0],[1,-1,0,1,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[[0,1,0,0,0],[1,-1,0,1,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[[0,0,1,0,0],[1,0,-1,1,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[[0,1,0,0,0],[0,0,0,1,0],[1,-1,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[[0,0,1,0,0],[0,1,-1,1,0],[1,-1,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 4
[[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 4
[[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 4
[[1,0,0,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 4
[[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 4
[[0,0,1,0,0],[1,0,-1,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 4
[[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] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 4
[[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] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 4
[[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] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 4
[[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] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 4
[[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] => [1,2,3,4] => ([(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] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[[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] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],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] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[[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]]
=> [1,3,5,2,4] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[[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]]
=> [1,3,5,2,4] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[[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]]
=> [1,3,5,2,4] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[[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]]
=> [1,3,5,2,4] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[[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]]
=> [1,3,5,2,4] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,5,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[[0,1,0,0,0],[1,-1,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,5,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[[0,0,1,0,0],[1,0,-1,0,1],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,5,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
Description
The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice.
Matching statistic: St000924
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00002: Alternating sign matrices —to left key permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St000924: Perfect matchings ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 43%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St000924: Perfect matchings ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 43%
Values
[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> 4 = 3 + 1
[[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> 4 = 3 + 1
[[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> 4 = 3 + 1
[[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> 4 = 3 + 1
[[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> 4 = 3 + 1
[[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
=> [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> 4 = 3 + 1
[[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
=> [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> 4 = 3 + 1
[[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5)]
=> 4 = 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] => [1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> 5 = 4 + 1
[[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,10)]
=> 5 = 4 + 1
[[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,10)]
=> 5 = 4 + 1
[[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,10),(8,9)]
=> 5 = 4 + 1
[[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9)]
=> 5 = 4 + 1
[[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9)]
=> 5 = 4 + 1
[[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,10),(8,9)]
=> 5 = 4 + 1
[[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,10),(8,9)]
=> 5 = 4 + 1
[[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,10),(8,9)]
=> 5 = 4 + 1
[[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9)]
=> 5 = 4 + 1
[[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9)]
=> 5 = 4 + 1
[[0,0,1,0,0],[1,0,-1,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9)]
=> 5 = 4 + 1
[[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9)]
=> 5 = 4 + 1
[[0,1,0,0,0],[1,-1,1,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9)]
=> 5 = 4 + 1
[[0,1,0,0,0],[0,0,1,0,0],[1,-1,0,1,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9)]
=> 5 = 4 + 1
[[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9)]
=> 5 = 4 + 1
[[0,1,0,0,0],[1,-1,0,1,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9)]
=> 5 = 4 + 1
[[0,0,1,0,0],[1,0,-1,1,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9)]
=> 5 = 4 + 1
[[0,1,0,0,0],[0,0,0,1,0],[1,-1,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9)]
=> 5 = 4 + 1
[[0,0,1,0,0],[0,1,-1,1,0],[1,-1,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9)]
=> 5 = 4 + 1
[[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,10),(6,9),(7,8)]
=> 5 = 4 + 1
[[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,10),(6,9),(7,8)]
=> 5 = 4 + 1
[[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,10),(6,9),(7,8)]
=> 5 = 4 + 1
[[1,0,0,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,10),(6,9),(7,8)]
=> 5 = 4 + 1
[[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,10),(6,9),(7,8)]
=> 5 = 4 + 1
[[0,0,1,0,0],[1,0,-1,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,10),(6,9),(7,8)]
=> 5 = 4 + 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] => [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> 5 = 4 + 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] => [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> 5 = 4 + 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] => [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> 5 = 4 + 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] => [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> 5 = 4 + 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] => [1,1,1,1,1,0,0,0,0,0]
=> [(1,10),(2,9),(3,8),(4,7),(5,6)]
=> 5 = 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] => [1,0,1,1,0,1,1,0,0,0]
=> [(1,2),(3,10),(4,5),(6,9),(7,8)]
=> 5 = 4 + 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] => [1,0,1,1,0,1,1,0,0,0]
=> [(1,2),(3,10),(4,5),(6,9),(7,8)]
=> 5 = 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] => [1,0,1,1,0,1,1,0,0,0]
=> [(1,2),(3,10),(4,5),(6,9),(7,8)]
=> 5 = 4 + 1
[[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]]
=> [1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [(1,2),(3,10),(4,5),(6,9),(7,8)]
=> 5 = 4 + 1
[[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]]
=> [1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [(1,2),(3,10),(4,5),(6,9),(7,8)]
=> 5 = 4 + 1
[[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]]
=> [1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [(1,2),(3,10),(4,5),(6,9),(7,8)]
=> 5 = 4 + 1
[[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]]
=> [1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [(1,2),(3,10),(4,5),(6,9),(7,8)]
=> 5 = 4 + 1
[[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]]
=> [1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [(1,2),(3,10),(4,5),(6,9),(7,8)]
=> 5 = 4 + 1
[[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> 5 = 4 + 1
[[0,1,0,0,0],[1,-1,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> 5 = 4 + 1
[[0,0,1,0,0],[1,0,-1,0,1],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> 5 = 4 + 1
[[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [1,3,2,4,5,6] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,10),(11,12)]
=> ? = 5 + 1
[[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [1,3,2,4,5,6] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,10),(11,12)]
=> ? = 5 + 1
[[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [1,4,2,3,5,6] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [(1,2),(3,8),(4,7),(5,6),(9,10),(11,12)]
=> ? = 4 + 1
[[0,1,0,0,0,0],[1,-1,0,1,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [1,4,2,3,5,6] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [(1,2),(3,8),(4,7),(5,6),(9,10),(11,12)]
=> ? = 4 + 1
[[0,0,1,0,0,0],[1,0,-1,1,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [1,4,2,3,5,6] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [(1,2),(3,8),(4,7),(5,6),(9,10),(11,12)]
=> ? = 4 + 1
[[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [1,3,4,2,5,6] => [1,0,1,1,0,1,0,0,1,0,1,0]
=> [(1,2),(3,8),(4,5),(6,7),(9,10),(11,12)]
=> ? = 4 + 1
[[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [1,3,4,2,5,6] => [1,0,1,1,0,1,0,0,1,0,1,0]
=> [(1,2),(3,8),(4,5),(6,7),(9,10),(11,12)]
=> ? = 4 + 1
[[0,1,0,0,0,0],[0,0,1,0,0,0],[1,-1,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [1,3,4,2,5,6] => [1,0,1,1,0,1,0,0,1,0,1,0]
=> [(1,2),(3,8),(4,5),(6,7),(9,10),(11,12)]
=> ? = 4 + 1
[[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]]
=> [1,4,3,2,5,6] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [(1,2),(3,8),(4,7),(5,6),(9,10),(11,12)]
=> ? = 1 + 1
[[0,1,0,0,0,0],[1,-1,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [1,4,3,2,5,6] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [(1,2),(3,8),(4,7),(5,6),(9,10),(11,12)]
=> ? = 1 + 1
[[0,0,1,0,0,0],[1,0,-1,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [1,4,3,2,5,6] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [(1,2),(3,8),(4,7),(5,6),(9,10),(11,12)]
=> ? = 1 + 1
[[0,1,0,0,0,0],[0,0,0,1,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [1,4,3,2,5,6] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [(1,2),(3,8),(4,7),(5,6),(9,10),(11,12)]
=> ? = 1 + 1
[[0,0,1,0,0,0],[0,1,-1,1,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [1,4,3,2,5,6] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [(1,2),(3,8),(4,7),(5,6),(9,10),(11,12)]
=> ? = 1 + 1
[[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]]
=> [1,3,2,4,6,5] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,12),(10,11)]
=> ? = 5 + 1
[[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [1,3,2,4,6,5] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,12),(10,11)]
=> ? = 5 + 1
[[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]]
=> [1,4,2,3,6,5] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [(1,2),(3,8),(4,7),(5,6),(9,12),(10,11)]
=> ? = 4 + 1
[[0,1,0,0,0,0],[1,-1,0,1,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [1,4,2,3,6,5] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [(1,2),(3,8),(4,7),(5,6),(9,12),(10,11)]
=> ? = 4 + 1
[[0,0,1,0,0,0],[1,0,-1,1,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [1,4,2,3,6,5] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [(1,2),(3,8),(4,7),(5,6),(9,12),(10,11)]
=> ? = 4 + 1
[[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [1,3,4,2,6,5] => [1,0,1,1,0,1,0,0,1,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7),(9,12),(10,11)]
=> ? = 4 + 1
[[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [1,3,4,2,6,5] => [1,0,1,1,0,1,0,0,1,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7),(9,12),(10,11)]
=> ? = 4 + 1
[[0,1,0,0,0,0],[0,0,1,0,0,0],[1,-1,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [1,3,4,2,6,5] => [1,0,1,1,0,1,0,0,1,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7),(9,12),(10,11)]
=> ? = 4 + 1
[[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [1,4,3,2,6,5] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [(1,2),(3,8),(4,7),(5,6),(9,12),(10,11)]
=> ? = 1 + 1
[[0,1,0,0,0,0],[1,-1,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [1,4,3,2,6,5] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [(1,2),(3,8),(4,7),(5,6),(9,12),(10,11)]
=> ? = 1 + 1
[[0,0,1,0,0,0],[1,0,-1,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [1,4,3,2,6,5] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [(1,2),(3,8),(4,7),(5,6),(9,12),(10,11)]
=> ? = 1 + 1
[[0,1,0,0,0,0],[0,0,0,1,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [1,4,3,2,6,5] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [(1,2),(3,8),(4,7),(5,6),(9,12),(10,11)]
=> ? = 1 + 1
[[0,0,1,0,0,0],[0,1,-1,1,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [1,4,3,2,6,5] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [(1,2),(3,8),(4,7),(5,6),(9,12),(10,11)]
=> ? = 1 + 1
[[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,-1,1],[0,0,0,0,1,0]]
=> [1,3,2,4,6,5] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,12),(10,11)]
=> ? = 5 + 1
[[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [1,3,2,4,6,5] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,12),(10,11)]
=> ? = 5 + 1
[[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,1,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [1,3,2,4,6,5] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,12),(10,11)]
=> ? = 5 + 1
[[0,1,0,0,0,0],[1,-1,0,1,0,0],[0,1,0,0,0,0],[0,0,1,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [1,3,2,4,6,5] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,12),(10,11)]
=> ? = 5 + 1
[[0,0,1,0,0,0],[1,0,-1,1,0,0],[0,1,0,0,0,0],[0,0,1,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [1,3,2,4,6,5] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,12),(10,11)]
=> ? = 5 + 1
[[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,1,0,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [1,3,2,4,6,5] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,12),(10,11)]
=> ? = 5 + 1
[[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,0,0,1,0,0],[0,1,0,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [1,3,2,4,6,5] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,12),(10,11)]
=> ? = 5 + 1
[[0,1,0,0,0,0],[0,0,1,0,0,0],[1,-1,0,1,0,0],[0,1,0,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [1,3,2,4,6,5] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,12),(10,11)]
=> ? = 5 + 1
[[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [1,3,2,4,6,5] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,12),(10,11)]
=> ? = 5 + 1
[[0,1,0,0,0,0],[1,-1,0,1,0,0],[0,0,1,0,0,0],[0,1,0,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [1,3,2,4,6,5] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,12),(10,11)]
=> ? = 5 + 1
[[0,0,1,0,0,0],[1,0,-1,1,0,0],[0,0,1,0,0,0],[0,1,0,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [1,3,2,4,6,5] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,12),(10,11)]
=> ? = 5 + 1
[[0,1,0,0,0,0],[0,0,0,1,0,0],[1,-1,1,0,0,0],[0,1,0,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [1,3,2,4,6,5] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,12),(10,11)]
=> ? = 5 + 1
[[0,0,1,0,0,0],[0,1,-1,1,0,0],[1,-1,1,0,0,0],[0,1,0,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [1,3,2,4,6,5] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,12),(10,11)]
=> ? = 5 + 1
[[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,-1,1],[0,0,0,0,1,0]]
=> [1,4,2,3,6,5] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [(1,2),(3,8),(4,7),(5,6),(9,12),(10,11)]
=> ? = 4 + 1
[[0,1,0,0,0,0],[1,-1,0,0,1,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [1,4,2,3,6,5] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [(1,2),(3,8),(4,7),(5,6),(9,12),(10,11)]
=> ? = 4 + 1
[[0,0,1,0,0,0],[1,0,-1,0,1,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [1,4,2,3,6,5] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [(1,2),(3,8),(4,7),(5,6),(9,12),(10,11)]
=> ? = 4 + 1
[[0,0,0,1,0,0],[1,0,0,-1,1,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [1,4,2,3,6,5] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [(1,2),(3,8),(4,7),(5,6),(9,12),(10,11)]
=> ? = 4 + 1
[[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [1,3,4,2,6,5] => [1,0,1,1,0,1,0,0,1,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7),(9,12),(10,11)]
=> ? = 4 + 1
[[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [1,3,4,2,6,5] => [1,0,1,1,0,1,0,0,1,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7),(9,12),(10,11)]
=> ? = 4 + 1
[[0,1,0,0,0,0],[0,0,1,0,0,0],[1,-1,0,0,1,0],[0,1,0,0,0,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [1,3,4,2,6,5] => [1,0,1,1,0,1,0,0,1,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7),(9,12),(10,11)]
=> ? = 4 + 1
[[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,-1,1,0],[0,1,0,0,0,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [1,3,4,2,6,5] => [1,0,1,1,0,1,0,0,1,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7),(9,12),(10,11)]
=> ? = 4 + 1
[[0,1,0,0,0,0],[1,-1,0,1,0,0],[0,0,1,-1,1,0],[0,1,0,0,0,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [1,3,4,2,6,5] => [1,0,1,1,0,1,0,0,1,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7),(9,12),(10,11)]
=> ? = 4 + 1
[[0,0,1,0,0,0],[1,0,-1,1,0,0],[0,0,1,-1,1,0],[0,1,0,0,0,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [1,3,4,2,6,5] => [1,0,1,1,0,1,0,0,1,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7),(9,12),(10,11)]
=> ? = 4 + 1
[[0,1,0,0,0,0],[0,0,0,1,0,0],[1,-1,1,-1,1,0],[0,1,0,0,0,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [1,3,4,2,6,5] => [1,0,1,1,0,1,0,0,1,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7),(9,12),(10,11)]
=> ? = 4 + 1
Description
The number of topologically connected components of a perfect matching.
For example, the perfect matching $\{\{1,4\},\{2,3\}\}$ has the two connected components $\{1,4\}$ and $\{2,3\}$.
The number of perfect matchings with only one block is [[oeis:A000699]].
Matching statistic: St001879
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00002: Alternating sign matrices —to left key permutation⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St001879: Posets ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 57%
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St001879: Posets ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 57%
Values
[[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)
=> 3
[[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [1,2,4,3] => [.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,2,4,3] => [.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,2,4,3] => [.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> [1,4,2,3] => [.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
=> [1,4,2,3] => [.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
=> [1,4,2,3] => [.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [4,1,2,3] => [[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 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)
=> 4
[[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 4
[[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 4
[[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 4
[[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 4
[[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 4
[[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 4
[[0,0,1,0,0],[1,0,-1,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 4
[[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 4
[[0,1,0,0,0],[1,-1,1,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 4
[[0,1,0,0,0],[0,0,1,0,0],[1,-1,0,1,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 4
[[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 4
[[0,1,0,0,0],[1,-1,0,1,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 4
[[0,0,1,0,0],[1,0,-1,1,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 4
[[0,1,0,0,0],[0,0,0,1,0],[1,-1,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 4
[[0,0,1,0,0],[0,1,-1,1,0],[1,-1,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 4
[[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[[1,0,0,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[[0,0,1,0,0],[1,0,-1,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[[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] => [.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[[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] => [.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[[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] => [.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[[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] => [.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[[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] => [[.,[.,[.,[.,.]]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 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] => [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 4
[[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] => [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 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] => [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 4
[[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]]
=> [1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 4
[[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]]
=> [1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 4
[[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]]
=> [1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 4
[[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]]
=> [1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 4
[[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]]
=> [1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 4
[[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,5,3,2,4] => [.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ? = 4
[[0,1,0,0,0],[1,-1,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,5,3,2,4] => [.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ? = 4
[[0,0,1,0,0],[1,0,-1,0,1],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,5,3,2,4] => [.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ? = 4
[[0,0,0,1,0],[1,0,0,-1,1],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,5,3,2,4] => [.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ? = 4
[[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [5,1,3,2,4] => [[.,[[.,.],[.,.]]],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ? = 4
[[0,1,0,0,0],[0,0,0,0,1],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,5,3,2,4] => [.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ? = 4
[[0,0,1,0,0],[0,1,-1,0,1],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,5,3,2,4] => [.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ? = 4
[[0,0,0,1,0],[0,1,0,-1,1],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,5,3,2,4] => [.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ? = 4
[[0,0,0,0,1],[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [5,1,3,2,4] => [[.,[[.,.],[.,.]]],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ? = 4
[[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [1,2,3,4,5,6] => [.,[.,[.,[.,[.,[.,.]]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [1,3,2,4,5,6] => [.,[[.,.],[.,[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ? = 5
[[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [1,3,2,4,5,6] => [.,[[.,.],[.,[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ? = 5
[[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [1,2,4,3,5,6] => [.,[.,[[.,.],[.,[.,.]]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ? = 5
[[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [1,2,4,3,5,6] => [.,[.,[[.,.],[.,[.,.]]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ? = 5
[[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [1,2,4,3,5,6] => [.,[.,[[.,.],[.,[.,.]]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ? = 5
[[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [1,4,2,3,5,6] => [.,[[.,[.,.]],[.,[.,.]]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ? = 4
[[0,1,0,0,0,0],[1,-1,0,1,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [1,4,2,3,5,6] => [.,[[.,[.,.]],[.,[.,.]]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ? = 4
[[0,0,1,0,0,0],[1,0,-1,1,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [1,4,2,3,5,6] => [.,[[.,[.,.]],[.,[.,.]]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ? = 4
[[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [1,3,4,2,5,6] => [.,[[.,.],[.,[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ? = 4
[[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [1,3,4,2,5,6] => [.,[[.,.],[.,[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ? = 4
[[0,1,0,0,0,0],[0,0,1,0,0,0],[1,-1,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [1,3,4,2,5,6] => [.,[[.,.],[.,[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ? = 4
[[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]]
=> [1,4,3,2,5,6] => [.,[[[.,.],.],[.,[.,.]]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ? = 1
[[0,1,0,0,0,0],[1,-1,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [1,4,3,2,5,6] => [.,[[[.,.],.],[.,[.,.]]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ? = 1
[[0,0,1,0,0,0],[1,0,-1,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [1,4,3,2,5,6] => [.,[[[.,.],.],[.,[.,.]]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ? = 1
[[0,1,0,0,0,0],[0,0,0,1,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [1,4,3,2,5,6] => [.,[[[.,.],.],[.,[.,.]]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ? = 1
[[0,0,1,0,0,0],[0,1,-1,1,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [1,4,3,2,5,6] => [.,[[[.,.],.],[.,[.,.]]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ? = 1
[[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [1,2,3,4,6,5] => [.,[.,[.,[.,[[.,.],.]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [1,3,2,4,6,5] => [.,[[.,.],[.,[[.,.],.]]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ? = 5
[[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [1,3,2,4,6,5] => [.,[[.,.],[.,[[.,.],.]]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ? = 5
[[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [1,2,3,4,6,5] => [.,[.,[.,[.,[[.,.],.]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [1,2,3,4,6,5] => [.,[.,[.,[.,[[.,.],.]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [1,2,3,4,6,5] => [.,[.,[.,[.,[[.,.],.]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [1,2,3,4,6,5] => [.,[.,[.,[.,[[.,.],.]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,2,3,6,4,5] => [.,[.,[.,[[.,[.,.]],.]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,-1,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,2,3,6,4,5] => [.,[.,[.,[[.,[.,.]],.]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,-1,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,2,3,6,4,5] => [.,[.,[.,[[.,[.,.]],.]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,-1,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,2,3,6,4,5] => [.,[.,[.,[[.,[.,.]],.]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,-1,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,2,3,6,4,5] => [.,[.,[.,[[.,[.,.]],.]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,-1,0,1,0],[0,0,1,0,-1,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,2,3,6,4,5] => [.,[.,[.,[[.,[.,.]],.]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,-1,0,1,0],[0,0,1,0,-1,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,2,3,6,4,5] => [.,[.,[.,[[.,[.,.]],.]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,-1,1,0],[0,0,1,0,-1,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,2,3,6,4,5] => [.,[.,[.,[[.,[.,.]],.]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[[0,1,0,0,0,0],[1,-1,0,1,0,0],[0,1,0,-1,1,0],[0,0,1,0,-1,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,2,3,6,4,5] => [.,[.,[.,[[.,[.,.]],.]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[[0,0,1,0,0,0],[1,0,-1,1,0,0],[0,1,0,-1,1,0],[0,0,1,0,-1,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,2,3,6,4,5] => [.,[.,[.,[[.,[.,.]],.]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[[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]]
=> [1,2,6,3,4,5] => [.,[.,[[.,[.,[.,.]]],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,-1,0,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,2,6,3,4,5] => [.,[.,[[.,[.,[.,.]]],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,-1,0,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,2,6,3,4,5] => [.,[.,[[.,[.,[.,.]]],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,-1,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,2,6,3,4,5] => [.,[.,[[.,[.,[.,.]]],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[[0,1,0,0,0,0],[1,-1,0,1,0,0],[0,1,0,-1,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,2,6,3,4,5] => [.,[.,[[.,[.,[.,.]]],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[[0,0,1,0,0,0],[1,0,-1,1,0,0],[0,1,0,-1,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,2,6,3,4,5] => [.,[.,[[.,[.,[.,.]]],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[[1,0,0,0,0,0],[0,0,0,0,1,0],[0,1,0,0,-1,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,2,6,3,4,5] => [.,[.,[[.,[.,[.,.]]],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[[0,1,0,0,0,0],[1,-1,0,0,1,0],[0,1,0,0,-1,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,2,6,3,4,5] => [.,[.,[[.,[.,[.,.]]],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[[0,0,1,0,0,0],[1,0,-1,0,1,0],[0,1,0,0,-1,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,2,6,3,4,5] => [.,[.,[[.,[.,[.,.]]],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[[0,0,0,1,0,0],[1,0,0,-1,1,0],[0,1,0,0,-1,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,2,6,3,4,5] => [.,[.,[[.,[.,[.,.]]],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
Description
The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice.
Matching statistic: St001645
Mp00002: Alternating sign matrices —to left key permutation⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001645: Graphs ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 43%
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001645: Graphs ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 43%
Values
[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? = 3 + 1
[[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [1,2,4,3] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ? = 3 + 1
[[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,2,4,3] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ? = 3 + 1
[[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,2,4,3] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ? = 3 + 1
[[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> [1,4,2,3] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 4 = 3 + 1
[[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
=> [1,4,2,3] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 4 = 3 + 1
[[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
=> [1,4,2,3] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 4 = 3 + 1
[[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [4,1,2,3] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> ? = 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] => [1,2,3,4,5] => ([],5)
=> ? = 4 + 1
[[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,3,2,4,5] => [3,1,2,4,5] => ([(2,4),(3,4)],5)
=> ? = 4 + 1
[[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,3,2,4,5] => [3,1,2,4,5] => ([(2,4),(3,4)],5)
=> ? = 4 + 1
[[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,2,3,5,4] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? = 4 + 1
[[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 4 + 1
[[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 4 + 1
[[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,2,3,5,4] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? = 4 + 1
[[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,2,3,5,4] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? = 4 + 1
[[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,2,3,5,4] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? = 4 + 1
[[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 4 + 1
[[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 4 + 1
[[0,0,1,0,0],[1,0,-1,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 4 + 1
[[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 4 + 1
[[0,1,0,0,0],[1,-1,1,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 4 + 1
[[0,1,0,0,0],[0,0,1,0,0],[1,-1,0,1,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 4 + 1
[[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 4 + 1
[[0,1,0,0,0],[1,-1,0,1,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 4 + 1
[[0,0,1,0,0],[1,0,-1,1,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 4 + 1
[[0,1,0,0,0],[0,0,0,1,0],[1,-1,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 4 + 1
[[0,0,1,0,0],[0,1,-1,1,0],[1,-1,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 4 + 1
[[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ? = 4 + 1
[[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ? = 4 + 1
[[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ? = 4 + 1
[[1,0,0,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ? = 4 + 1
[[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ? = 4 + 1
[[0,0,1,0,0],[1,0,-1,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ? = 4 + 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] => [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ? = 4 + 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] => [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ? = 4 + 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] => [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ? = 4 + 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] => [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ? = 4 + 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] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? = 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] => [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ? = 4 + 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] => [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ? = 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] => [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ? = 4 + 1
[[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]]
=> [1,3,5,2,4] => [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ? = 4 + 1
[[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]]
=> [1,3,5,2,4] => [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ? = 4 + 1
[[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]]
=> [1,3,5,2,4] => [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ? = 4 + 1
[[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]]
=> [1,3,5,2,4] => [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ? = 4 + 1
[[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]]
=> [1,3,5,2,4] => [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ? = 4 + 1
[[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,5,3,2,4] => [4,3,5,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ? = 4 + 1
[[0,1,0,0,0],[1,-1,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,5,3,2,4] => [4,3,5,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ? = 4 + 1
[[0,0,1,0,0],[1,0,-1,0,1],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,5,3,2,4] => [4,3,5,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ? = 4 + 1
[[0,0,0,1,0],[1,0,0,-1,1],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,5,3,2,4] => [4,3,5,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ? = 4 + 1
[[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [5,1,3,2,4] => [4,2,3,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4 + 1
[[0,1,0,0,0],[0,0,0,0,1],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,5,3,2,4] => [4,3,5,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ? = 4 + 1
[[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]]
=> [1,2,6,3,4,5] => [4,5,6,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> 6 = 5 + 1
[[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,-1,0,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,2,6,3,4,5] => [4,5,6,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> 6 = 5 + 1
[[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,-1,0,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,2,6,3,4,5] => [4,5,6,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> 6 = 5 + 1
[[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,-1,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,2,6,3,4,5] => [4,5,6,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> 6 = 5 + 1
[[0,1,0,0,0,0],[1,-1,0,1,0,0],[0,1,0,-1,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,2,6,3,4,5] => [4,5,6,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> 6 = 5 + 1
[[0,0,1,0,0,0],[1,0,-1,1,0,0],[0,1,0,-1,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,2,6,3,4,5] => [4,5,6,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> 6 = 5 + 1
[[1,0,0,0,0,0],[0,0,0,0,1,0],[0,1,0,0,-1,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,2,6,3,4,5] => [4,5,6,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> 6 = 5 + 1
[[0,1,0,0,0,0],[1,-1,0,0,1,0],[0,1,0,0,-1,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,2,6,3,4,5] => [4,5,6,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> 6 = 5 + 1
[[0,0,1,0,0,0],[1,0,-1,0,1,0],[0,1,0,0,-1,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,2,6,3,4,5] => [4,5,6,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> 6 = 5 + 1
[[0,0,0,1,0,0],[1,0,0,-1,1,0],[0,1,0,0,-1,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,2,6,3,4,5] => [4,5,6,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> 6 = 5 + 1
[[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,2,6,4,3,5] => [5,4,6,1,2,3] => ([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 5 + 1
[[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,-1,0,0,1],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,2,6,4,3,5] => [5,4,6,1,2,3] => ([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 5 + 1
[[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,-1,0,0,1],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,2,6,4,3,5] => [5,4,6,1,2,3] => ([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 5 + 1
[[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,-1,0,1],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,2,6,4,3,5] => [5,4,6,1,2,3] => ([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 5 + 1
[[0,1,0,0,0,0],[1,-1,0,1,0,0],[0,1,0,-1,0,1],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,2,6,4,3,5] => [5,4,6,1,2,3] => ([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 5 + 1
[[0,0,1,0,0,0],[1,0,-1,1,0,0],[0,1,0,-1,0,1],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,2,6,4,3,5] => [5,4,6,1,2,3] => ([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 5 + 1
[[1,0,0,0,0,0],[0,0,0,0,1,0],[0,1,0,0,-1,1],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,2,6,4,3,5] => [5,4,6,1,2,3] => ([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 5 + 1
[[0,1,0,0,0,0],[1,-1,0,0,1,0],[0,1,0,0,-1,1],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,2,6,4,3,5] => [5,4,6,1,2,3] => ([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 5 + 1
[[0,0,1,0,0,0],[1,0,-1,0,1,0],[0,1,0,0,-1,1],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,2,6,4,3,5] => [5,4,6,1,2,3] => ([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 5 + 1
[[0,0,0,1,0,0],[1,0,0,-1,1,0],[0,1,0,0,-1,1],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,2,6,4,3,5] => [5,4,6,1,2,3] => ([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 5 + 1
[[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,1,-1,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,2,6,4,3,5] => [5,4,6,1,2,3] => ([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 5 + 1
[[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,0,0,0,0,1],[0,1,-1,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,2,6,4,3,5] => [5,4,6,1,2,3] => ([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 5 + 1
[[0,1,0,0,0,0],[0,0,1,0,0,0],[1,-1,0,0,0,1],[0,1,-1,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,2,6,4,3,5] => [5,4,6,1,2,3] => ([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 5 + 1
[[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,-1,0,1],[0,1,-1,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,2,6,4,3,5] => [5,4,6,1,2,3] => ([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 5 + 1
[[0,1,0,0,0,0],[1,-1,0,1,0,0],[0,0,1,-1,0,1],[0,1,-1,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,2,6,4,3,5] => [5,4,6,1,2,3] => ([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 5 + 1
[[0,0,1,0,0,0],[1,0,-1,1,0,0],[0,0,1,-1,0,1],[0,1,-1,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,2,6,4,3,5] => [5,4,6,1,2,3] => ([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 5 + 1
[[0,1,0,0,0,0],[0,0,0,1,0,0],[1,-1,1,-1,0,1],[0,1,-1,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,2,6,4,3,5] => [5,4,6,1,2,3] => ([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 5 + 1
[[0,0,1,0,0,0],[0,1,-1,1,0,0],[1,-1,1,-1,0,1],[0,1,-1,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,2,6,4,3,5] => [5,4,6,1,2,3] => ([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 5 + 1
[[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,-1,1],[0,1,-1,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,2,6,4,3,5] => [5,4,6,1,2,3] => ([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 5 + 1
[[0,1,0,0,0,0],[1,-1,0,0,1,0],[0,0,1,0,-1,1],[0,1,-1,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,2,6,4,3,5] => [5,4,6,1,2,3] => ([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 5 + 1
[[0,0,1,0,0,0],[1,0,-1,0,1,0],[0,0,1,0,-1,1],[0,1,-1,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,2,6,4,3,5] => [5,4,6,1,2,3] => ([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 5 + 1
[[0,0,0,1,0,0],[1,0,0,-1,1,0],[0,0,1,0,-1,1],[0,1,-1,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,2,6,4,3,5] => [5,4,6,1,2,3] => ([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 5 + 1
[[0,1,0,0,0,0],[0,0,0,0,1,0],[1,-1,1,0,-1,1],[0,1,-1,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,2,6,4,3,5] => [5,4,6,1,2,3] => ([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 5 + 1
[[0,0,1,0,0,0],[0,1,-1,0,1,0],[1,-1,1,0,-1,1],[0,1,-1,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,2,6,4,3,5] => [5,4,6,1,2,3] => ([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 5 + 1
[[0,0,0,1,0,0],[0,1,0,-1,1,0],[1,-1,1,0,-1,1],[0,1,-1,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [1,2,6,4,3,5] => [5,4,6,1,2,3] => ([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 5 + 1
[[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,0,1],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> [1,2,3,7,4,5,6] => [5,6,7,1,2,3,4] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> 7 = 6 + 1
[[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,-1,0,0,1],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> [1,2,3,7,4,5,6] => [5,6,7,1,2,3,4] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> 7 = 6 + 1
[[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,1,0,-1,0,1],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> [1,2,3,7,4,5,6] => [5,6,7,1,2,3,4] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> 7 = 6 + 1
[[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,1,0,0,-1,1],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> [1,2,3,7,4,5,6] => [5,6,7,1,2,3,4] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> 7 = 6 + 1
[[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,0,0,1],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> [1,2,7,3,4,5,6] => [4,5,6,7,1,2,3] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> 7 = 6 + 1
[[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,1,-1,1,0,0,0],[0,0,1,-1,0,0,1],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> [1,2,3,7,4,5,6] => [5,6,7,1,2,3,4] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> 7 = 6 + 1
[[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,1,-1,0,1,0,0],[0,0,1,0,-1,0,1],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> [1,2,3,7,4,5,6] => [5,6,7,1,2,3,4] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> 7 = 6 + 1
[[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,1,-1,0,0,1,0],[0,0,1,0,0,-1,1],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> [1,2,3,7,4,5,6] => [5,6,7,1,2,3,4] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> 7 = 6 + 1
[[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,1,-1,0,0,0,1],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> [1,2,7,3,4,5,6] => [4,5,6,7,1,2,3] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> 7 = 6 + 1
[[1,0,0,0,0,0,0],[0,0,0,1,0,0,0],[0,1,0,-1,1,0,0],[0,0,1,0,-1,0,1],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> [1,2,3,7,4,5,6] => [5,6,7,1,2,3,4] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> 7 = 6 + 1
[[1,0,0,0,0,0,0],[0,0,0,1,0,0,0],[0,1,0,-1,0,1,0],[0,0,1,0,0,-1,1],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> [1,2,3,7,4,5,6] => [5,6,7,1,2,3,4] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> 7 = 6 + 1
[[1,0,0,0,0,0,0],[0,0,0,1,0,0,0],[0,1,0,-1,0,0,1],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> [1,2,7,3,4,5,6] => [4,5,6,7,1,2,3] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> 7 = 6 + 1
Description
The pebbling number of a connected graph.
Matching statistic: St000019
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00007: Alternating sign matrices —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
St000019: Permutations ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 71%
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
St000019: Permutations ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 71%
Values
[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [5,1,2,3,4] => 4 = 3 + 1
[[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [5,1,2,4,3] => 4 = 3 + 1
[[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [4,1,5,3,2] => 4 = 3 + 1
[[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => [4,5,2,3,1] => 4 = 3 + 1
[[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [5,1,3,4,2] => 4 = 3 + 1
[[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
=> [1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => [3,5,2,4,1] => 4 = 3 + 1
[[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
=> [1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [3,5,4,2,1] => 4 = 3 + 1
[[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5,2,3,4,1] => 4 = 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]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => [6,1,2,3,4,5] => 5 = 4 + 1
[[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [6,1,3,2,4,5] => 5 = 4 + 1
[[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => [3,6,2,1,4,5] => 5 = 4 + 1
[[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => [6,1,2,3,5,4] => 5 = 4 + 1
[[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => [6,1,3,2,5,4] => 5 = 4 + 1
[[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => [3,6,2,1,5,4] => 5 = 4 + 1
[[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => [5,1,2,6,4,3] => 5 = 4 + 1
[[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => [5,1,6,3,4,2] => 5 = 4 + 1
[[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => [6,5,2,3,4,1] => 5 = 4 + 1
[[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => [5,1,3,6,4,2] => 5 = 4 + 1
[[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => [3,5,2,6,4,1] => 5 = 4 + 1
[[0,0,1,0,0],[1,0,-1,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,1,1,0,1,0,0,1,0,0]
=> [6,3,5,1,2,4] => [3,6,5,2,4,1] => 5 = 4 + 1
[[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => [5,1,6,3,4,2] => 5 = 4 + 1
[[0,1,0,0,0],[1,-1,1,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => [6,5,2,3,4,1] => 5 = 4 + 1
[[0,1,0,0,0],[0,0,1,0,0],[1,-1,0,1,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => [6,5,2,3,4,1] => 5 = 4 + 1
[[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => [5,1,3,6,4,2] => 5 = 4 + 1
[[0,1,0,0,0],[1,-1,0,1,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => [3,5,2,6,4,1] => 5 = 4 + 1
[[0,0,1,0,0],[1,0,-1,1,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [1,1,1,0,1,0,0,1,0,0]
=> [6,3,5,1,2,4] => [3,6,5,2,4,1] => 5 = 4 + 1
[[0,1,0,0,0],[0,0,0,1,0],[1,-1,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => [3,5,2,6,4,1] => 5 = 4 + 1
[[0,0,1,0,0],[0,1,-1,1,0],[1,-1,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [1,1,1,0,1,0,0,1,0,0]
=> [6,3,5,1,2,4] => [3,6,5,2,4,1] => 5 = 4 + 1
[[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [6,1,2,4,5,3] => 5 = 4 + 1
[[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => [4,1,6,3,5,2] => 5 = 4 + 1
[[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => [4,6,2,3,5,1] => 5 = 4 + 1
[[1,0,0,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => [4,1,6,5,3,2] => 5 = 4 + 1
[[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => [4,6,2,5,3,1] => 5 = 4 + 1
[[0,0,1,0,0],[1,0,-1,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [6,5,4,1,2,3] => [4,5,6,3,2,1] => 5 = 4 + 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]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => [6,1,3,4,5,2] => 5 = 4 + 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]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => [3,6,2,4,5,1] => 5 = 4 + 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]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [5,3,4,1,6,2] => [3,6,4,2,5,1] => 5 = 4 + 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]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [6,3,4,5,1,2] => [3,6,4,5,2,1] => 5 = 4 + 1
[[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [6,2,3,4,5,1] => 5 = 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]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => [4,1,6,3,5,2] => 5 = 4 + 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]]
=> [1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => [4,6,2,3,5,1] => 5 = 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]]
=> [1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => [4,6,2,3,5,1] => 5 = 4 + 1
[[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]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => [4,1,6,5,3,2] => 5 = 4 + 1
[[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]]
=> [1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => [4,6,2,5,3,1] => 5 = 4 + 1
[[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]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [6,5,4,1,2,3] => [4,5,6,3,2,1] => 5 = 4 + 1
[[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]]
=> [1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => [4,6,2,5,3,1] => 5 = 4 + 1
[[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]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [6,5,4,1,2,3] => [4,5,6,3,2,1] => 5 = 4 + 1
[[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => [6,1,3,4,5,2] => 5 = 4 + 1
[[0,1,0,0,0],[1,-1,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => [3,6,2,4,5,1] => 5 = 4 + 1
[[0,0,1,0,0],[1,0,-1,0,1],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [5,3,4,1,6,2] => [3,6,4,2,5,1] => 5 = 4 + 1
[[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [3,1,7,2,4,5,6] => [7,1,3,2,4,5,6] => ? = 5 + 1
[[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> [7,3,1,2,4,5,6] => [3,7,2,1,4,5,6] => ? = 5 + 1
[[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [4,1,2,7,3,5,6] => [7,1,2,4,3,5,6] => ? = 5 + 1
[[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [7,1,4,2,3,5,6] => [4,1,7,3,2,5,6] => ? = 5 + 1
[[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> [7,4,1,2,3,5,6] => [4,7,2,3,1,5,6] => ? = 5 + 1
[[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [3,1,4,7,2,5,6] => [7,1,3,4,2,5,6] => ? = 4 + 1
[[0,1,0,0,0,0],[1,-1,0,1,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [1,1,0,1,1,0,0,0,1,0,1,0]
=> [4,3,1,7,2,5,6] => [3,7,2,4,1,5,6] => ? = 4 + 1
[[0,0,1,0,0,0],[1,0,-1,1,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> [7,3,4,1,2,5,6] => [3,7,4,2,1,5,6] => ? = 4 + 1
[[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [7,1,4,2,3,5,6] => [4,1,7,3,2,5,6] => ? = 4 + 1
[[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> [7,4,1,2,3,5,6] => [4,7,2,3,1,5,6] => ? = 4 + 1
[[0,1,0,0,0,0],[0,0,1,0,0,0],[1,-1,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> [7,4,1,2,3,5,6] => [4,7,2,3,1,5,6] => ? = 4 + 1
[[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]]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [3,1,4,7,2,5,6] => [7,1,3,4,2,5,6] => ? = 1 + 1
[[0,1,0,0,0,0],[1,-1,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [1,1,0,1,1,0,0,0,1,0,1,0]
=> [4,3,1,7,2,5,6] => [3,7,2,4,1,5,6] => ? = 1 + 1
[[0,0,1,0,0,0],[1,0,-1,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> [7,3,4,1,2,5,6] => [3,7,4,2,1,5,6] => ? = 1 + 1
[[0,1,0,0,0,0],[0,0,0,1,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [1,1,0,1,1,0,0,0,1,0,1,0]
=> [4,3,1,7,2,5,6] => [3,7,2,4,1,5,6] => ? = 1 + 1
[[0,0,1,0,0,0],[0,1,-1,1,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> [7,3,4,1,2,5,6] => [3,7,4,2,1,5,6] => ? = 1 + 1
[[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,1,2,3,4,7,5] => [7,1,2,3,4,6,5] => ? = 5 + 1
[[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]]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> [3,1,6,2,4,7,5] => [7,1,3,2,4,6,5] => ? = 5 + 1
[[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [1,1,0,1,0,0,1,0,1,1,0,0]
=> [6,3,1,2,4,7,5] => [3,7,2,1,4,6,5] => ? = 5 + 1
[[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]]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,1,2,6,3,7,5] => [7,1,2,4,3,6,5] => ? = 5 + 1
[[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [1,0,1,1,0,1,0,0,1,1,0,0]
=> [6,1,4,2,3,7,5] => [4,1,7,3,2,6,5] => ? = 5 + 1
[[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [1,1,0,1,0,1,0,0,1,1,0,0]
=> [6,4,1,2,3,7,5] => [4,7,2,3,1,6,5] => ? = 5 + 1
[[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]]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> [3,1,4,6,2,7,5] => [7,1,3,4,2,6,5] => ? = 4 + 1
[[0,1,0,0,0,0],[1,-1,0,1,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [1,1,0,1,1,0,0,0,1,1,0,0]
=> [4,3,1,6,2,7,5] => [3,7,2,4,1,6,5] => ? = 4 + 1
[[0,0,1,0,0,0],[1,0,-1,1,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [1,1,1,0,1,0,0,0,1,1,0,0]
=> [6,3,4,1,2,7,5] => [3,7,4,2,1,6,5] => ? = 4 + 1
[[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [1,0,1,1,0,1,0,0,1,1,0,0]
=> [6,1,4,2,3,7,5] => [4,1,7,3,2,6,5] => ? = 4 + 1
[[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [1,1,0,1,0,1,0,0,1,1,0,0]
=> [6,4,1,2,3,7,5] => [4,7,2,3,1,6,5] => ? = 4 + 1
[[0,1,0,0,0,0],[0,0,1,0,0,0],[1,-1,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [1,1,0,1,0,1,0,0,1,1,0,0]
=> [6,4,1,2,3,7,5] => [4,7,2,3,1,6,5] => ? = 4 + 1
[[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> [3,1,4,6,2,7,5] => [7,1,3,4,2,6,5] => ? = 1 + 1
[[0,1,0,0,0,0],[1,-1,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [1,1,0,1,1,0,0,0,1,1,0,0]
=> [4,3,1,6,2,7,5] => [3,7,2,4,1,6,5] => ? = 1 + 1
[[0,0,1,0,0,0],[1,0,-1,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [1,1,1,0,1,0,0,0,1,1,0,0]
=> [6,3,4,1,2,7,5] => [3,7,4,2,1,6,5] => ? = 1 + 1
[[0,1,0,0,0,0],[0,0,0,1,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [1,1,0,1,1,0,0,0,1,1,0,0]
=> [4,3,1,6,2,7,5] => [3,7,2,4,1,6,5] => ? = 1 + 1
[[0,0,1,0,0,0],[0,1,-1,1,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [1,1,1,0,1,0,0,0,1,1,0,0]
=> [6,3,4,1,2,7,5] => [3,7,4,2,1,6,5] => ? = 1 + 1
[[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,1,2,3,6,4,5] => [6,1,2,3,7,5,4] => ? = 5 + 1
[[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,-1,1],[0,0,0,0,1,0]]
=> [1,0,1,1,0,0,1,1,0,1,0,0]
=> [3,1,7,2,6,4,5] => [6,1,3,2,7,5,4] => ? = 5 + 1
[[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> [7,3,1,2,6,4,5] => [3,6,2,1,7,5,4] => ? = 5 + 1
[[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [7,1,2,6,3,4,5] => [6,1,2,7,4,5,3] => ? = 5 + 1
[[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [6,1,7,2,3,4,5] => [7,1,6,3,4,5,2] => ? = 5 + 1
[[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [7,6,1,2,3,4,5] => [6,7,2,3,4,5,1] => ? = 5 + 1
[[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,1,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [3,1,7,6,2,4,5] => [6,1,3,7,4,5,2] => ? = 5 + 1
[[0,1,0,0,0,0],[1,-1,0,1,0,0],[0,1,0,0,0,0],[0,0,1,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [7,3,1,6,2,4,5] => [3,6,2,7,4,5,1] => ? = 5 + 1
[[0,0,1,0,0,0],[1,0,-1,1,0,0],[0,1,0,0,0,0],[0,0,1,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [6,3,7,1,2,4,5] => [6,7,3,2,4,5,1] => ? = 5 + 1
[[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,1,0,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [6,1,7,2,3,4,5] => [7,1,6,3,4,5,2] => ? = 5 + 1
[[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,0,0,1,0,0],[0,1,0,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [7,6,1,2,3,4,5] => [6,7,2,3,4,5,1] => ? = 5 + 1
[[0,1,0,0,0,0],[0,0,1,0,0,0],[1,-1,0,1,0,0],[0,1,0,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [7,6,1,2,3,4,5] => [6,7,2,3,4,5,1] => ? = 5 + 1
[[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [3,1,7,6,2,4,5] => [6,1,3,7,4,5,2] => ? = 5 + 1
[[0,1,0,0,0,0],[1,-1,0,1,0,0],[0,0,1,0,0,0],[0,1,0,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [7,3,1,6,2,4,5] => [3,6,2,7,4,5,1] => ? = 5 + 1
[[0,0,1,0,0,0],[1,0,-1,1,0,0],[0,0,1,0,0,0],[0,1,0,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [6,3,7,1,2,4,5] => [6,7,3,2,4,5,1] => ? = 5 + 1
[[0,1,0,0,0,0],[0,0,0,1,0,0],[1,-1,1,0,0,0],[0,1,0,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [7,3,1,6,2,4,5] => [3,6,2,7,4,5,1] => ? = 5 + 1
[[0,0,1,0,0,0],[0,1,-1,1,0,0],[1,-1,1,0,0,0],[0,1,0,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [6,3,7,1,2,4,5] => [6,7,3,2,4,5,1] => ? = 5 + 1
Description
The cardinality of the support of a permutation.
A permutation $\sigma$ may be written as a product $\sigma = s_{i_1}\dots s_{i_k}$ with $k$ minimal, where $s_i = (i,i+1)$ denotes the simple transposition swapping the entries in positions $i$ and $i+1$.
The set of indices $\{i_1,\dots,i_k\}$ is the '''support''' of $\sigma$ and independent of the chosen way to write $\sigma$ as such a product.
See [2], Definition 1 and Proposition 10.
The '''connectivity set''' of $\sigma$ of length $n$ is the set of indices $1 \leq i < n$ such that $\sigma(k) < i$ for all $k < i$.
Thus, the connectivity set is the complement of the support.
Matching statistic: St000197
(load all 37 compositions to match this statistic)
(load all 37 compositions to match this statistic)
Mp00002: Alternating sign matrices —to left key permutation⟶ Permutations
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
St000197: Alternating sign matrices ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 29%
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
St000197: Alternating sign matrices ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 29%
Values
[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> 4 = 3 + 1
[[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [1,2,4,3] => [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> 4 = 3 + 1
[[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,2,4,3] => [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> 4 = 3 + 1
[[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,2,4,3] => [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> 4 = 3 + 1
[[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> [1,4,2,3] => [[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]]
=> 4 = 3 + 1
[[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
=> [1,4,2,3] => [[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]]
=> 4 = 3 + 1
[[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
=> [1,4,2,3] => [[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]]
=> 4 = 3 + 1
[[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [4,1,2,3] => [[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> 4 = 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] => [[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]]
=> 5 = 4 + 1
[[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,3,2,4,5] => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> 5 = 4 + 1
[[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,3,2,4,5] => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> 5 = 4 + 1
[[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,2,3,5,4] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> 5 = 4 + 1
[[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> 5 = 4 + 1
[[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> 5 = 4 + 1
[[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,2,3,5,4] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> 5 = 4 + 1
[[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,2,3,5,4] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> 5 = 4 + 1
[[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,2,3,5,4] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> 5 = 4 + 1
[[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> 5 = 4 + 1
[[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> 5 = 4 + 1
[[0,0,1,0,0],[1,0,-1,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> 5 = 4 + 1
[[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> 5 = 4 + 1
[[0,1,0,0,0],[1,-1,1,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> 5 = 4 + 1
[[0,1,0,0,0],[0,0,1,0,0],[1,-1,0,1,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> 5 = 4 + 1
[[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> 5 = 4 + 1
[[0,1,0,0,0],[1,-1,0,1,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> 5 = 4 + 1
[[0,0,1,0,0],[1,0,-1,1,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> 5 = 4 + 1
[[0,1,0,0,0],[0,0,0,1,0],[1,-1,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> 5 = 4 + 1
[[0,0,1,0,0],[0,1,-1,1,0],[1,-1,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> 5 = 4 + 1
[[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,0,1,0,0]]
=> 5 = 4 + 1
[[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,0,1,0,0]]
=> 5 = 4 + 1
[[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,0,1,0,0]]
=> 5 = 4 + 1
[[1,0,0,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,0,1,0,0]]
=> 5 = 4 + 1
[[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,0,1,0,0]]
=> 5 = 4 + 1
[[0,0,1,0,0],[1,0,-1,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,0,1,0,0]]
=> 5 = 4 + 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] => [[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]]
=> 5 = 4 + 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] => [[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]]
=> 5 = 4 + 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] => [[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]]
=> 5 = 4 + 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] => [[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]]
=> 5 = 4 + 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] => [[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1],[1,0,0,0,0]]
=> 5 = 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] => [[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]]
=> 5 = 4 + 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] => [[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]]
=> 5 = 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] => [[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]]
=> 5 = 4 + 1
[[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]]
=> [1,3,5,2,4] => [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0]]
=> 5 = 4 + 1
[[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]]
=> [1,3,5,2,4] => [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0]]
=> 5 = 4 + 1
[[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]]
=> [1,3,5,2,4] => [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0]]
=> 5 = 4 + 1
[[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]]
=> [1,3,5,2,4] => [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0]]
=> 5 = 4 + 1
[[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]]
=> [1,3,5,2,4] => [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0]]
=> 5 = 4 + 1
[[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,5,3,2,4] => [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1],[0,1,0,0,0]]
=> 5 = 4 + 1
[[0,1,0,0,0],[1,-1,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,5,3,2,4] => [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1],[0,1,0,0,0]]
=> 5 = 4 + 1
[[0,0,1,0,0],[1,0,-1,0,1],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,5,3,2,4] => [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1],[0,1,0,0,0]]
=> 5 = 4 + 1
[[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [1,2,3,4,5,6] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> ? = 5 + 1
[[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [1,3,2,4,5,6] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> ? = 5 + 1
[[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [1,3,2,4,5,6] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> ? = 5 + 1
[[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [1,2,4,3,5,6] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> ? = 5 + 1
[[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [1,2,4,3,5,6] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> ? = 5 + 1
[[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [1,2,4,3,5,6] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> ? = 5 + 1
[[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [1,4,2,3,5,6] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> ? = 4 + 1
[[0,1,0,0,0,0],[1,-1,0,1,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [1,4,2,3,5,6] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> ? = 4 + 1
[[0,0,1,0,0,0],[1,0,-1,1,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [1,4,2,3,5,6] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> ? = 4 + 1
[[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [1,3,4,2,5,6] => [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> ? = 4 + 1
[[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [1,3,4,2,5,6] => [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> ? = 4 + 1
[[0,1,0,0,0,0],[0,0,1,0,0,0],[1,-1,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [1,3,4,2,5,6] => [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> ? = 4 + 1
[[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]]
=> [1,4,3,2,5,6] => [[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
[[0,1,0,0,0,0],[1,-1,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [1,4,3,2,5,6] => [[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
[[0,0,1,0,0,0],[1,0,-1,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [1,4,3,2,5,6] => [[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
[[0,1,0,0,0,0],[0,0,0,1,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [1,4,3,2,5,6] => [[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
[[0,0,1,0,0,0],[0,1,-1,1,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [1,4,3,2,5,6] => [[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
[[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [1,2,3,4,6,5] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> ? = 5 + 1
[[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]]
=> [1,3,2,4,6,5] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> ? = 5 + 1
[[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [1,3,2,4,6,5] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> ? = 5 + 1
[[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]]
=> [1,2,4,3,6,5] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> ? = 5 + 1
[[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [1,2,4,3,6,5] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> ? = 5 + 1
[[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [1,2,4,3,6,5] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> ? = 5 + 1
[[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]]
=> [1,4,2,3,6,5] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> ? = 4 + 1
[[0,1,0,0,0,0],[1,-1,0,1,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [1,4,2,3,6,5] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> ? = 4 + 1
[[0,0,1,0,0,0],[1,0,-1,1,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [1,4,2,3,6,5] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> ? = 4 + 1
[[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [1,3,4,2,6,5] => [[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]]
=> ? = 4 + 1
[[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [1,3,4,2,6,5] => [[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]]
=> ? = 4 + 1
[[0,1,0,0,0,0],[0,0,1,0,0,0],[1,-1,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [1,3,4,2,6,5] => [[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]]
=> ? = 4 + 1
[[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [1,4,3,2,6,5] => [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> ? = 1 + 1
[[0,1,0,0,0,0],[1,-1,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [1,4,3,2,6,5] => [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> ? = 1 + 1
[[0,0,1,0,0,0],[1,0,-1,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [1,4,3,2,6,5] => [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> ? = 1 + 1
[[0,1,0,0,0,0],[0,0,0,1,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [1,4,3,2,6,5] => [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> ? = 1 + 1
[[0,0,1,0,0,0],[0,1,-1,1,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [1,4,3,2,6,5] => [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> ? = 1 + 1
[[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [1,2,3,4,6,5] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> ? = 5 + 1
[[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,-1,1],[0,0,0,0,1,0]]
=> [1,3,2,4,6,5] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> ? = 5 + 1
[[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [1,3,2,4,6,5] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> ? = 5 + 1
[[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [1,2,3,4,6,5] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> ? = 5 + 1
[[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [1,2,3,4,6,5] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> ? = 5 + 1
[[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [1,2,3,4,6,5] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> ? = 5 + 1
[[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,1,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [1,3,2,4,6,5] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> ? = 5 + 1
[[0,1,0,0,0,0],[1,-1,0,1,0,0],[0,1,0,0,0,0],[0,0,1,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [1,3,2,4,6,5] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> ? = 5 + 1
[[0,0,1,0,0,0],[1,0,-1,1,0,0],[0,1,0,0,0,0],[0,0,1,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [1,3,2,4,6,5] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> ? = 5 + 1
[[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,1,0,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [1,3,2,4,6,5] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> ? = 5 + 1
[[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,0,0,1,0,0],[0,1,0,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [1,3,2,4,6,5] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> ? = 5 + 1
[[0,1,0,0,0,0],[0,0,1,0,0,0],[1,-1,0,1,0,0],[0,1,0,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [1,3,2,4,6,5] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> ? = 5 + 1
[[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [1,3,2,4,6,5] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> ? = 5 + 1
[[0,1,0,0,0,0],[1,-1,0,1,0,0],[0,0,1,0,0,0],[0,1,0,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [1,3,2,4,6,5] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> ? = 5 + 1
[[0,0,1,0,0,0],[1,0,-1,1,0,0],[0,0,1,0,0,0],[0,1,0,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [1,3,2,4,6,5] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> ? = 5 + 1
[[0,1,0,0,0,0],[0,0,0,1,0,0],[1,-1,1,0,0,0],[0,1,0,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [1,3,2,4,6,5] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> ? = 5 + 1
Description
The number of entries equal to positive one in the alternating sign matrix.
Matching statistic: St001480
(load all 28 compositions to match this statistic)
(load all 28 compositions to match this statistic)
Mp00007: Alternating sign matrices —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001480: Dyck paths ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 29%
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001480: Dyck paths ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 29%
Values
[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 4 = 3 + 1
[[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 4 = 3 + 1
[[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 4 = 3 + 1
[[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 4 = 3 + 1
[[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
=> [1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 4 = 3 + 1
[[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4 = 3 + 1
[[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 4 = 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]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 5 = 4 + 1
[[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> 5 = 4 + 1
[[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> 5 = 4 + 1
[[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> 5 = 4 + 1
[[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> 5 = 4 + 1
[[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> 5 = 4 + 1
[[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> 5 = 4 + 1
[[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> 5 = 4 + 1
[[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> 5 = 4 + 1
[[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> 5 = 4 + 1
[[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> 5 = 4 + 1
[[0,0,1,0,0],[1,0,-1,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> 5 = 4 + 1
[[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> 5 = 4 + 1
[[0,1,0,0,0],[1,-1,1,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> 5 = 4 + 1
[[0,1,0,0,0],[0,0,1,0,0],[1,-1,0,1,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> 5 = 4 + 1
[[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> 5 = 4 + 1
[[0,1,0,0,0],[1,-1,0,1,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> 5 = 4 + 1
[[0,0,1,0,0],[1,0,-1,1,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> 5 = 4 + 1
[[0,1,0,0,0],[0,0,0,1,0],[1,-1,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> 5 = 4 + 1
[[0,0,1,0,0],[0,1,-1,1,0],[1,-1,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> 5 = 4 + 1
[[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> 5 = 4 + 1
[[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> 5 = 4 + 1
[[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> 5 = 4 + 1
[[1,0,0,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> 5 = 4 + 1
[[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> 5 = 4 + 1
[[0,0,1,0,0],[1,0,-1,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> 5 = 4 + 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]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 5 = 4 + 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]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> 5 = 4 + 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]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> 5 = 4 + 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]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 5 = 4 + 1
[[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 5 = 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]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> 5 = 4 + 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]]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> 5 = 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]]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> 5 = 4 + 1
[[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]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> 5 = 4 + 1
[[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]]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> 5 = 4 + 1
[[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]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> 5 = 4 + 1
[[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]]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> 5 = 4 + 1
[[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]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> 5 = 4 + 1
[[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 5 = 4 + 1
[[0,1,0,0,0],[1,-1,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> 5 = 4 + 1
[[0,0,1,0,0],[1,0,-1,0,1],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> 5 = 4 + 1
[[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 5 + 1
[[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> ? = 5 + 1
[[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> ? = 5 + 1
[[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> ? = 5 + 1
[[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> ? = 5 + 1
[[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 5 + 1
[[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> ? = 4 + 1
[[0,1,0,0,0,0],[1,-1,0,1,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [1,1,0,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> ? = 4 + 1
[[0,0,1,0,0,0],[1,0,-1,1,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> ? = 4 + 1
[[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> ? = 4 + 1
[[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 4 + 1
[[0,1,0,0,0,0],[0,0,1,0,0,0],[1,-1,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 4 + 1
[[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]]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> ? = 1 + 1
[[0,1,0,0,0,0],[1,-1,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [1,1,0,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> ? = 1 + 1
[[0,0,1,0,0,0],[1,0,-1,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> ? = 1 + 1
[[0,1,0,0,0,0],[0,0,0,1,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [1,1,0,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> ? = 1 + 1
[[0,0,1,0,0,0],[0,1,-1,1,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> ? = 1 + 1
[[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 5 + 1
[[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]]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,1,0,0,0]
=> ? = 5 + 1
[[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [1,1,0,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,1,0,0,0]
=> ? = 5 + 1
[[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]]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> ? = 5 + 1
[[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,1,1,0,0,0]
=> ? = 5 + 1
[[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,0,1,1,0,0,0]
=> ? = 5 + 1
[[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]]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> ? = 4 + 1
[[0,1,0,0,0,0],[1,-1,0,1,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [1,1,0,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,1,1,0,0,0,1,1,0,0,0]
=> ? = 4 + 1
[[0,0,1,0,0,0],[1,0,-1,1,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [1,1,1,0,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,1,1,0,0,0]
=> ? = 4 + 1
[[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,1,1,0,0,0]
=> ? = 4 + 1
[[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,0,1,1,0,0,0]
=> ? = 4 + 1
[[0,1,0,0,0,0],[0,0,1,0,0,0],[1,-1,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,0,1,1,0,0,0]
=> ? = 4 + 1
[[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> ? = 1 + 1
[[0,1,0,0,0,0],[1,-1,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [1,1,0,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,1,1,0,0,0,1,1,0,0,0]
=> ? = 1 + 1
[[0,0,1,0,0,0],[1,0,-1,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [1,1,1,0,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,1,1,0,0,0]
=> ? = 1 + 1
[[0,1,0,0,0,0],[0,0,0,1,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [1,1,0,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,1,1,0,0,0,1,1,0,0,0]
=> ? = 1 + 1
[[0,0,1,0,0,0],[0,1,-1,1,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [1,1,1,0,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,1,1,0,0,0]
=> ? = 1 + 1
[[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 5 + 1
[[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,-1,1],[0,0,0,0,1,0]]
=> [1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,1,0,0,0]
=> ? = 5 + 1
[[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,1,0,0,0]
=> ? = 5 + 1
[[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> ? = 5 + 1
[[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> ? = 5 + 1
[[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 5 + 1
[[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,1,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,1,0,0,0]
=> ? = 5 + 1
[[0,1,0,0,0,0],[1,-1,0,1,0,0],[0,1,0,0,0,0],[0,0,1,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,1,0,0,0]
=> ? = 5 + 1
[[0,0,1,0,0,0],[1,0,-1,1,0,0],[0,1,0,0,0,0],[0,0,1,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,1,0,0,0]
=> ? = 5 + 1
[[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,1,0,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> ? = 5 + 1
[[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,0,0,1,0,0],[0,1,0,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 5 + 1
[[0,1,0,0,0,0],[0,0,1,0,0,0],[1,-1,0,1,0,0],[0,1,0,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 5 + 1
[[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,1,0,0,0]
=> ? = 5 + 1
[[0,1,0,0,0,0],[1,-1,0,1,0,0],[0,0,1,0,0,0],[0,1,0,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,1,0,0,0]
=> ? = 5 + 1
[[0,0,1,0,0,0],[1,0,-1,1,0,0],[0,0,1,0,0,0],[0,1,0,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,1,0,0,0]
=> ? = 5 + 1
[[0,1,0,0,0,0],[0,0,0,1,0,0],[1,-1,1,0,0,0],[0,1,0,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,1,0,0,0]
=> ? = 5 + 1
Description
The number of simple summands of the module J^2/J^3. Here J is the Jacobson radical of the Nakayama algebra algebra corresponding to the Dyck path.
Matching statistic: St001557
(load all 17 compositions to match this statistic)
(load all 17 compositions to match this statistic)
Mp00003: Alternating sign matrices —rotate counterclockwise⟶ Alternating sign matrices
Mp00002: Alternating sign matrices —to left key permutation⟶ Permutations
St001557: Permutations ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 29%
Mp00002: Alternating sign matrices —to left key permutation⟶ Permutations
St001557: Permutations ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 29%
Values
[[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]]
=> [4,3,2,1] => 2 = 3 - 1
[[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [[0,0,1,0],[0,0,0,1],[0,1,0,0],[1,0,0,0]]
=> [3,4,2,1] => 2 = 3 - 1
[[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> [[0,0,1,0],[0,1,-1,1],[0,0,1,0],[1,0,0,0]]
=> [2,4,3,1] => 2 = 3 - 1
[[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> [[0,0,1,0],[0,1,-1,1],[1,-1,1,0],[0,1,0,0]]
=> [1,4,3,2] => 2 = 3 - 1
[[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> [[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]]
=> [2,4,3,1] => 2 = 3 - 1
[[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
=> [[0,1,0,0],[0,0,0,1],[1,-1,1,0],[0,1,0,0]]
=> [1,4,3,2] => 2 = 3 - 1
[[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
=> [[0,1,0,0],[1,-1,0,1],[0,0,1,0],[0,1,0,0]]
=> [1,4,3,2] => 2 = 3 - 1
[[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> [1,4,3,2] => 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]]
=> [[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]]
=> [5,4,3,2,1] => 3 = 4 - 1
[[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[0,0,0,0,1],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0]]
=> [5,4,2,3,1] => 3 = 4 - 1
[[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[0,0,0,0,1],[0,0,0,1,0],[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0]]
=> [5,4,1,3,2] => 3 = 4 - 1
[[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[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]]
=> [4,5,3,2,1] => 3 = 4 - 1
[[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[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]]
=> [4,5,2,3,1] => 3 = 4 - 1
[[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0]]
=> [4,5,1,3,2] => 3 = 4 - 1
[[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0]]
=> [3,5,4,2,1] => 3 = 4 - 1
[[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[0,0,0,1,0],[0,0,1,-1,1],[0,1,-1,1,0],[0,0,1,0,0],[1,0,0,0,0]]
=> [2,5,4,3,1] => 3 = 4 - 1
[[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[0,0,0,1,0],[0,0,1,-1,1],[0,1,-1,1,0],[1,-1,1,0,0],[0,1,0,0,0]]
=> [1,5,4,3,2] => 3 = 4 - 1
[[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[0,0,0,1,0],[0,1,0,-1,1],[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0]]
=> [2,5,4,3,1] => 3 = 4 - 1
[[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[0,0,0,1,0],[0,1,0,-1,1],[0,0,0,1,0],[1,-1,1,0,0],[0,1,0,0,0]]
=> [1,5,4,3,2] => 3 = 4 - 1
[[0,0,1,0,0],[1,0,-1,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[0,0,0,1,0],[0,1,0,-1,1],[1,-1,0,1,0],[0,0,1,0,0],[0,1,0,0,0]]
=> [1,5,4,3,2] => 3 = 4 - 1
[[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [[0,0,0,1,0],[0,0,1,-1,1],[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0]]
=> [3,5,2,4,1] => 3 = 4 - 1
[[0,1,0,0,0],[1,-1,1,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [[0,0,0,1,0],[0,0,1,-1,1],[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0]]
=> [3,5,1,4,2] => 3 = 4 - 1
[[0,1,0,0,0],[0,0,1,0,0],[1,-1,0,1,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [[0,0,0,1,0],[0,0,1,-1,1],[0,1,0,0,0],[1,0,-1,1,0],[0,0,1,0,0]]
=> [2,5,1,4,3] => 3 = 4 - 1
[[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [[0,0,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0]]
=> [2,5,3,4,1] => 3 = 4 - 1
[[0,1,0,0,0],[1,-1,0,1,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [[0,0,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[1,-1,0,1,0],[0,1,0,0,0]]
=> [1,5,3,4,2] => 3 = 4 - 1
[[0,0,1,0,0],[1,0,-1,1,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [[0,0,0,1,0],[0,1,0,-1,1],[1,-1,1,0,0],[0,0,0,1,0],[0,1,0,0,0]]
=> [1,5,3,4,2] => 3 = 4 - 1
[[0,1,0,0,0],[0,0,0,1,0],[1,-1,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [[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]]
=> [2,5,1,4,3] => 3 = 4 - 1
[[0,0,1,0,0],[0,1,-1,1,0],[1,-1,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [[0,0,0,1,0],[0,1,0,-1,1],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,0,0]]
=> [1,5,2,4,3] => 3 = 4 - 1
[[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0]]
=> [3,5,4,2,1] => 3 = 4 - 1
[[1,0,0,0,0],[0,0,1,0,0],[0,1,-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],[0,0,1,0,0],[1,0,0,0,0]]
=> [2,5,4,3,1] => 3 = 4 - 1
[[0,1,0,0,0],[1,-1,1,0,0],[0,1,-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,-1,1,0,0],[0,1,0,0,0]]
=> [1,5,4,3,2] => 3 = 4 - 1
[[1,0,0,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [[0,0,1,0,0],[0,1,-1,0,1],[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0]]
=> [2,5,4,3,1] => 3 = 4 - 1
[[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [[0,0,1,0,0],[0,1,-1,0,1],[0,0,0,1,0],[1,-1,1,0,0],[0,1,0,0,0]]
=> [1,5,4,3,2] => 3 = 4 - 1
[[0,0,1,0,0],[1,0,-1,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [[0,0,1,0,0],[0,1,-1,0,1],[1,-1,0,1,0],[0,0,1,0,0],[0,1,0,0,0]]
=> [1,5,4,3,2] => 3 = 4 - 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]]
=> [[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]]
=> [2,5,4,3,1] => 3 = 4 - 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]]
=> [[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]]
=> [1,5,4,3,2] => 3 = 4 - 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]]
=> [[0,1,0,0,0],[0,0,0,0,1],[1,-1,0,1,0],[0,0,1,0,0],[0,1,0,0,0]]
=> [1,5,4,3,2] => 3 = 4 - 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]]
=> [[0,1,0,0,0],[1,-1,0,0,1],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0]]
=> [1,5,4,3,2] => 3 = 4 - 1
[[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0]]
=> [1,5,4,3,2] => 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]]
=> [[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]]
=> [3,5,2,4,1] => 3 = 4 - 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]]
=> [[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]]
=> [3,5,1,4,2] => 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]]
=> [[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]]
=> [2,5,1,4,3] => 3 = 4 - 1
[[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]]
=> [[0,0,1,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0]]
=> [2,5,3,4,1] => 3 = 4 - 1
[[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]]
=> [[0,0,1,0,0],[0,1,-1,0,1],[0,0,1,0,0],[1,-1,0,1,0],[0,1,0,0,0]]
=> [1,5,3,4,2] => 3 = 4 - 1
[[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]]
=> [[0,0,1,0,0],[0,1,-1,0,1],[1,-1,1,0,0],[0,0,0,1,0],[0,1,0,0,0]]
=> [1,5,3,4,2] => 3 = 4 - 1
[[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]]
=> [2,5,1,4,3] => 3 = 4 - 1
[[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,-1,0,1],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,0,0]]
=> [1,5,2,4,3] => 3 = 4 - 1
[[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0]]
=> [2,5,3,4,1] => 3 = 4 - 1
[[0,1,0,0,0],[1,-1,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[1,-1,0,1,0],[0,1,0,0,0]]
=> [1,5,3,4,2] => 3 = 4 - 1
[[0,0,1,0,0],[1,0,-1,0,1],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [[0,1,0,0,0],[0,0,0,0,1],[1,-1,1,0,0],[0,0,0,1,0],[0,1,0,0,0]]
=> [1,5,3,4,2] => 3 = 4 - 1
[[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0]]
=> [6,5,4,3,2,1] => ? = 5 - 1
[[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[1,0,0,0,0,0]]
=> [6,5,4,2,3,1] => ? = 5 - 1
[[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0]]
=> [6,5,4,1,3,2] => ? = 5 - 1
[[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0]]
=> [6,5,3,4,2,1] => ? = 5 - 1
[[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,0,0,0],[1,0,0,0,0,0]]
=> [6,5,2,4,3,1] => ? = 5 - 1
[[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,1,-1,1,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0]]
=> [6,5,1,4,3,2] => ? = 5 - 1
[[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[0,0,0,0,0,1],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[1,0,0,0,0,0]]
=> [6,5,2,4,3,1] => ? = 4 - 1
[[0,1,0,0,0,0],[1,-1,0,1,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[0,0,0,0,0,1],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0]]
=> [6,5,1,4,3,2] => ? = 4 - 1
[[0,0,1,0,0,0],[1,0,-1,1,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[0,0,0,0,0,1],[0,0,0,0,1,0],[0,1,0,0,0,0],[1,-1,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0]]
=> [6,5,1,4,3,2] => ? = 4 - 1
[[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[1,0,0,0,0,0]]
=> [6,5,3,2,4,1] => ? = 4 - 1
[[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[1,-1,0,1,0,0],[0,1,0,0,0,0]]
=> [6,5,3,1,4,2] => ? = 4 - 1
[[0,1,0,0,0,0],[0,0,1,0,0,0],[1,-1,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,-1,1,0,0],[0,0,1,0,0,0]]
=> [6,5,2,1,4,3] => ? = 4 - 1
[[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]]
=> [[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]]
=> [6,5,2,3,4,1] => ? = 1 - 1
[[0,1,0,0,0,0],[1,-1,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[0,0,0,0,0,1],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[1,-1,0,1,0,0],[0,1,0,0,0,0]]
=> [6,5,1,3,4,2] => ? = 1 - 1
[[0,0,1,0,0,0],[1,0,-1,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[0,0,0,0,0,1],[0,0,0,0,1,0],[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0]]
=> [6,5,1,3,4,2] => ? = 1 - 1
[[0,1,0,0,0,0],[0,0,0,1,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[0,0,0,0,0,1],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[1,0,-1,1,0,0],[0,0,1,0,0,0]]
=> [6,5,2,1,4,3] => ? = 1 - 1
[[0,0,1,0,0,0],[0,1,-1,1,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[0,0,0,0,0,1],[0,0,0,0,1,0],[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,0,0,0]]
=> [6,5,1,2,4,3] => ? = 1 - 1
[[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0]]
=> [5,6,4,3,2,1] => ? = 5 - 1
[[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]]
=> [[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]]
=> [5,6,4,2,3,1] => ? = 5 - 1
[[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0]]
=> [5,6,4,1,3,2] => ? = 5 - 1
[[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]]
=> [[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]]
=> [5,6,3,4,2,1] => ? = 5 - 1
[[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,0,0,0],[1,0,0,0,0,0]]
=> [5,6,2,4,3,1] => ? = 5 - 1
[[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,1,-1,1,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0]]
=> [5,6,1,4,3,2] => ? = 5 - 1
[[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]]
=> [[0,0,0,0,1,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[1,0,0,0,0,0]]
=> [5,6,2,4,3,1] => ? = 4 - 1
[[0,1,0,0,0,0],[1,-1,0,1,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[0,0,0,0,1,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,1,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0]]
=> [5,6,1,4,3,2] => ? = 4 - 1
[[0,0,1,0,0,0],[1,0,-1,1,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[0,0,0,0,1,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[1,-1,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0]]
=> [5,6,1,4,3,2] => ? = 4 - 1
[[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[1,0,0,0,0,0]]
=> [5,6,3,2,4,1] => ? = 4 - 1
[[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,1,0,0,0,0],[1,-1,0,1,0,0],[0,1,0,0,0,0]]
=> [5,6,3,1,4,2] => ? = 4 - 1
[[0,1,0,0,0,0],[0,0,1,0,0,0],[1,-1,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,-1,1,0,0],[0,0,1,0,0,0]]
=> [5,6,2,1,4,3] => ? = 4 - 1
[[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[0,0,0,0,1,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[1,0,0,0,0,0]]
=> [5,6,2,3,4,1] => ? = 1 - 1
[[0,1,0,0,0,0],[1,-1,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[0,0,0,0,1,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,1,0,0,0],[1,-1,0,1,0,0],[0,1,0,0,0,0]]
=> [5,6,1,3,4,2] => ? = 1 - 1
[[0,0,1,0,0,0],[1,0,-1,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[0,0,0,0,1,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0]]
=> [5,6,1,3,4,2] => ? = 1 - 1
[[0,1,0,0,0,0],[0,0,0,1,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[0,0,0,0,1,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,1,0,0,0],[1,0,-1,1,0,0],[0,0,1,0,0,0]]
=> [5,6,2,1,4,3] => ? = 1 - 1
[[0,0,1,0,0,0],[0,1,-1,1,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[0,0,0,0,1,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,0,0,0]]
=> [5,6,1,2,4,3] => ? = 1 - 1
[[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [[0,0,0,0,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0]]
=> [4,6,5,3,2,1] => ? = 5 - 1
[[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,-1,1],[0,0,0,0,1,0]]
=> [[0,0,0,0,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[1,0,0,0,0,0]]
=> [4,6,5,2,3,1] => ? = 5 - 1
[[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [[0,0,0,0,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0],[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0]]
=> [4,6,5,1,3,2] => ? = 5 - 1
[[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [[0,0,0,0,1,0],[0,0,0,1,-1,1],[0,0,1,-1,1,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0]]
=> [3,6,5,4,2,1] => ? = 5 - 1
[[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [[0,0,0,0,1,0],[0,0,0,1,-1,1],[0,0,1,-1,1,0],[0,1,-1,1,0,0],[0,0,1,0,0,0],[1,0,0,0,0,0]]
=> [2,6,5,4,3,1] => ? = 5 - 1
[[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [[0,0,0,0,1,0],[0,0,0,1,-1,1],[0,0,1,-1,1,0],[0,1,-1,1,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0]]
=> [1,6,5,4,3,2] => ? = 5 - 1
[[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,1,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [[0,0,0,0,1,0],[0,0,0,1,-1,1],[0,1,0,-1,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[1,0,0,0,0,0]]
=> [2,6,5,4,3,1] => ? = 5 - 1
[[0,1,0,0,0,0],[1,-1,0,1,0,0],[0,1,0,0,0,0],[0,0,1,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [[0,0,0,0,1,0],[0,0,0,1,-1,1],[0,1,0,-1,1,0],[0,0,0,1,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0]]
=> [1,6,5,4,3,2] => ? = 5 - 1
[[0,0,1,0,0,0],[1,0,-1,1,0,0],[0,1,0,0,0,0],[0,0,1,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [[0,0,0,0,1,0],[0,0,0,1,-1,1],[0,1,0,-1,1,0],[1,-1,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0]]
=> [1,6,5,4,3,2] => ? = 5 - 1
[[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,1,0,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [[0,0,0,0,1,0],[0,0,0,1,-1,1],[0,0,1,-1,1,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[1,0,0,0,0,0]]
=> [3,6,5,2,4,1] => ? = 5 - 1
[[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,0,0,1,0,0],[0,1,0,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [[0,0,0,0,1,0],[0,0,0,1,-1,1],[0,0,1,-1,1,0],[0,1,0,0,0,0],[1,-1,0,1,0,0],[0,1,0,0,0,0]]
=> [3,6,5,1,4,2] => ? = 5 - 1
[[0,1,0,0,0,0],[0,0,1,0,0,0],[1,-1,0,1,0,0],[0,1,0,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [[0,0,0,0,1,0],[0,0,0,1,-1,1],[0,0,1,-1,1,0],[0,1,0,0,0,0],[1,0,-1,1,0,0],[0,0,1,0,0,0]]
=> [2,6,5,1,4,3] => ? = 5 - 1
[[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [[0,0,0,0,1,0],[0,0,0,1,-1,1],[0,1,0,-1,1,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[1,0,0,0,0,0]]
=> [2,6,5,3,4,1] => ? = 5 - 1
[[0,1,0,0,0,0],[1,-1,0,1,0,0],[0,0,1,0,0,0],[0,1,0,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [[0,0,0,0,1,0],[0,0,0,1,-1,1],[0,1,0,-1,1,0],[0,0,1,0,0,0],[1,-1,0,1,0,0],[0,1,0,0,0,0]]
=> [1,6,5,3,4,2] => ? = 5 - 1
[[0,0,1,0,0,0],[1,0,-1,1,0,0],[0,0,1,0,0,0],[0,1,0,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [[0,0,0,0,1,0],[0,0,0,1,-1,1],[0,1,0,-1,1,0],[1,-1,1,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0]]
=> [1,6,5,3,4,2] => ? = 5 - 1
[[0,1,0,0,0,0],[0,0,0,1,0,0],[1,-1,1,0,0,0],[0,1,0,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [[0,0,0,0,1,0],[0,0,0,1,-1,1],[0,1,0,-1,1,0],[0,0,1,0,0,0],[1,0,-1,1,0,0],[0,0,1,0,0,0]]
=> [2,6,5,1,4,3] => ? = 5 - 1
Description
The number of inversions of the second entry of a permutation.
This is, for a permutation $\pi$ of length $n$,
$$\# \{2 < k \leq n \mid \pi(2) > \pi(k)\}.$$
The number of inversions of the first entry is [[St000054]] and the number of inversions of the third entry is [[St001556]]. The sequence of inversions of all the entries define the [[http://www.findstat.org/Permutations#The_Lehmer_code_and_the_major_code_of_a_permutation|Lehmer code]] of a permutation.
Matching statistic: St001958
(load all 32 compositions to match this statistic)
(load all 32 compositions to match this statistic)
Mp00007: Alternating sign matrices —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001958: Permutations ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 29%
Mp00201: Dyck paths —Ringel⟶ Permutations
St001958: Permutations ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 29%
Values
[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 4 = 3 + 1
[[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 4 = 3 + 1
[[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 4 = 3 + 1
[[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => 4 = 3 + 1
[[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 4 = 3 + 1
[[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
=> [1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => 4 = 3 + 1
[[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
=> [1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => 4 = 3 + 1
[[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 4 = 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]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => 5 = 4 + 1
[[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => 5 = 4 + 1
[[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => 5 = 4 + 1
[[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => 5 = 4 + 1
[[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => 5 = 4 + 1
[[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => 5 = 4 + 1
[[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => 5 = 4 + 1
[[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => 5 = 4 + 1
[[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => 5 = 4 + 1
[[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => 5 = 4 + 1
[[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => 5 = 4 + 1
[[0,0,1,0,0],[1,0,-1,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,1,1,0,1,0,0,1,0,0]
=> [6,3,5,1,2,4] => 5 = 4 + 1
[[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => 5 = 4 + 1
[[0,1,0,0,0],[1,-1,1,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => 5 = 4 + 1
[[0,1,0,0,0],[0,0,1,0,0],[1,-1,0,1,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => 5 = 4 + 1
[[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => 5 = 4 + 1
[[0,1,0,0,0],[1,-1,0,1,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => 5 = 4 + 1
[[0,0,1,0,0],[1,0,-1,1,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [1,1,1,0,1,0,0,1,0,0]
=> [6,3,5,1,2,4] => 5 = 4 + 1
[[0,1,0,0,0],[0,0,0,1,0],[1,-1,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => 5 = 4 + 1
[[0,0,1,0,0],[0,1,-1,1,0],[1,-1,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [1,1,1,0,1,0,0,1,0,0]
=> [6,3,5,1,2,4] => 5 = 4 + 1
[[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => 5 = 4 + 1
[[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => 5 = 4 + 1
[[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => 5 = 4 + 1
[[1,0,0,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => 5 = 4 + 1
[[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => 5 = 4 + 1
[[0,0,1,0,0],[1,0,-1,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [6,5,4,1,2,3] => 5 = 4 + 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]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => 5 = 4 + 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]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => 5 = 4 + 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]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [5,3,4,1,6,2] => 5 = 4 + 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]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [6,3,4,5,1,2] => 5 = 4 + 1
[[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => 5 = 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]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => 5 = 4 + 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]]
=> [1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => 5 = 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]]
=> [1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => 5 = 4 + 1
[[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]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => 5 = 4 + 1
[[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]]
=> [1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => 5 = 4 + 1
[[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]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [6,5,4,1,2,3] => 5 = 4 + 1
[[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]]
=> [1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => 5 = 4 + 1
[[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]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [6,5,4,1,2,3] => 5 = 4 + 1
[[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => 5 = 4 + 1
[[0,1,0,0,0],[1,-1,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => 5 = 4 + 1
[[0,0,1,0,0],[1,0,-1,0,1],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [5,3,4,1,6,2] => 5 = 4 + 1
[[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,1,2,3,4,5,6] => ? = 5 + 1
[[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [3,1,7,2,4,5,6] => ? = 5 + 1
[[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> [7,3,1,2,4,5,6] => ? = 5 + 1
[[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [4,1,2,7,3,5,6] => ? = 5 + 1
[[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [7,1,4,2,3,5,6] => ? = 5 + 1
[[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> [7,4,1,2,3,5,6] => ? = 5 + 1
[[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [3,1,4,7,2,5,6] => ? = 4 + 1
[[0,1,0,0,0,0],[1,-1,0,1,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [1,1,0,1,1,0,0,0,1,0,1,0]
=> [4,3,1,7,2,5,6] => ? = 4 + 1
[[0,0,1,0,0,0],[1,0,-1,1,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> [7,3,4,1,2,5,6] => ? = 4 + 1
[[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [7,1,4,2,3,5,6] => ? = 4 + 1
[[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> [7,4,1,2,3,5,6] => ? = 4 + 1
[[0,1,0,0,0,0],[0,0,1,0,0,0],[1,-1,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> [7,4,1,2,3,5,6] => ? = 4 + 1
[[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]]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [3,1,4,7,2,5,6] => ? = 1 + 1
[[0,1,0,0,0,0],[1,-1,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [1,1,0,1,1,0,0,0,1,0,1,0]
=> [4,3,1,7,2,5,6] => ? = 1 + 1
[[0,0,1,0,0,0],[1,0,-1,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> [7,3,4,1,2,5,6] => ? = 1 + 1
[[0,1,0,0,0,0],[0,0,0,1,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [1,1,0,1,1,0,0,0,1,0,1,0]
=> [4,3,1,7,2,5,6] => ? = 1 + 1
[[0,0,1,0,0,0],[0,1,-1,1,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> [7,3,4,1,2,5,6] => ? = 1 + 1
[[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,1,2,3,4,7,5] => ? = 5 + 1
[[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]]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> [3,1,6,2,4,7,5] => ? = 5 + 1
[[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [1,1,0,1,0,0,1,0,1,1,0,0]
=> [6,3,1,2,4,7,5] => ? = 5 + 1
[[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]]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,1,2,6,3,7,5] => ? = 5 + 1
[[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [1,0,1,1,0,1,0,0,1,1,0,0]
=> [6,1,4,2,3,7,5] => ? = 5 + 1
[[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [1,1,0,1,0,1,0,0,1,1,0,0]
=> [6,4,1,2,3,7,5] => ? = 5 + 1
[[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]]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> [3,1,4,6,2,7,5] => ? = 4 + 1
[[0,1,0,0,0,0],[1,-1,0,1,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [1,1,0,1,1,0,0,0,1,1,0,0]
=> [4,3,1,6,2,7,5] => ? = 4 + 1
[[0,0,1,0,0,0],[1,0,-1,1,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [1,1,1,0,1,0,0,0,1,1,0,0]
=> [6,3,4,1,2,7,5] => ? = 4 + 1
[[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [1,0,1,1,0,1,0,0,1,1,0,0]
=> [6,1,4,2,3,7,5] => ? = 4 + 1
[[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [1,1,0,1,0,1,0,0,1,1,0,0]
=> [6,4,1,2,3,7,5] => ? = 4 + 1
[[0,1,0,0,0,0],[0,0,1,0,0,0],[1,-1,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [1,1,0,1,0,1,0,0,1,1,0,0]
=> [6,4,1,2,3,7,5] => ? = 4 + 1
[[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> [3,1,4,6,2,7,5] => ? = 1 + 1
[[0,1,0,0,0,0],[1,-1,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [1,1,0,1,1,0,0,0,1,1,0,0]
=> [4,3,1,6,2,7,5] => ? = 1 + 1
[[0,0,1,0,0,0],[1,0,-1,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [1,1,1,0,1,0,0,0,1,1,0,0]
=> [6,3,4,1,2,7,5] => ? = 1 + 1
[[0,1,0,0,0,0],[0,0,0,1,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [1,1,0,1,1,0,0,0,1,1,0,0]
=> [4,3,1,6,2,7,5] => ? = 1 + 1
[[0,0,1,0,0,0],[0,1,-1,1,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [1,1,1,0,1,0,0,0,1,1,0,0]
=> [6,3,4,1,2,7,5] => ? = 1 + 1
[[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,1,2,3,6,4,5] => ? = 5 + 1
[[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,-1,1],[0,0,0,0,1,0]]
=> [1,0,1,1,0,0,1,1,0,1,0,0]
=> [3,1,7,2,6,4,5] => ? = 5 + 1
[[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> [7,3,1,2,6,4,5] => ? = 5 + 1
[[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [7,1,2,6,3,4,5] => ? = 5 + 1
[[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [6,1,7,2,3,4,5] => ? = 5 + 1
[[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [7,6,1,2,3,4,5] => ? = 5 + 1
[[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,1,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [3,1,7,6,2,4,5] => ? = 5 + 1
[[0,1,0,0,0,0],[1,-1,0,1,0,0],[0,1,0,0,0,0],[0,0,1,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [7,3,1,6,2,4,5] => ? = 5 + 1
[[0,0,1,0,0,0],[1,0,-1,1,0,0],[0,1,0,0,0,0],[0,0,1,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [6,3,7,1,2,4,5] => ? = 5 + 1
[[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,1,0,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [6,1,7,2,3,4,5] => ? = 5 + 1
[[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,0,0,1,0,0],[0,1,0,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [7,6,1,2,3,4,5] => ? = 5 + 1
[[0,1,0,0,0,0],[0,0,1,0,0,0],[1,-1,0,1,0,0],[0,1,0,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [7,6,1,2,3,4,5] => ? = 5 + 1
[[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [3,1,7,6,2,4,5] => ? = 5 + 1
[[0,1,0,0,0,0],[1,-1,0,1,0,0],[0,0,1,0,0,0],[0,1,0,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [7,3,1,6,2,4,5] => ? = 5 + 1
[[0,0,1,0,0,0],[1,0,-1,1,0,0],[0,0,1,0,0,0],[0,1,0,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [6,3,7,1,2,4,5] => ? = 5 + 1
[[0,1,0,0,0,0],[0,0,0,1,0,0],[1,-1,1,0,0,0],[0,1,0,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [7,3,1,6,2,4,5] => ? = 5 + 1
Description
The degree of the polynomial interpolating the values of a permutation.
Given a permutation $\pi\in\mathfrak S_n$ there is a polynomial $p$ of minimal degree such that $p(n)=\pi(n)$ for $n\in\{1,\dots,n\}$.
This statistic records the degree of $p$.
Matching statistic: St000501
(load all 41 compositions to match this statistic)
(load all 41 compositions to match this statistic)
Mp00007: Alternating sign matrices —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000501: Permutations ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 29%
Mp00201: Dyck paths —Ringel⟶ Permutations
St000501: Permutations ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 29%
Values
[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 5 = 3 + 2
[[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 5 = 3 + 2
[[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 5 = 3 + 2
[[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => 5 = 3 + 2
[[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 5 = 3 + 2
[[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
=> [1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => 5 = 3 + 2
[[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
=> [1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => 5 = 3 + 2
[[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 5 = 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]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => 6 = 4 + 2
[[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => 6 = 4 + 2
[[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => 6 = 4 + 2
[[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => 6 = 4 + 2
[[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => 6 = 4 + 2
[[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => 6 = 4 + 2
[[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => 6 = 4 + 2
[[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => 6 = 4 + 2
[[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => 6 = 4 + 2
[[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => 6 = 4 + 2
[[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => 6 = 4 + 2
[[0,0,1,0,0],[1,0,-1,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,1,1,0,1,0,0,1,0,0]
=> [6,3,5,1,2,4] => 6 = 4 + 2
[[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => 6 = 4 + 2
[[0,1,0,0,0],[1,-1,1,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => 6 = 4 + 2
[[0,1,0,0,0],[0,0,1,0,0],[1,-1,0,1,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => 6 = 4 + 2
[[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => 6 = 4 + 2
[[0,1,0,0,0],[1,-1,0,1,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => 6 = 4 + 2
[[0,0,1,0,0],[1,0,-1,1,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [1,1,1,0,1,0,0,1,0,0]
=> [6,3,5,1,2,4] => 6 = 4 + 2
[[0,1,0,0,0],[0,0,0,1,0],[1,-1,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => 6 = 4 + 2
[[0,0,1,0,0],[0,1,-1,1,0],[1,-1,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [1,1,1,0,1,0,0,1,0,0]
=> [6,3,5,1,2,4] => 6 = 4 + 2
[[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => 6 = 4 + 2
[[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => 6 = 4 + 2
[[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => 6 = 4 + 2
[[1,0,0,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => 6 = 4 + 2
[[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => 6 = 4 + 2
[[0,0,1,0,0],[1,0,-1,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [6,5,4,1,2,3] => 6 = 4 + 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]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => 6 = 4 + 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]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => 6 = 4 + 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]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [5,3,4,1,6,2] => 6 = 4 + 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]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [6,3,4,5,1,2] => 6 = 4 + 2
[[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => 6 = 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]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => 6 = 4 + 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]]
=> [1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => 6 = 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]]
=> [1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => 6 = 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]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => 6 = 4 + 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]]
=> [1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => 6 = 4 + 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]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [6,5,4,1,2,3] => 6 = 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]]
=> [1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => 6 = 4 + 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]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [6,5,4,1,2,3] => 6 = 4 + 2
[[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => 6 = 4 + 2
[[0,1,0,0,0],[1,-1,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => 6 = 4 + 2
[[0,0,1,0,0],[1,0,-1,0,1],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [5,3,4,1,6,2] => 6 = 4 + 2
[[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,1,2,3,4,5,6] => ? = 5 + 2
[[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [3,1,7,2,4,5,6] => ? = 5 + 2
[[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> [7,3,1,2,4,5,6] => ? = 5 + 2
[[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [4,1,2,7,3,5,6] => ? = 5 + 2
[[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [7,1,4,2,3,5,6] => ? = 5 + 2
[[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> [7,4,1,2,3,5,6] => ? = 5 + 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],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [3,1,4,7,2,5,6] => ? = 4 + 2
[[0,1,0,0,0,0],[1,-1,0,1,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [1,1,0,1,1,0,0,0,1,0,1,0]
=> [4,3,1,7,2,5,6] => ? = 4 + 2
[[0,0,1,0,0,0],[1,0,-1,1,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> [7,3,4,1,2,5,6] => ? = 4 + 2
[[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [7,1,4,2,3,5,6] => ? = 4 + 2
[[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> [7,4,1,2,3,5,6] => ? = 4 + 2
[[0,1,0,0,0,0],[0,0,1,0,0,0],[1,-1,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> [7,4,1,2,3,5,6] => ? = 4 + 2
[[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]]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [3,1,4,7,2,5,6] => ? = 1 + 2
[[0,1,0,0,0,0],[1,-1,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [1,1,0,1,1,0,0,0,1,0,1,0]
=> [4,3,1,7,2,5,6] => ? = 1 + 2
[[0,0,1,0,0,0],[1,0,-1,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> [7,3,4,1,2,5,6] => ? = 1 + 2
[[0,1,0,0,0,0],[0,0,0,1,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [1,1,0,1,1,0,0,0,1,0,1,0]
=> [4,3,1,7,2,5,6] => ? = 1 + 2
[[0,0,1,0,0,0],[0,1,-1,1,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> [7,3,4,1,2,5,6] => ? = 1 + 2
[[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,1,2,3,4,7,5] => ? = 5 + 2
[[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]]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> [3,1,6,2,4,7,5] => ? = 5 + 2
[[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [1,1,0,1,0,0,1,0,1,1,0,0]
=> [6,3,1,2,4,7,5] => ? = 5 + 2
[[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]]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,1,2,6,3,7,5] => ? = 5 + 2
[[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [1,0,1,1,0,1,0,0,1,1,0,0]
=> [6,1,4,2,3,7,5] => ? = 5 + 2
[[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [1,1,0,1,0,1,0,0,1,1,0,0]
=> [6,4,1,2,3,7,5] => ? = 5 + 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]]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> [3,1,4,6,2,7,5] => ? = 4 + 2
[[0,1,0,0,0,0],[1,-1,0,1,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [1,1,0,1,1,0,0,0,1,1,0,0]
=> [4,3,1,6,2,7,5] => ? = 4 + 2
[[0,0,1,0,0,0],[1,0,-1,1,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [1,1,1,0,1,0,0,0,1,1,0,0]
=> [6,3,4,1,2,7,5] => ? = 4 + 2
[[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [1,0,1,1,0,1,0,0,1,1,0,0]
=> [6,1,4,2,3,7,5] => ? = 4 + 2
[[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [1,1,0,1,0,1,0,0,1,1,0,0]
=> [6,4,1,2,3,7,5] => ? = 4 + 2
[[0,1,0,0,0,0],[0,0,1,0,0,0],[1,-1,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [1,1,0,1,0,1,0,0,1,1,0,0]
=> [6,4,1,2,3,7,5] => ? = 4 + 2
[[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> [3,1,4,6,2,7,5] => ? = 1 + 2
[[0,1,0,0,0,0],[1,-1,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [1,1,0,1,1,0,0,0,1,1,0,0]
=> [4,3,1,6,2,7,5] => ? = 1 + 2
[[0,0,1,0,0,0],[1,0,-1,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [1,1,1,0,1,0,0,0,1,1,0,0]
=> [6,3,4,1,2,7,5] => ? = 1 + 2
[[0,1,0,0,0,0],[0,0,0,1,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [1,1,0,1,1,0,0,0,1,1,0,0]
=> [4,3,1,6,2,7,5] => ? = 1 + 2
[[0,0,1,0,0,0],[0,1,-1,1,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [1,1,1,0,1,0,0,0,1,1,0,0]
=> [6,3,4,1,2,7,5] => ? = 1 + 2
[[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,1,2,3,6,4,5] => ? = 5 + 2
[[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,-1,1],[0,0,0,0,1,0]]
=> [1,0,1,1,0,0,1,1,0,1,0,0]
=> [3,1,7,2,6,4,5] => ? = 5 + 2
[[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> [7,3,1,2,6,4,5] => ? = 5 + 2
[[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [7,1,2,6,3,4,5] => ? = 5 + 2
[[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [6,1,7,2,3,4,5] => ? = 5 + 2
[[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [7,6,1,2,3,4,5] => ? = 5 + 2
[[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,1,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [3,1,7,6,2,4,5] => ? = 5 + 2
[[0,1,0,0,0,0],[1,-1,0,1,0,0],[0,1,0,0,0,0],[0,0,1,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [7,3,1,6,2,4,5] => ? = 5 + 2
[[0,0,1,0,0,0],[1,0,-1,1,0,0],[0,1,0,0,0,0],[0,0,1,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [6,3,7,1,2,4,5] => ? = 5 + 2
[[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,1,0,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [6,1,7,2,3,4,5] => ? = 5 + 2
[[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,0,0,1,0,0],[0,1,0,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [7,6,1,2,3,4,5] => ? = 5 + 2
[[0,1,0,0,0,0],[0,0,1,0,0,0],[1,-1,0,1,0,0],[0,1,0,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [7,6,1,2,3,4,5] => ? = 5 + 2
[[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [3,1,7,6,2,4,5] => ? = 5 + 2
[[0,1,0,0,0,0],[1,-1,0,1,0,0],[0,0,1,0,0,0],[0,1,0,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [7,3,1,6,2,4,5] => ? = 5 + 2
[[0,0,1,0,0,0],[1,0,-1,1,0,0],[0,0,1,0,0,0],[0,1,0,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [6,3,7,1,2,4,5] => ? = 5 + 2
[[0,1,0,0,0,0],[0,0,0,1,0,0],[1,-1,1,0,0,0],[0,1,0,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [7,3,1,6,2,4,5] => ? = 5 + 2
Description
The size of the first part in the decomposition of a permutation.
For a permutation $\pi$ of $\{1,\ldots,n\}$, this is defined to be the smallest $k > 0$ such that $\{\pi(1),\ldots,\pi(k)\} = \{1,\ldots,k\}$. This statistic is undefined for the empty permutation.
For the number of parts in the decomposition see [[St000056]].
The following 82 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000673The number of non-fixed points of a permutation. St000844The size of the largest block in the direct sum decomposition of a permutation. St001556The number of inversions of the third entry of a permutation. St001190Number of simple modules with projective dimension at most 4 in the corresponding Nakayama algebra. St001468The smallest fixpoint of a permutation. St000199The column of the unique '1' in the last row of the alternating sign matrix. St000327The number of cover relations in a poset. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000829The Ulam distance of a permutation to the identity permutation. St001430The number of positive entries in a signed permutation. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St001925The minimal number of zeros in a row of an alternating sign matrix. St001927Sparre Andersen's number of positives of a signed permutation. St000029The depth of a permutation. St000051The size of the left subtree of a binary tree. St000067The inversion number of the alternating sign matrix. St000209Maximum difference of elements in cycles. St000210Minimum over maximum difference of elements in cycles. St000216The absolute length of a permutation. St000653The last descent of a permutation. St000809The reduced reflection length of the permutation. St000890The number of nonzero entries in an alternating sign matrix. St000956The maximal displacement of a permutation. St000957The number of Bruhat lower covers of a permutation. St001002Number of indecomposable modules with projective and injective dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). St001077The prefix exchange distance of a permutation. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001429The number of negative entries in a signed permutation. St001511The minimal number of transpositions needed to sort a permutation in either direction. St001558The number of transpositions that are smaller or equal to a permutation in Bruhat order. St001579The number of cyclically simple transpositions decreasing the number of cyclic descents needed to sort a permutation. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001827The number of two-component spanning forests of a graph. St001869The maximum cut size of a graph. St001873For a Nakayama algebra corresponding to a Dyck path, we define the matrix C with entries the Hom-spaces between $e_i J$ and $e_j J$ (the radical of the indecomposable projective modules). St001907The number of Bastidas - Hohlweg - Saliola excedances of a signed permutation. St000144The pyramid weight of the Dyck path. St000189The number of elements in the poset. St000235The number of indices that are not cyclical small weak excedances. St000240The number of indices that are not small excedances. St000288The number of ones in a binary word. St000336The leg major index of a standard tableau. St000485The length of the longest cycle of a permutation. St000487The length of the shortest cycle of a permutation. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000949Gives the number of generalised tilting modules of the corresponding LNakayama algebra. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001018Sum of projective dimension of the indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001040The depth of the decreasing labelled binary unordered tree associated with the perfect matching. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001520The number of strict 3-descents. St001948The number of augmented double ascents of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St000026The position of the first return of a Dyck path. St001003The number of indecomposable modules with projective dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St001023Number of simple modules with projective dimension at most 3 in the Nakayama algebra corresponding to the Dyck path. St001065Number of indecomposable reflexive modules in the corresponding Nakayama algebra. St001179Number of indecomposable injective modules with projective dimension at most 2 in the corresponding Nakayama algebra. St001237The number of simple modules with injective dimension at most one or dominant dimension at least one. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St001255The vector space dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001650The order of Ringel's homological bijection associated to the linear Nakayama algebra corresponding to the Dyck path. St000967The value p(1) for the Coxeterpolynomial p of the corresponding LNakayama algebra. St001218Smallest index k greater than or equal to one such that the Coxeter matrix C of the corresponding Nakayama algebra has C^k=1. St001817The number of flag weak exceedances of a signed permutation. St001892The flag excedance statistic of a signed permutation. St001896The number of right descents of a signed permutations. St001864The number of excedances of a signed permutation. St001866The nesting alignments of a signed permutation. St001964The interval resolution global dimension of a poset. St001637The number of (upper) dissectors of a poset. St001668The number of points of the poset minus the width of the poset. St000010The length of the partition. St000784The maximum of the length and the largest part of the integer partition. St000147The largest part of an integer partition. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment.
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