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Your data matches 72 different statistics following compositions of up to 3 maps.
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Matching statistic: St000101
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(load all 3 compositions to match this statistic)
St000101: Semistandard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
=> 0
[[2,2]]
=> 0
[[1],[2]]
=> 1
[[1,1,2]]
=> 0
[[1,2,2]]
=> 0
[[2,2,2]]
=> 0
[[1,1],[2]]
=> 1
[[1,2],[2]]
=> 1
[[1,1,3]]
=> 0
[[1,2,3]]
=> 0
[[1,3,3]]
=> 0
[[2,2,3]]
=> 0
[[2,3,3]]
=> 0
[[3,3,3]]
=> 0
[[1,1],[3]]
=> 1
[[1,2],[3]]
=> 1
[[1,3],[2]]
=> 2
[[1,3],[3]]
=> 1
[[2,2],[3]]
=> 1
[[2,3],[3]]
=> 1
[[1],[2],[3]]
=> 3
[[1,1,1,2]]
=> 0
[[1,1,2,2]]
=> 0
[[1,2,2,2]]
=> 0
[[2,2,2,2]]
=> 0
[[1,1,1],[2]]
=> 1
[[1,1,2],[2]]
=> 1
[[1,2,2],[2]]
=> 1
[[1,1],[2,2]]
=> 2
[[1,1,1,3]]
=> 0
[[1,1,2,3]]
=> 0
[[1,1,3,3]]
=> 0
[[1,2,2,3]]
=> 0
[[1,2,3,3]]
=> 0
[[1,3,3,3]]
=> 0
[[2,2,2,3]]
=> 0
[[2,2,3,3]]
=> 0
[[2,3,3,3]]
=> 0
[[3,3,3,3]]
=> 0
[[1,1,1],[3]]
=> 1
[[1,1,2],[3]]
=> 1
[[1,1,3],[2]]
=> 2
[[1,1,3],[3]]
=> 1
[[1,2,2],[3]]
=> 1
[[1,2,3],[2]]
=> 2
[[1,2,3],[3]]
=> 1
[[1,3,3],[2]]
=> 2
[[1,3,3],[3]]
=> 1
[[2,2,2],[3]]
=> 1
[[2,2,3],[3]]
=> 1
Description
The cocharge of a semistandard tableau.
Matching statistic: St000454
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(load all 12 compositions to match this statistic)
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00318: Graphs —dual on components⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 45% ●values known / values provided: 49%●distinct values known / distinct values provided: 45%
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00318: Graphs —dual on components⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 45% ●values known / values provided: 49%●distinct values known / distinct values provided: 45%
Values
[[1,2]]
=> [1,2] => ([],2)
=> ([],2)
=> 0
[[2,2]]
=> [1,2] => ([],2)
=> ([],2)
=> 0
[[1],[2]]
=> [2,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[1,1,2]]
=> [1,2,3] => ([],3)
=> ([],3)
=> 0
[[1,2,2]]
=> [1,2,3] => ([],3)
=> ([],3)
=> 0
[[2,2,2]]
=> [1,2,3] => ([],3)
=> ([],3)
=> 0
[[1,1],[2]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ? = 1
[[1,2],[2]]
=> [2,1,3] => ([(1,2)],3)
=> ([(1,2)],3)
=> 1
[[1,1,3]]
=> [1,2,3] => ([],3)
=> ([],3)
=> 0
[[1,2,3]]
=> [1,2,3] => ([],3)
=> ([],3)
=> 0
[[1,3,3]]
=> [1,2,3] => ([],3)
=> ([],3)
=> 0
[[2,2,3]]
=> [1,2,3] => ([],3)
=> ([],3)
=> 0
[[2,3,3]]
=> [1,2,3] => ([],3)
=> ([],3)
=> 0
[[3,3,3]]
=> [1,2,3] => ([],3)
=> ([],3)
=> 0
[[1,1],[3]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ? ∊ {1,1,3}
[[1,2],[3]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ? ∊ {1,1,3}
[[1,3],[2]]
=> [2,1,3] => ([(1,2)],3)
=> ([(1,2)],3)
=> 1
[[1,3],[3]]
=> [2,1,3] => ([(1,2)],3)
=> ([(1,2)],3)
=> 1
[[2,2],[3]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ? ∊ {1,1,3}
[[2,3],[3]]
=> [2,1,3] => ([(1,2)],3)
=> ([(1,2)],3)
=> 1
[[1],[2],[3]]
=> [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
[[1,1,1,2]]
=> [1,2,3,4] => ([],4)
=> ([],4)
=> 0
[[1,1,2,2]]
=> [1,2,3,4] => ([],4)
=> ([],4)
=> 0
[[1,2,2,2]]
=> [1,2,3,4] => ([],4)
=> ([],4)
=> 0
[[2,2,2,2]]
=> [1,2,3,4] => ([],4)
=> ([],4)
=> 0
[[1,1,1],[2]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {1,1}
[[1,1,2],[2]]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? ∊ {1,1}
[[1,2,2],[2]]
=> [2,1,3,4] => ([(2,3)],4)
=> ([(2,3)],4)
=> 1
[[1,1],[2,2]]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[[1,1,1,3]]
=> [1,2,3,4] => ([],4)
=> ([],4)
=> 0
[[1,1,2,3]]
=> [1,2,3,4] => ([],4)
=> ([],4)
=> 0
[[1,1,3,3]]
=> [1,2,3,4] => ([],4)
=> ([],4)
=> 0
[[1,2,2,3]]
=> [1,2,3,4] => ([],4)
=> ([],4)
=> 0
[[1,2,3,3]]
=> [1,2,3,4] => ([],4)
=> ([],4)
=> 0
[[1,3,3,3]]
=> [1,2,3,4] => ([],4)
=> ([],4)
=> 0
[[2,2,2,3]]
=> [1,2,3,4] => ([],4)
=> ([],4)
=> 0
[[2,2,3,3]]
=> [1,2,3,4] => ([],4)
=> ([],4)
=> 0
[[2,3,3,3]]
=> [1,2,3,4] => ([],4)
=> ([],4)
=> 0
[[3,3,3,3]]
=> [1,2,3,4] => ([],4)
=> ([],4)
=> 0
[[1,1,1],[3]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,2,2,2,3,3,3}
[[1,1,2],[3]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,2,2,2,3,3,3}
[[1,1,3],[2]]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,2,2,2,3,3,3}
[[1,1,3],[3]]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,2,2,2,3,3,3}
[[1,2,2],[3]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,2,2,2,3,3,3}
[[1,2,3],[2]]
=> [2,1,3,4] => ([(2,3)],4)
=> ([(2,3)],4)
=> 1
[[1,2,3],[3]]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,2,2,2,3,3,3}
[[1,3,3],[2]]
=> [2,1,3,4] => ([(2,3)],4)
=> ([(2,3)],4)
=> 1
[[1,3,3],[3]]
=> [2,1,3,4] => ([(2,3)],4)
=> ([(2,3)],4)
=> 1
[[2,2,2],[3]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,2,2,2,3,3,3}
[[2,2,3],[3]]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,2,2,2,3,3,3}
[[2,3,3],[3]]
=> [2,1,3,4] => ([(2,3)],4)
=> ([(2,3)],4)
=> 1
[[1,1],[2,3]]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[[1,1],[3,3]]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[[1,2],[2,3]]
=> [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ? ∊ {1,1,1,1,1,2,2,2,3,3,3}
[[1,2],[3,3]]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[[2,2],[3,3]]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[[1,1],[2],[3]]
=> [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,2,2,2,3,3,3}
[[1,2],[2],[3]]
=> [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,2,2,2,3,3,3}
[[1,3],[2],[3]]
=> [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[[1,1,1,1,2]]
=> [1,2,3,4,5] => ([],5)
=> ([],5)
=> 0
[[1,1,1,2,2]]
=> [1,2,3,4,5] => ([],5)
=> ([],5)
=> 0
[[1,1,2,2,2]]
=> [1,2,3,4,5] => ([],5)
=> ([],5)
=> 0
[[1,2,2,2,2]]
=> [1,2,3,4,5] => ([],5)
=> ([],5)
=> 0
[[2,2,2,2,2]]
=> [1,2,3,4,5] => ([],5)
=> ([],5)
=> 0
[[1,1,1,1],[2]]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[[1,1,1,2],[2]]
=> [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {1,1,1}
[[1,1,2,2],[2]]
=> [3,1,2,4,5] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? ∊ {1,1,1}
[[1,2,2,2],[2]]
=> [2,1,3,4,5] => ([(3,4)],5)
=> ([(3,4)],5)
=> 1
[[1,1,1],[2,2]]
=> [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ? ∊ {1,1,1}
[[1,1,2],[2,2]]
=> [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[[1,1,1],[4]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[[1,1,2],[4]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[[1,1,4],[2]]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[[1,1,3],[4]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[[1,1,4],[3]]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[[1,1,4],[4]]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[[1,2,2],[4]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[[1,2,3],[4]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[[1,2,4],[3]]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[[1,2,4],[4]]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[[1,3,3],[4]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[[1,3,4],[4]]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[[2,2,2],[4]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[[2,2,3],[4]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[[2,2,4],[3]]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[[2,2,4],[4]]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[[2,3,3],[4]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[[2,3,4],[4]]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[[3,3,3],[4]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[[3,3,4],[4]]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[[1,2],[2,4]]
=> [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[[1,3],[2,4]]
=> [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[[1,3],[3,4]]
=> [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[[2,3],[3,4]]
=> [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[[1,1],[2],[4]]
=> [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[[1,1],[3],[4]]
=> [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[[1,2],[2],[4]]
=> [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[[1,2],[3],[4]]
=> [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[[1,3],[2],[4]]
=> [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
[[1,3],[3],[4]]
=> [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,5,6}
Description
The largest eigenvalue of a graph if it is integral.
If a graph is $d$-regular, then its largest eigenvalue equals $d$. One can show that the largest eigenvalue always lies between the average degree and the maximal degree.
This statistic is undefined if the largest eigenvalue of the graph is not integral.
Matching statistic: St001232
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 36% ●values known / values provided: 36%●distinct values known / distinct values provided: 45%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 36% ●values known / values provided: 36%●distinct values known / distinct values provided: 45%
Values
[[1,2]]
=> [2]
=> [1,1,0,0,1,0]
=> 1 = 0 + 1
[[2,2]]
=> [2]
=> [1,1,0,0,1,0]
=> 1 = 0 + 1
[[1],[2]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[[1,1,2]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[[1,2,2]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[[2,2,2]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[[1,1],[2]]
=> [2,1]
=> [1,0,1,0,1,0]
=> ? ∊ {1,1} + 1
[[1,2],[2]]
=> [2,1]
=> [1,0,1,0,1,0]
=> ? ∊ {1,1} + 1
[[1,1,3]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[[1,2,3]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[[1,3,3]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[[2,2,3]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[[2,3,3]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[[3,3,3]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[[1,1],[3]]
=> [2,1]
=> [1,0,1,0,1,0]
=> ? ∊ {1,1,1,1,1,3} + 1
[[1,2],[3]]
=> [2,1]
=> [1,0,1,0,1,0]
=> ? ∊ {1,1,1,1,1,3} + 1
[[1,3],[2]]
=> [2,1]
=> [1,0,1,0,1,0]
=> ? ∊ {1,1,1,1,1,3} + 1
[[1,3],[3]]
=> [2,1]
=> [1,0,1,0,1,0]
=> ? ∊ {1,1,1,1,1,3} + 1
[[2,2],[3]]
=> [2,1]
=> [1,0,1,0,1,0]
=> ? ∊ {1,1,1,1,1,3} + 1
[[2,3],[3]]
=> [2,1]
=> [1,0,1,0,1,0]
=> ? ∊ {1,1,1,1,1,3} + 1
[[1],[2],[3]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[[1,1,1,2]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[[1,1,2,2]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[[1,2,2,2]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[[2,2,2,2]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[[1,1,1],[2]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> ? ∊ {1,1,2} + 1
[[1,1,2],[2]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> ? ∊ {1,1,2} + 1
[[1,2,2],[2]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> ? ∊ {1,1,2} + 1
[[1,1],[2,2]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[[1,1,1,3]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[[1,1,2,3]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[[1,1,3,3]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[[1,2,2,3]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[[1,2,3,3]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[[1,3,3,3]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[[2,2,2,3]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[[2,2,3,3]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[[2,3,3,3]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[[3,3,3,3]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[[1,1,1],[3]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,3,3,3} + 1
[[1,1,2],[3]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,3,3,3} + 1
[[1,1,3],[2]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,3,3,3} + 1
[[1,1,3],[3]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,3,3,3} + 1
[[1,2,2],[3]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,3,3,3} + 1
[[1,2,3],[2]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,3,3,3} + 1
[[1,2,3],[3]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,3,3,3} + 1
[[1,3,3],[2]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,3,3,3} + 1
[[1,3,3],[3]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,3,3,3} + 1
[[2,2,2],[3]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,3,3,3} + 1
[[2,2,3],[3]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,3,3,3} + 1
[[2,3,3],[3]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,3,3,3} + 1
[[1,1],[2,3]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[[1,1],[3,3]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[[1,2],[2,3]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[[1,2],[3,3]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[[2,2],[3,3]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[[1,1],[2],[3]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,3,3,3} + 1
[[1,2],[2],[3]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,3,3,3} + 1
[[1,3],[2],[3]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,3,3,3} + 1
[[1,1,1,1,2]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1 = 0 + 1
[[1,1,1,2,2]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1 = 0 + 1
[[1,1,2,2,2]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1 = 0 + 1
[[1,2,2,2,2]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1 = 0 + 1
[[2,2,2,2,2]]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1 = 0 + 1
[[1,1,1,1],[2]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> ? ∊ {1,1,1,1,2,2} + 1
[[1,1,1,2],[2]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> ? ∊ {1,1,1,1,2,2} + 1
[[1,1,2,2],[2]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> ? ∊ {1,1,1,1,2,2} + 1
[[1,2,2,2],[2]]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> ? ∊ {1,1,1,1,2,2} + 1
[[1,1,1],[2,2]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> ? ∊ {1,1,1,1,2,2} + 1
[[1,1,2],[2,2]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> ? ∊ {1,1,1,1,2,2} + 1
[[1,1,1,4]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[[1,1,2,4]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[[1,1,3,4]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[[1,1,4,4]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[[1,2,2,4]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[[1,2,3,4]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[[1,2,4,4]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[[1,3,3,4]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[[1,3,4,4]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[[1,4,4,4]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[[2,2,2,4]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[[2,2,3,4]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[[1,1,1],[4]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,5,6} + 1
[[1,1,2],[4]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,5,6} + 1
[[1,1,4],[2]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,5,6} + 1
[[1,1,3],[4]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,5,6} + 1
[[1,1,4],[3]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,5,6} + 1
[[1,1,4],[4]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,5,6} + 1
[[1,2,2],[4]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,5,6} + 1
[[1,2,4],[2]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,5,6} + 1
[[1,2,3],[4]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,5,6} + 1
[[1,2,4],[3]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,5,6} + 1
[[1,3,4],[2]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,5,6} + 1
[[1,2,4],[4]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,5,6} + 1
[[1,4,4],[2]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,5,6} + 1
[[1,3,3],[4]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,5,6} + 1
[[1,3,4],[3]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,5,6} + 1
[[1,3,4],[4]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,5,6} + 1
[[1,4,4],[3]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,5,6} + 1
[[1,4,4],[4]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,5,6} + 1
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Matching statistic: St000771
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00107: Semistandard tableaux —catabolism⟶ Semistandard tableaux
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000771: Graphs ⟶ ℤResult quality: 23% ●values known / values provided: 23%●distinct values known / distinct values provided: 27%
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000771: Graphs ⟶ ℤResult quality: 23% ●values known / values provided: 23%●distinct values known / distinct values provided: 27%
Values
[[1,2]]
=> [[1,2]]
=> [1,2] => ([],2)
=> ? ∊ {0,0,1}
[[2,2]]
=> [[2,2]]
=> [1,2] => ([],2)
=> ? ∊ {0,0,1}
[[1],[2]]
=> [[1,2]]
=> [1,2] => ([],2)
=> ? ∊ {0,0,1}
[[1,1,2]]
=> [[1,1,2]]
=> [1,2,3] => ([],3)
=> ? ∊ {0,0,0,1,1}
[[1,2,2]]
=> [[1,2,2]]
=> [1,2,3] => ([],3)
=> ? ∊ {0,0,0,1,1}
[[2,2,2]]
=> [[2,2,2]]
=> [1,2,3] => ([],3)
=> ? ∊ {0,0,0,1,1}
[[1,1],[2]]
=> [[1,1,2]]
=> [1,2,3] => ([],3)
=> ? ∊ {0,0,0,1,1}
[[1,2],[2]]
=> [[1,2,2]]
=> [1,2,3] => ([],3)
=> ? ∊ {0,0,0,1,1}
[[1,1,3]]
=> [[1,1,3]]
=> [1,2,3] => ([],3)
=> ? ∊ {0,0,0,0,0,0,1,1,1,2,3}
[[1,2,3]]
=> [[1,2,3]]
=> [1,2,3] => ([],3)
=> ? ∊ {0,0,0,0,0,0,1,1,1,2,3}
[[1,3,3]]
=> [[1,3,3]]
=> [1,2,3] => ([],3)
=> ? ∊ {0,0,0,0,0,0,1,1,1,2,3}
[[2,2,3]]
=> [[2,2,3]]
=> [1,2,3] => ([],3)
=> ? ∊ {0,0,0,0,0,0,1,1,1,2,3}
[[2,3,3]]
=> [[2,3,3]]
=> [1,2,3] => ([],3)
=> ? ∊ {0,0,0,0,0,0,1,1,1,2,3}
[[3,3,3]]
=> [[3,3,3]]
=> [1,2,3] => ([],3)
=> ? ∊ {0,0,0,0,0,0,1,1,1,2,3}
[[1,1],[3]]
=> [[1,1,3]]
=> [1,2,3] => ([],3)
=> ? ∊ {0,0,0,0,0,0,1,1,1,2,3}
[[1,2],[3]]
=> [[1,2,3]]
=> [1,2,3] => ([],3)
=> ? ∊ {0,0,0,0,0,0,1,1,1,2,3}
[[1,3],[2]]
=> [[1,2],[3]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 1
[[1,3],[3]]
=> [[1,3,3]]
=> [1,2,3] => ([],3)
=> ? ∊ {0,0,0,0,0,0,1,1,1,2,3}
[[2,2],[3]]
=> [[2,2,3]]
=> [1,2,3] => ([],3)
=> ? ∊ {0,0,0,0,0,0,1,1,1,2,3}
[[2,3],[3]]
=> [[2,3,3]]
=> [1,2,3] => ([],3)
=> ? ∊ {0,0,0,0,0,0,1,1,1,2,3}
[[1],[2],[3]]
=> [[1,2],[3]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 1
[[1,1,1,2]]
=> [[1,1,1,2]]
=> [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,1,1,1,2}
[[1,1,2,2]]
=> [[1,1,2,2]]
=> [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,1,1,1,2}
[[1,2,2,2]]
=> [[1,2,2,2]]
=> [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,1,1,1,2}
[[2,2,2,2]]
=> [[2,2,2,2]]
=> [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,1,1,1,2}
[[1,1,1],[2]]
=> [[1,1,1,2]]
=> [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,1,1,1,2}
[[1,1,2],[2]]
=> [[1,1,2,2]]
=> [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,1,1,1,2}
[[1,2,2],[2]]
=> [[1,2,2,2]]
=> [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,1,1,1,2}
[[1,1],[2,2]]
=> [[1,1,2,2]]
=> [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,1,1,1,2}
[[1,1,1,3]]
=> [[1,1,1,3]]
=> [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3}
[[1,1,2,3]]
=> [[1,1,2,3]]
=> [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3}
[[1,1,3,3]]
=> [[1,1,3,3]]
=> [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3}
[[1,2,2,3]]
=> [[1,2,2,3]]
=> [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3}
[[1,2,3,3]]
=> [[1,2,3,3]]
=> [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3}
[[1,3,3,3]]
=> [[1,3,3,3]]
=> [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3}
[[2,2,2,3]]
=> [[2,2,2,3]]
=> [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3}
[[2,2,3,3]]
=> [[2,2,3,3]]
=> [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3}
[[2,3,3,3]]
=> [[2,3,3,3]]
=> [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3}
[[3,3,3,3]]
=> [[3,3,3,3]]
=> [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3}
[[1,1,1],[3]]
=> [[1,1,1,3]]
=> [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3}
[[1,1,2],[3]]
=> [[1,1,2,3]]
=> [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3}
[[1,1,3],[2]]
=> [[1,1,2],[3]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2
[[1,1,3],[3]]
=> [[1,1,3,3]]
=> [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3}
[[1,2,2],[3]]
=> [[1,2,2,3]]
=> [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3}
[[1,2,3],[2]]
=> [[1,2,2],[3]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2
[[1,2,3],[3]]
=> [[1,2,3,3]]
=> [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3}
[[1,3,3],[2]]
=> [[1,2,3],[3]]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3}
[[1,3,3],[3]]
=> [[1,3,3,3]]
=> [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3}
[[2,2,2],[3]]
=> [[2,2,2,3]]
=> [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3}
[[2,2,3],[3]]
=> [[2,2,3,3]]
=> [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3}
[[2,3,3],[3]]
=> [[2,3,3,3]]
=> [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3}
[[1,1],[2,3]]
=> [[1,1,2,3]]
=> [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3}
[[1,1],[3,3]]
=> [[1,1,3,3]]
=> [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3}
[[1,2],[2,3]]
=> [[1,2,2,3]]
=> [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3}
[[1,1],[2],[3]]
=> [[1,1,2],[3]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2
[[1,2],[2],[3]]
=> [[1,2,2],[3]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2
[[1,1,4],[2]]
=> [[1,1,2],[4]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2
[[1,1,4],[3]]
=> [[1,1,3],[4]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2
[[1,2,4],[2]]
=> [[1,2,2],[4]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2
[[1,2,4],[3]]
=> [[1,2,3],[4]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2
[[1,3,4],[3]]
=> [[1,3,3],[4]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2
[[2,2,4],[3]]
=> [[2,2,3],[4]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2
[[2,3,4],[3]]
=> [[2,3,3],[4]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2
[[1,1],[2],[4]]
=> [[1,1,2],[4]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2
[[1,1],[3],[4]]
=> [[1,1,3],[4]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2
[[1,2],[2],[4]]
=> [[1,2,2],[4]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2
[[1,2],[3],[4]]
=> [[1,2,3],[4]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2
[[1,4],[2],[3]]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[[1,3],[3],[4]]
=> [[1,3,3],[4]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2
[[2,2],[3],[4]]
=> [[2,2,3],[4]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2
[[2,3],[3],[4]]
=> [[2,3,3],[4]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2
[[1],[2],[3],[4]]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[[1,1,1,3],[2]]
=> [[1,1,1,2],[3]]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[[1,1,2,3],[2]]
=> [[1,1,2,2],[3]]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[[1,2,2,3],[2]]
=> [[1,2,2,2],[3]]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[[1,1,3],[2,2]]
=> [[1,1,2,2],[3]]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[[1,1,1],[2],[3]]
=> [[1,1,1,2],[3]]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[[1,1,2],[2],[3]]
=> [[1,1,2,2],[3]]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[[1,2,2],[2],[3]]
=> [[1,2,2,2],[3]]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[[1,1],[2,2],[3]]
=> [[1,1,2,2],[3]]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[[1,1,1,4],[2]]
=> [[1,1,1,2],[4]]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[[1,1,1,4],[3]]
=> [[1,1,1,3],[4]]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[[1,1,2,4],[2]]
=> [[1,1,2,2],[4]]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[[1,1,2,4],[3]]
=> [[1,1,2,3],[4]]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[[1,1,3,4],[3]]
=> [[1,1,3,3],[4]]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[[1,2,2,4],[2]]
=> [[1,2,2,2],[4]]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[[1,2,2,4],[3]]
=> [[1,2,2,3],[4]]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[[1,2,3,4],[3]]
=> [[1,2,3,3],[4]]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[[1,3,3,4],[3]]
=> [[1,3,3,3],[4]]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[[2,2,2,4],[3]]
=> [[2,2,2,3],[4]]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[[2,2,3,4],[3]]
=> [[2,2,3,3],[4]]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[[2,3,3,4],[3]]
=> [[2,3,3,3],[4]]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[[1,1,4],[2,2]]
=> [[1,1,2,2],[4]]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[[1,1,4],[2,3]]
=> [[1,1,2,3],[4]]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[[1,1,4],[3,3]]
=> [[1,1,3,3],[4]]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[[1,2,4],[2,3]]
=> [[1,2,2,3],[4]]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[[1,2,4],[3,3]]
=> [[1,2,3,3],[4]]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[[2,2,4],[3,3]]
=> [[2,2,3,3],[4]]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[[1,1,1],[2],[4]]
=> [[1,1,1,2],[4]]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[[1,1,1],[3],[4]]
=> [[1,1,1,3],[4]]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
Description
The largest multiplicity of a distance Laplacian eigenvalue in a connected graph.
The distance Laplacian of a graph is the (symmetric) matrix with row and column sums $0$, which has the negative distances between two vertices as its off-diagonal entries. This statistic is the largest multiplicity of an eigenvalue.
For example, the cycle on four vertices has distance Laplacian
$$
\left(\begin{array}{rrrr}
4 & -1 & -2 & -1 \\
-1 & 4 & -1 & -2 \\
-2 & -1 & 4 & -1 \\
-1 & -2 & -1 & 4
\end{array}\right).
$$
Its eigenvalues are $0,4,4,6$, so the statistic is $2$.
The path on four vertices has eigenvalues $0, 4.7\dots, 6, 9.2\dots$ and therefore statistic $1$.
Matching statistic: St000772
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00107: Semistandard tableaux —catabolism⟶ Semistandard tableaux
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000772: Graphs ⟶ ℤResult quality: 23% ●values known / values provided: 23%●distinct values known / distinct values provided: 27%
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000772: Graphs ⟶ ℤResult quality: 23% ●values known / values provided: 23%●distinct values known / distinct values provided: 27%
Values
[[1,2]]
=> [[1,2]]
=> [1,2] => ([],2)
=> ? ∊ {0,0,1}
[[2,2]]
=> [[2,2]]
=> [1,2] => ([],2)
=> ? ∊ {0,0,1}
[[1],[2]]
=> [[1,2]]
=> [1,2] => ([],2)
=> ? ∊ {0,0,1}
[[1,1,2]]
=> [[1,1,2]]
=> [1,2,3] => ([],3)
=> ? ∊ {0,0,0,1,1}
[[1,2,2]]
=> [[1,2,2]]
=> [1,2,3] => ([],3)
=> ? ∊ {0,0,0,1,1}
[[2,2,2]]
=> [[2,2,2]]
=> [1,2,3] => ([],3)
=> ? ∊ {0,0,0,1,1}
[[1,1],[2]]
=> [[1,1,2]]
=> [1,2,3] => ([],3)
=> ? ∊ {0,0,0,1,1}
[[1,2],[2]]
=> [[1,2,2]]
=> [1,2,3] => ([],3)
=> ? ∊ {0,0,0,1,1}
[[1,1,3]]
=> [[1,1,3]]
=> [1,2,3] => ([],3)
=> ? ∊ {0,0,0,0,0,0,1,1,1,2,3}
[[1,2,3]]
=> [[1,2,3]]
=> [1,2,3] => ([],3)
=> ? ∊ {0,0,0,0,0,0,1,1,1,2,3}
[[1,3,3]]
=> [[1,3,3]]
=> [1,2,3] => ([],3)
=> ? ∊ {0,0,0,0,0,0,1,1,1,2,3}
[[2,2,3]]
=> [[2,2,3]]
=> [1,2,3] => ([],3)
=> ? ∊ {0,0,0,0,0,0,1,1,1,2,3}
[[2,3,3]]
=> [[2,3,3]]
=> [1,2,3] => ([],3)
=> ? ∊ {0,0,0,0,0,0,1,1,1,2,3}
[[3,3,3]]
=> [[3,3,3]]
=> [1,2,3] => ([],3)
=> ? ∊ {0,0,0,0,0,0,1,1,1,2,3}
[[1,1],[3]]
=> [[1,1,3]]
=> [1,2,3] => ([],3)
=> ? ∊ {0,0,0,0,0,0,1,1,1,2,3}
[[1,2],[3]]
=> [[1,2,3]]
=> [1,2,3] => ([],3)
=> ? ∊ {0,0,0,0,0,0,1,1,1,2,3}
[[1,3],[2]]
=> [[1,2],[3]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 1
[[1,3],[3]]
=> [[1,3,3]]
=> [1,2,3] => ([],3)
=> ? ∊ {0,0,0,0,0,0,1,1,1,2,3}
[[2,2],[3]]
=> [[2,2,3]]
=> [1,2,3] => ([],3)
=> ? ∊ {0,0,0,0,0,0,1,1,1,2,3}
[[2,3],[3]]
=> [[2,3,3]]
=> [1,2,3] => ([],3)
=> ? ∊ {0,0,0,0,0,0,1,1,1,2,3}
[[1],[2],[3]]
=> [[1,2],[3]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 1
[[1,1,1,2]]
=> [[1,1,1,2]]
=> [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,1,1,1,2}
[[1,1,2,2]]
=> [[1,1,2,2]]
=> [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,1,1,1,2}
[[1,2,2,2]]
=> [[1,2,2,2]]
=> [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,1,1,1,2}
[[2,2,2,2]]
=> [[2,2,2,2]]
=> [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,1,1,1,2}
[[1,1,1],[2]]
=> [[1,1,1,2]]
=> [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,1,1,1,2}
[[1,1,2],[2]]
=> [[1,1,2,2]]
=> [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,1,1,1,2}
[[1,2,2],[2]]
=> [[1,2,2,2]]
=> [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,1,1,1,2}
[[1,1],[2,2]]
=> [[1,1,2,2]]
=> [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,1,1,1,2}
[[1,1,1,3]]
=> [[1,1,1,3]]
=> [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3}
[[1,1,2,3]]
=> [[1,1,2,3]]
=> [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3}
[[1,1,3,3]]
=> [[1,1,3,3]]
=> [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3}
[[1,2,2,3]]
=> [[1,2,2,3]]
=> [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3}
[[1,2,3,3]]
=> [[1,2,3,3]]
=> [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3}
[[1,3,3,3]]
=> [[1,3,3,3]]
=> [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3}
[[2,2,2,3]]
=> [[2,2,2,3]]
=> [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3}
[[2,2,3,3]]
=> [[2,2,3,3]]
=> [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3}
[[2,3,3,3]]
=> [[2,3,3,3]]
=> [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3}
[[3,3,3,3]]
=> [[3,3,3,3]]
=> [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3}
[[1,1,1],[3]]
=> [[1,1,1,3]]
=> [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3}
[[1,1,2],[3]]
=> [[1,1,2,3]]
=> [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3}
[[1,1,3],[2]]
=> [[1,1,2],[3]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2
[[1,1,3],[3]]
=> [[1,1,3,3]]
=> [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3}
[[1,2,2],[3]]
=> [[1,2,2,3]]
=> [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3}
[[1,2,3],[2]]
=> [[1,2,2],[3]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2
[[1,2,3],[3]]
=> [[1,2,3,3]]
=> [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3}
[[1,3,3],[2]]
=> [[1,2,3],[3]]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3}
[[1,3,3],[3]]
=> [[1,3,3,3]]
=> [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3}
[[2,2,2],[3]]
=> [[2,2,2,3]]
=> [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3}
[[2,2,3],[3]]
=> [[2,2,3,3]]
=> [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3}
[[2,3,3],[3]]
=> [[2,3,3,3]]
=> [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3}
[[1,1],[2,3]]
=> [[1,1,2,3]]
=> [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3}
[[1,1],[3,3]]
=> [[1,1,3,3]]
=> [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3}
[[1,2],[2,3]]
=> [[1,2,2,3]]
=> [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3}
[[1,1],[2],[3]]
=> [[1,1,2],[3]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2
[[1,2],[2],[3]]
=> [[1,2,2],[3]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2
[[1,1,4],[2]]
=> [[1,1,2],[4]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2
[[1,1,4],[3]]
=> [[1,1,3],[4]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2
[[1,2,4],[2]]
=> [[1,2,2],[4]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2
[[1,2,4],[3]]
=> [[1,2,3],[4]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2
[[1,3,4],[3]]
=> [[1,3,3],[4]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2
[[2,2,4],[3]]
=> [[2,2,3],[4]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2
[[2,3,4],[3]]
=> [[2,3,3],[4]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2
[[1,1],[2],[4]]
=> [[1,1,2],[4]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2
[[1,1],[3],[4]]
=> [[1,1,3],[4]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2
[[1,2],[2],[4]]
=> [[1,2,2],[4]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2
[[1,2],[3],[4]]
=> [[1,2,3],[4]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2
[[1,4],[2],[3]]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[[1,3],[3],[4]]
=> [[1,3,3],[4]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2
[[2,2],[3],[4]]
=> [[2,2,3],[4]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2
[[2,3],[3],[4]]
=> [[2,3,3],[4]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2
[[1],[2],[3],[4]]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[[1,1,1,3],[2]]
=> [[1,1,1,2],[3]]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[[1,1,2,3],[2]]
=> [[1,1,2,2],[3]]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[[1,2,2,3],[2]]
=> [[1,2,2,2],[3]]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[[1,1,3],[2,2]]
=> [[1,1,2,2],[3]]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[[1,1,1],[2],[3]]
=> [[1,1,1,2],[3]]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[[1,1,2],[2],[3]]
=> [[1,1,2,2],[3]]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[[1,2,2],[2],[3]]
=> [[1,2,2,2],[3]]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[[1,1],[2,2],[3]]
=> [[1,1,2,2],[3]]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[[1,1,1,4],[2]]
=> [[1,1,1,2],[4]]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[[1,1,1,4],[3]]
=> [[1,1,1,3],[4]]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[[1,1,2,4],[2]]
=> [[1,1,2,2],[4]]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[[1,1,2,4],[3]]
=> [[1,1,2,3],[4]]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[[1,1,3,4],[3]]
=> [[1,1,3,3],[4]]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[[1,2,2,4],[2]]
=> [[1,2,2,2],[4]]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[[1,2,2,4],[3]]
=> [[1,2,2,3],[4]]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[[1,2,3,4],[3]]
=> [[1,2,3,3],[4]]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[[1,3,3,4],[3]]
=> [[1,3,3,3],[4]]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[[2,2,2,4],[3]]
=> [[2,2,2,3],[4]]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[[2,2,3,4],[3]]
=> [[2,2,3,3],[4]]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[[2,3,3,4],[3]]
=> [[2,3,3,3],[4]]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[[1,1,4],[2,2]]
=> [[1,1,2,2],[4]]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[[1,1,4],[2,3]]
=> [[1,1,2,3],[4]]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[[1,1,4],[3,3]]
=> [[1,1,3,3],[4]]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[[1,2,4],[2,3]]
=> [[1,2,2,3],[4]]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[[1,2,4],[3,3]]
=> [[1,2,3,3],[4]]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[[2,2,4],[3,3]]
=> [[2,2,3,3],[4]]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[[1,1,1],[2],[4]]
=> [[1,1,1,2],[4]]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[[1,1,1],[3],[4]]
=> [[1,1,1,3],[4]]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
Description
The multiplicity of the largest distance Laplacian eigenvalue in a connected graph.
The distance Laplacian of a graph is the (symmetric) matrix with row and column sums $0$, which has the negative distances between two vertices as its off-diagonal entries. This statistic is the largest multiplicity of an eigenvalue.
For example, the cycle on four vertices has distance Laplacian
$$
\left(\begin{array}{rrrr}
4 & -1 & -2 & -1 \\
-1 & 4 & -1 & -2 \\
-2 & -1 & 4 & -1 \\
-1 & -2 & -1 & 4
\end{array}\right).
$$
Its eigenvalues are $0,4,4,6$, so the statistic is $1$.
The path on four vertices has eigenvalues $0, 4.7\dots, 6, 9.2\dots$ and therefore also statistic $1$.
The graphs with statistic $n-1$, $n-2$ and $n-3$ have been characterised, see [1].
Matching statistic: St001060
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001060: Graphs ⟶ ℤResult quality: 18% ●values known / values provided: 20%●distinct values known / distinct values provided: 18%
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001060: Graphs ⟶ ℤResult quality: 18% ●values known / values provided: 20%●distinct values known / distinct values provided: 18%
Values
[[1,2]]
=> [1,2] => [1,2] => ([],2)
=> ? ∊ {0,0,1}
[[2,2]]
=> [1,2] => [1,2] => ([],2)
=> ? ∊ {0,0,1}
[[1],[2]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> ? ∊ {0,0,1}
[[1,1,2]]
=> [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {0,0,0,1,1}
[[1,2,2]]
=> [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {0,0,0,1,1}
[[2,2,2]]
=> [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {0,0,0,1,1}
[[1,1],[2]]
=> [3,1,2] => [1,3,2] => ([(1,2)],3)
=> ? ∊ {0,0,0,1,1}
[[1,2],[2]]
=> [2,1,3] => [2,1,3] => ([(1,2)],3)
=> ? ∊ {0,0,0,1,1}
[[1,1,3]]
=> [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,2}
[[1,2,3]]
=> [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,2}
[[1,3,3]]
=> [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,2}
[[2,2,3]]
=> [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,2}
[[2,3,3]]
=> [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,2}
[[3,3,3]]
=> [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,2}
[[1,1],[3]]
=> [3,1,2] => [1,3,2] => ([(1,2)],3)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,2}
[[1,2],[3]]
=> [3,1,2] => [1,3,2] => ([(1,2)],3)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,2}
[[1,3],[2]]
=> [2,1,3] => [2,1,3] => ([(1,2)],3)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,2}
[[1,3],[3]]
=> [2,1,3] => [2,1,3] => ([(1,2)],3)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,2}
[[2,2],[3]]
=> [3,1,2] => [1,3,2] => ([(1,2)],3)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,2}
[[2,3],[3]]
=> [2,1,3] => [2,1,3] => ([(1,2)],3)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,2}
[[1],[2],[3]]
=> [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,1,1,2]]
=> [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,1,1,1}
[[1,1,2,2]]
=> [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,1,1,1}
[[1,2,2,2]]
=> [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,1,1,1}
[[2,2,2,2]]
=> [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,1,1,1}
[[1,1,1],[2]]
=> [4,1,2,3] => [1,2,4,3] => ([(2,3)],4)
=> ? ∊ {0,0,0,0,1,1,1}
[[1,1,2],[2]]
=> [3,1,2,4] => [1,3,2,4] => ([(2,3)],4)
=> ? ∊ {0,0,0,0,1,1,1}
[[1,2,2],[2]]
=> [2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
=> ? ∊ {0,0,0,0,1,1,1}
[[1,1],[2,2]]
=> [3,4,1,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 2
[[1,1,1,3]]
=> [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,3}
[[1,1,2,3]]
=> [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,3}
[[1,1,3,3]]
=> [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,3}
[[1,2,2,3]]
=> [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,3}
[[1,2,3,3]]
=> [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,3}
[[1,3,3,3]]
=> [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,3}
[[2,2,2,3]]
=> [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,3}
[[2,2,3,3]]
=> [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,3}
[[2,3,3,3]]
=> [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,3}
[[3,3,3,3]]
=> [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,3}
[[1,1,1],[3]]
=> [4,1,2,3] => [1,2,4,3] => ([(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,3}
[[1,1,2],[3]]
=> [4,1,2,3] => [1,2,4,3] => ([(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,3}
[[1,1,3],[2]]
=> [3,1,2,4] => [1,3,2,4] => ([(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,3}
[[1,1,3],[3]]
=> [3,1,2,4] => [1,3,2,4] => ([(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,3}
[[1,2,2],[3]]
=> [4,1,2,3] => [1,2,4,3] => ([(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,3}
[[1,2,3],[2]]
=> [2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,3}
[[1,2,3],[3]]
=> [3,1,2,4] => [1,3,2,4] => ([(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,3}
[[1,3,3],[2]]
=> [2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,3}
[[1,3,3],[3]]
=> [2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,3}
[[2,2,2],[3]]
=> [4,1,2,3] => [1,2,4,3] => ([(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,3}
[[2,2,3],[3]]
=> [3,1,2,4] => [1,3,2,4] => ([(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,3}
[[2,3,3],[3]]
=> [2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,3}
[[1,1],[2,3]]
=> [3,4,1,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 2
[[1,1],[3,3]]
=> [3,4,1,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 2
[[1,2],[2,3]]
=> [2,4,1,3] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> 2
[[1,2],[3,3]]
=> [3,4,1,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 2
[[2,2],[3,3]]
=> [3,4,1,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 2
[[1,1],[2],[3]]
=> [4,3,1,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 3
[[1,2],[2],[3]]
=> [4,2,1,3] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 2
[[1,3],[2],[3]]
=> [3,2,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 3
[[1,1,1,1,2]]
=> [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2}
[[1,1],[2,4]]
=> [3,4,1,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 2
[[1,1],[3,4]]
=> [3,4,1,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 2
[[1,1],[4,4]]
=> [3,4,1,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 2
[[1,2],[2,4]]
=> [2,4,1,3] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> 2
[[1,2],[3,4]]
=> [3,4,1,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 2
[[1,3],[2,4]]
=> [2,4,1,3] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> 2
[[1,2],[4,4]]
=> [3,4,1,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 2
[[1,3],[3,4]]
=> [2,4,1,3] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> 2
[[1,3],[4,4]]
=> [3,4,1,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 2
[[2,2],[3,4]]
=> [3,4,1,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 2
[[2,2],[4,4]]
=> [3,4,1,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 2
[[2,3],[3,4]]
=> [2,4,1,3] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> 2
[[2,3],[4,4]]
=> [3,4,1,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 2
[[3,3],[4,4]]
=> [3,4,1,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 2
[[1,1],[2],[4]]
=> [4,3,1,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 3
[[1,1],[3],[4]]
=> [4,3,1,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 3
[[1,2],[2],[4]]
=> [4,2,1,3] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 2
[[1,2],[3],[4]]
=> [4,3,1,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 3
[[1,3],[2],[4]]
=> [4,2,1,3] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 2
[[1,4],[2],[3]]
=> [3,2,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 3
[[1,4],[2],[4]]
=> [3,2,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 3
[[1,3],[3],[4]]
=> [4,2,1,3] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 2
[[1,4],[3],[4]]
=> [3,2,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 3
[[2,2],[3],[4]]
=> [4,3,1,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 3
[[2,3],[3],[4]]
=> [4,2,1,3] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 2
[[2,4],[3],[4]]
=> [3,2,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 3
[[1],[2],[3],[4]]
=> [4,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[[1,1,2],[2],[3]]
=> [5,3,1,2,4] => [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> 2
[[1,2,3],[2],[3]]
=> [4,2,1,3,5] => [3,1,4,2,5] => ([(1,4),(2,3),(3,4)],5)
=> 2
[[1,1],[2,2],[3]]
=> [5,3,4,1,2] => [1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[1,1],[2,3],[3]]
=> [4,3,5,1,2] => [1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[1,2],[2,3],[3]]
=> [4,2,5,1,3] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 3
[[1,1,2],[2],[4]]
=> [5,3,1,2,4] => [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> 2
[[1,1,3],[2],[4]]
=> [5,3,1,2,4] => [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> 2
[[1,1,3],[3],[4]]
=> [5,3,1,2,4] => [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> 2
[[1,2,4],[2],[3]]
=> [4,2,1,3,5] => [3,1,4,2,5] => ([(1,4),(2,3),(3,4)],5)
=> 2
[[1,2,4],[2],[4]]
=> [4,2,1,3,5] => [3,1,4,2,5] => ([(1,4),(2,3),(3,4)],5)
=> 2
[[1,2,3],[3],[4]]
=> [5,3,1,2,4] => [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> 2
[[1,3,4],[2],[4]]
=> [4,2,1,3,5] => [3,1,4,2,5] => ([(1,4),(2,3),(3,4)],5)
=> 2
[[1,3,4],[3],[4]]
=> [4,2,1,3,5] => [3,1,4,2,5] => ([(1,4),(2,3),(3,4)],5)
=> 2
Description
The distinguishing index of a graph.
This is the smallest number of colours such that there is a colouring of the edges which is not preserved by any automorphism.
If the graph has a connected component which is a single edge, or at least two isolated vertices, this statistic is undefined.
Matching statistic: St001330
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00203: Graphs —cone⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 45%
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00203: Graphs —cone⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 45%
Values
[[1,2]]
=> [1,2] => ([],2)
=> ([(0,2),(1,2)],3)
=> 2 = 0 + 2
[[2,2]]
=> [1,2] => ([],2)
=> ([(0,2),(1,2)],3)
=> 2 = 0 + 2
[[1],[2]]
=> [2,1] => ([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
[[1,1,2]]
=> [1,2,3] => ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[1,2,2]]
=> [1,2,3] => ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[2,2,2]]
=> [1,2,3] => ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[1,1],[2]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1} + 2
[[1,2],[2]]
=> [2,1,3] => ([(1,2)],3)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1} + 2
[[1,1,3]]
=> [1,2,3] => ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[1,2,3]]
=> [1,2,3] => ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[1,3,3]]
=> [1,2,3] => ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[2,2,3]]
=> [1,2,3] => ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[2,3,3]]
=> [1,2,3] => ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[3,3,3]]
=> [1,2,3] => ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 2 = 0 + 2
[[1,1],[3]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,3} + 2
[[1,2],[3]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,3} + 2
[[1,3],[2]]
=> [2,1,3] => ([(1,2)],3)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,3} + 2
[[1,3],[3]]
=> [2,1,3] => ([(1,2)],3)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,3} + 2
[[2,2],[3]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,3} + 2
[[2,3],[3]]
=> [2,1,3] => ([(1,2)],3)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,3} + 2
[[1],[2],[3]]
=> [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 2 + 2
[[1,1,1,2]]
=> [1,2,3,4] => ([],4)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[[1,1,2,2]]
=> [1,2,3,4] => ([],4)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[[1,2,2,2]]
=> [1,2,3,4] => ([],4)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[[2,2,2,2]]
=> [1,2,3,4] => ([],4)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[[1,1,1],[2]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,2} + 2
[[1,1,2],[2]]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,2} + 2
[[1,2,2],[2]]
=> [2,1,3,4] => ([(2,3)],4)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,2} + 2
[[1,1],[2,2]]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,2} + 2
[[1,1,1,3]]
=> [1,2,3,4] => ([],4)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[[1,1,2,3]]
=> [1,2,3,4] => ([],4)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[[1,1,3,3]]
=> [1,2,3,4] => ([],4)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[[1,2,2,3]]
=> [1,2,3,4] => ([],4)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[[1,2,3,3]]
=> [1,2,3,4] => ([],4)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[[1,3,3,3]]
=> [1,2,3,4] => ([],4)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[[2,2,2,3]]
=> [1,2,3,4] => ([],4)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[[2,2,3,3]]
=> [1,2,3,4] => ([],4)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[[2,3,3,3]]
=> [1,2,3,4] => ([],4)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[[3,3,3,3]]
=> [1,2,3,4] => ([],4)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[[1,1,1],[3]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3} + 2
[[1,1,2],[3]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3} + 2
[[1,1,3],[2]]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3} + 2
[[1,1,3],[3]]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3} + 2
[[1,2,2],[3]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3} + 2
[[1,2,3],[2]]
=> [2,1,3,4] => ([(2,3)],4)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3} + 2
[[1,2,3],[3]]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3} + 2
[[1,3,3],[2]]
=> [2,1,3,4] => ([(2,3)],4)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3} + 2
[[1,3,3],[3]]
=> [2,1,3,4] => ([(2,3)],4)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3} + 2
[[2,2,2],[3]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3} + 2
[[2,2,3],[3]]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3} + 2
[[2,3,3],[3]]
=> [2,1,3,4] => ([(2,3)],4)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3} + 2
[[1,1],[2,3]]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3} + 2
[[1,1],[3,3]]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3} + 2
[[1,2],[2,3]]
=> [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3} + 2
[[1,2],[3,3]]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3} + 2
[[2,2],[3,3]]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3} + 2
[[1,1],[2],[3]]
=> [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3} + 2
[[1,2],[2],[3]]
=> [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3} + 2
[[1,3],[2],[3]]
=> [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3} + 2
[[1,1,1,1,2]]
=> [1,2,3,4,5] => ([],5)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 0 + 2
[[1,1,1,2,2]]
=> [1,2,3,4,5] => ([],5)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 0 + 2
[[1,1,2,2,2]]
=> [1,2,3,4,5] => ([],5)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 0 + 2
[[1,2,2,2,2]]
=> [1,2,3,4,5] => ([],5)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 0 + 2
[[2,2,2,2,2]]
=> [1,2,3,4,5] => ([],5)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 0 + 2
[[1,1,1,1],[2]]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,1,2,2} + 2
[[1,1,1,2],[2]]
=> [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,1,2,2} + 2
[[1,1,2,2],[2]]
=> [3,1,2,4,5] => ([(2,4),(3,4)],5)
=> ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,1,2,2} + 2
[[1,2,2,2],[2]]
=> [2,1,3,4,5] => ([(3,4)],5)
=> ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,1,2,2} + 2
[[1,1,1],[2,2]]
=> [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,1,2,2} + 2
[[1,1,2],[2,2]]
=> [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,1,2,2} + 2
[[1,1,1,4]]
=> [1,2,3,4] => ([],4)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[[1,1,2,4]]
=> [1,2,3,4] => ([],4)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[[1,1,3,4]]
=> [1,2,3,4] => ([],4)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[[1,1,4,4]]
=> [1,2,3,4] => ([],4)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[[1,2,2,4]]
=> [1,2,3,4] => ([],4)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[[1,2,3,4]]
=> [1,2,3,4] => ([],4)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[[1,2,4,4]]
=> [1,2,3,4] => ([],4)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[[1,3,3,4]]
=> [1,2,3,4] => ([],4)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[[1,3,4,4]]
=> [1,2,3,4] => ([],4)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[[1,4,4,4]]
=> [1,2,3,4] => ([],4)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[[2,2,2,4]]
=> [1,2,3,4] => ([],4)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[[2,2,3,4]]
=> [1,2,3,4] => ([],4)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[[2,2,4,4]]
=> [1,2,3,4] => ([],4)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[[2,3,3,4]]
=> [1,2,3,4] => ([],4)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[[2,3,4,4]]
=> [1,2,3,4] => ([],4)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[[2,4,4,4]]
=> [1,2,3,4] => ([],4)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[[3,3,3,4]]
=> [1,2,3,4] => ([],4)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[[3,3,4,4]]
=> [1,2,3,4] => ([],4)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[[1,1,1],[4]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,5,6} + 2
[[1,1,2],[4]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,5,6} + 2
[[1,1,4],[2]]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,5,6} + 2
[[1,1,3],[4]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,5,6} + 2
[[1,1,4],[3]]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,5,6} + 2
[[1,1,4],[4]]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,5,6} + 2
[[1,2,2],[4]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,5,6} + 2
[[1,2,4],[2]]
=> [2,1,3,4] => ([(2,3)],4)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,5,6} + 2
[[1,2,3],[4]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,5,6} + 2
[[1,2,4],[3]]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,5,6} + 2
[[1,3,4],[2]]
=> [2,1,3,4] => ([(2,3)],4)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,5,6} + 2
[[1,2,4],[4]]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,5,6} + 2
Description
The hat guessing number of a graph.
Suppose that each vertex of a graph corresponds to a player, wearing a hat whose color is arbitrarily chosen from a set of $q$ possible colors. Each player can see the hat colors of his neighbors, but not his own hat color. All of the players are asked to guess their own hat colors simultaneously, according to a predetermined guessing strategy and the hat colors they see, where no communication between them is allowed. The hat guessing number $HG(G)$ of a graph $G$ is the largest integer $q$ such that there exists a guessing strategy guaranteeing at least one correct guess for any hat assignment of $q$ possible colors.
Because it suffices that a single player guesses correctly, the hat guessing number of a graph is the maximum of the hat guessing numbers of its connected components.
Matching statistic: St000097
Values
[[1,2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 1 = 0 + 1
[[2,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2 = 1 + 1
[[1],[2]]
=> ([],1)
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[1,1,2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 1 = 0 + 1
[[1,2,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2 = 1 + 1
[[2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[[1,1],[2]]
=> ([],1)
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[1,2],[2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 1 = 0 + 1
[[1,1,3]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2 = 1 + 1
[[1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
[[1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 3 + 1
[[2,2,3]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(0,6),(1,2),(1,4),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
=> ([(0,1),(0,4),(0,6),(1,3),(1,4),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,0,0,0} + 1
[[2,3,3]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ([(0,7),(1,3),(1,8),(2,7),(2,8),(3,5),(4,5),(4,6),(5,8),(6,7),(6,8)],9)
=> ([(0,3),(0,6),(0,7),(0,8),(1,2),(1,4),(1,5),(1,7),(1,8),(2,3),(2,4),(2,5),(2,8),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(4,8),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {0,0,0,0} + 1
[[3,3,3]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ([(0,8),(1,7),(2,5),(2,6),(3,7),(3,9),(4,8),(4,9),(5,7),(5,9),(6,8),(6,9)],10)
=> ([(0,3),(0,4),(0,7),(0,8),(0,9),(1,2),(1,4),(1,5),(1,6),(1,8),(1,9),(2,3),(2,5),(2,6),(2,8),(2,9),(3,4),(3,5),(3,7),(3,8),(4,6),(4,7),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? ∊ {0,0,0,0} + 1
[[1,1],[3]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 1 = 0 + 1
[[1,2],[3]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2 = 1 + 1
[[1,3],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2 = 1 + 1
[[1,3],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[[2,2],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[[2,3],[3]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ([(0,6),(0,7),(1,2),(1,3),(2,5),(3,4),(4,6),(5,7)],8)
=> ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,4),(1,5),(1,6),(1,7),(2,3),(2,5),(2,6),(2,7),(3,4),(3,5),(3,7),(4,6),(4,7),(5,6)],8)
=> ? ∊ {0,0,0,0} + 1
[[1],[2],[3]]
=> ([],1)
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[1,1,1,2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 1 = 0 + 1
[[1,1,2,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2 = 1 + 1
[[1,2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[[2,2,2,2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[[1,1,1],[2]]
=> ([],1)
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[1,1,2],[2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 1 = 0 + 1
[[1,2,2],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2 = 1 + 1
[[1,1],[2,2]]
=> ([],1)
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[1,1,1,3]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2 = 1 + 1
[[1,1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
[[1,1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 3 + 1
[[1,2,2,3]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(0,6),(1,2),(1,4),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
=> ([(0,1),(0,4),(0,6),(1,3),(1,4),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,0,0,0,0,0,1,2,2,2,2} + 1
[[1,2,3,3]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ([(0,7),(1,3),(1,8),(2,7),(2,8),(3,5),(4,5),(4,6),(5,8),(6,7),(6,8)],9)
=> ([(0,3),(0,6),(0,7),(0,8),(1,2),(1,4),(1,5),(1,7),(1,8),(2,3),(2,4),(2,5),(2,8),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(4,8),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {0,0,0,0,0,0,1,2,2,2,2} + 1
[[1,3,3,3]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ([(0,8),(1,7),(2,5),(2,6),(3,7),(3,9),(4,8),(4,9),(5,7),(5,9),(6,8),(6,9)],10)
=> ([(0,3),(0,4),(0,7),(0,8),(0,9),(1,2),(1,4),(1,5),(1,6),(1,8),(1,9),(2,3),(2,5),(2,6),(2,8),(2,9),(3,4),(3,5),(3,7),(3,8),(4,6),(4,7),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? ∊ {0,0,0,0,0,0,1,2,2,2,2} + 1
[[2,2,2,3]]
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> ([(0,8),(1,4),(1,8),(2,3),(2,6),(3,7),(4,5),(4,6),(5,7),(5,8),(6,7)],9)
=> ([(0,1),(0,2),(0,3),(0,5),(0,6),(1,2),(1,5),(1,6),(1,8),(2,5),(2,7),(2,8),(3,4),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {0,0,0,0,0,0,1,2,2,2,2} + 1
[[2,2,3,3]]
=> ([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
=> ([(0,9),(1,7),(1,8),(2,7),(2,11),(3,6),(3,8),(4,9),(4,11),(5,6),(5,9),(5,11),(6,10),(7,10),(8,10),(10,11)],12)
=> ([(0,2),(0,4),(0,5),(0,6),(0,9),(0,10),(0,11),(1,3),(1,6),(1,7),(1,8),(1,9),(1,10),(1,11),(2,3),(2,4),(2,5),(2,6),(2,7),(2,10),(2,11),(3,4),(3,7),(3,8),(3,9),(3,10),(3,11),(4,5),(4,6),(4,7),(4,8),(4,11),(5,6),(5,7),(5,8),(5,9),(5,11),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? ∊ {0,0,0,0,0,0,1,2,2,2,2} + 1
[[2,3,3,3]]
=> ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
=> ([(0,10),(1,4),(1,12),(2,10),(2,11),(3,11),(3,12),(4,8),(5,6),(5,7),(6,9),(6,13),(7,8),(7,13),(8,12),(9,10),(9,11),(11,13),(12,13)],14)
=> ([(0,3),(0,4),(0,5),(0,8),(0,9),(0,10),(0,11),(0,12),(0,13),(1,2),(1,3),(1,5),(1,6),(1,7),(1,9),(1,11),(1,12),(1,13),(2,4),(2,5),(2,6),(2,7),(2,10),(2,11),(2,12),(2,13),(3,4),(3,5),(3,6),(3,8),(3,9),(3,10),(3,12),(3,13),(4,7),(4,8),(4,9),(4,10),(4,11),(4,12),(4,13),(5,6),(5,7),(5,8),(5,9),(5,11),(5,12),(6,7),(6,8),(6,9),(6,10),(6,11),(6,13),(7,8),(7,9),(7,10),(7,11),(7,13),(8,9),(8,10),(8,11),(8,12),(8,13),(9,10),(9,12),(9,13),(10,11),(10,12),(10,13),(11,12),(11,13),(12,13)],14)
=> ? ∊ {0,0,0,0,0,0,1,2,2,2,2} + 1
[[3,3,3,3]]
=> ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
=> ([(0,11),(1,10),(2,8),(2,9),(3,10),(3,13),(4,11),(4,14),(5,13),(5,14),(6,8),(6,10),(6,13),(7,9),(7,11),(7,14),(8,12),(9,12),(12,13),(12,14)],15)
=> ([(0,3),(0,4),(0,5),(0,6),(0,9),(0,10),(0,11),(0,12),(0,13),(0,14),(1,2),(1,4),(1,5),(1,6),(1,7),(1,8),(1,11),(1,12),(1,13),(1,14),(2,3),(2,5),(2,6),(2,7),(2,8),(2,10),(2,12),(2,13),(2,14),(3,4),(3,5),(3,7),(3,9),(3,10),(3,11),(3,12),(3,13),(3,14),(4,6),(4,8),(4,9),(4,10),(4,11),(4,12),(4,13),(4,14),(5,6),(5,7),(5,8),(5,9),(5,10),(5,12),(5,13),(6,7),(6,8),(6,9),(6,11),(6,12),(6,14),(7,8),(7,9),(7,10),(7,11),(7,13),(7,14),(8,9),(8,10),(8,11),(8,13),(8,14),(9,10),(9,11),(9,12),(9,13),(9,14),(10,11),(10,12),(10,13),(10,14),(11,12),(11,13),(11,14),(12,13),(12,14),(13,14)],15)
=> ? ∊ {0,0,0,0,0,0,1,2,2,2,2} + 1
[[1,1,1],[3]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 1 = 0 + 1
[[1,1,2],[3]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2 = 1 + 1
[[1,1,3],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2 = 1 + 1
[[1,1,3],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[[1,2,2],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[[1,2,3],[2]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
[[1,2,3],[3]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ([(0,6),(0,7),(1,2),(1,3),(2,5),(3,4),(4,6),(5,7)],8)
=> ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,4),(1,5),(1,6),(1,7),(2,3),(2,5),(2,6),(2,7),(3,4),(3,5),(3,7),(4,6),(4,7),(5,6)],8)
=> ? ∊ {0,0,0,0,0,0,1,2,2,2,2} + 1
[[1,3,3],[2]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 3 + 1
[[1,3,3],[3]]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5)],7)
=> ([(0,1),(0,4),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(5,6)],7)
=> ? ∊ {0,0,0,0,0,0,1,2,2,2,2} + 1
[[2,2,2],[3]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[[2,2,3],[3]]
=> ([(0,6),(0,7),(1,9),(2,8),(3,5),(4,2),(5,1),(5,8),(6,3),(7,4),(8,9)],10)
=> ([(0,1),(0,3),(1,2),(2,4),(3,5),(4,8),(5,9),(6,7),(6,8),(7,9),(8,9)],10)
=> ([(0,3),(0,5),(0,6),(0,7),(0,8),(0,9),(1,2),(1,4),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,9),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,8),(7,9),(8,9)],10)
=> ? ∊ {0,0,0,0,0,0,1,2,2,2,2} + 1
[[2,3,3],[3]]
=> ([(0,9),(0,10),(1,11),(2,14),(3,12),(4,13),(5,4),(5,11),(6,5),(7,3),(8,1),(8,14),(9,6),(10,2),(10,8),(11,13),(13,12),(14,7)],15)
=> ([(0,4),(0,5),(1,2),(1,3),(2,10),(3,9),(4,11),(5,12),(6,9),(6,11),(7,10),(7,12),(8,13),(8,14),(9,13),(10,14),(11,13),(12,14)],15)
=> ([(0,1),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(0,11),(0,13),(0,14),(1,2),(1,3),(1,6),(1,7),(1,8),(1,9),(1,10),(1,11),(1,13),(1,14),(2,3),(2,4),(2,5),(2,6),(2,8),(2,9),(2,11),(2,12),(2,13),(2,14),(3,4),(3,5),(3,6),(3,8),(3,9),(3,10),(3,12),(3,13),(3,14),(4,5),(4,7),(4,8),(4,10),(4,11),(4,12),(4,13),(4,14),(5,7),(5,9),(5,10),(5,11),(5,12),(5,13),(5,14),(6,7),(6,8),(6,9),(6,10),(6,11),(6,12),(6,13),(6,14),(7,8),(7,9),(7,10),(7,11),(7,12),(7,13),(7,14),(8,9),(8,10),(8,11),(8,12),(8,13),(9,10),(9,11),(9,12),(9,14),(10,11),(10,12),(10,13),(11,12),(11,14),(12,13),(12,14),(13,14)],15)
=> ? ∊ {0,0,0,0,0,0,1,2,2,2,2} + 1
[[1,1],[2,3]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 1 = 0 + 1
[[1,1],[3,3]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2 = 1 + 1
[[1,2],[2,3]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2 = 1 + 1
[[1,2],[3,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
[[2,2],[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 3 + 1
[[1,1],[2],[3]]
=> ([],1)
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[1,2],[2],[3]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 1 = 0 + 1
[[1,3],[2],[3]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2 = 1 + 1
[[1,1,1,1,2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 1 = 0 + 1
[[1,1,1,2,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2 = 1 + 1
[[1,1,2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[[1,2,2,2,2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[[2,2,2,2,2]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,1),(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[[1,1,1,1],[2]]
=> ([],1)
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[1,1,2,4]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(0,6),(1,2),(1,4),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
=> ([(0,1),(0,4),(0,6),(1,3),(1,4),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,6} + 1
[[1,1,3,4]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ([(0,7),(1,3),(1,8),(2,7),(2,8),(3,5),(4,5),(4,6),(5,8),(6,7),(6,8)],9)
=> ([(0,3),(0,6),(0,7),(0,8),(1,2),(1,4),(1,5),(1,7),(1,8),(2,3),(2,4),(2,5),(2,8),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(4,8),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,6} + 1
[[1,1,4,4]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ([(0,8),(1,7),(2,5),(2,6),(3,7),(3,9),(4,8),(4,9),(5,7),(5,9),(6,8),(6,9)],10)
=> ([(0,3),(0,4),(0,7),(0,8),(0,9),(1,2),(1,4),(1,5),(1,6),(1,8),(1,9),(2,3),(2,5),(2,6),(2,8),(2,9),(3,4),(3,5),(3,7),(3,8),(4,6),(4,7),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,6} + 1
[[1,2,2,4]]
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> ([(0,8),(1,5),(1,7),(2,4),(2,6),(3,6),(3,7),(4,8),(4,9),(5,8),(5,9),(6,9),(7,9)],10)
=> ([(0,5),(0,6),(0,7),(0,8),(0,9),(1,2),(1,3),(1,4),(1,5),(1,7),(1,9),(2,3),(2,4),(2,5),(2,6),(2,9),(3,4),(3,6),(3,8),(3,9),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,6} + 1
[[1,2,3,4]]
=> ([(0,6),(1,9),(1,10),(2,8),(3,7),(4,3),(4,12),(5,2),(5,12),(6,4),(6,5),(7,9),(7,11),(8,10),(8,11),(9,13),(10,13),(11,13),(12,1),(12,7),(12,8)],14)
=> ([(0,10),(1,9),(1,12),(2,8),(2,11),(3,5),(3,6),(3,7),(4,5),(4,6),(4,13),(5,11),(6,12),(7,11),(7,12),(8,10),(8,13),(9,10),(9,13),(11,13),(12,13)],14)
=> ([(0,6),(0,7),(0,8),(0,9),(0,10),(0,11),(0,12),(0,13),(1,2),(1,3),(1,4),(1,5),(1,8),(1,9),(1,10),(1,12),(1,13),(2,3),(2,4),(2,5),(2,7),(2,9),(2,10),(2,11),(2,13),(3,4),(3,5),(3,6),(3,7),(3,8),(3,10),(3,11),(3,13),(4,5),(4,6),(4,7),(4,8),(4,10),(4,12),(4,13),(5,6),(5,9),(5,10),(5,11),(5,12),(5,13),(6,7),(6,8),(6,9),(6,11),(6,12),(6,13),(7,8),(7,9),(7,11),(7,12),(7,13),(8,9),(8,11),(8,12),(8,13),(9,10),(9,11),(9,12),(10,11),(10,12),(10,13),(11,12),(11,13),(12,13)],14)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,6} + 1
[[1,2,4,4]]
=> ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
=> ([(0,10),(1,8),(1,14),(2,9),(2,13),(3,4),(3,11),(4,12),(5,7),(5,11),(5,13),(6,11),(6,13),(6,14),(7,12),(7,15),(8,10),(8,15),(9,10),(9,15),(11,12),(12,14),(13,15),(14,15)],16)
=> ([(0,3),(0,4),(0,8),(0,9),(0,10),(0,11),(0,12),(0,13),(0,14),(0,15),(1,2),(1,4),(1,5),(1,6),(1,7),(1,9),(1,10),(1,11),(1,13),(1,14),(1,15),(2,3),(2,4),(2,5),(2,6),(2,7),(2,10),(2,12),(2,13),(2,14),(2,15),(3,5),(3,6),(3,8),(3,9),(3,10),(3,11),(3,12),(3,14),(3,15),(4,5),(4,6),(4,7),(4,10),(4,11),(4,12),(4,13),(4,15),(5,6),(5,7),(5,8),(5,9),(5,11),(5,13),(5,14),(5,15),(6,7),(6,8),(6,9),(6,12),(6,13),(6,14),(6,15),(7,8),(7,10),(7,11),(7,12),(7,13),(7,14),(7,15),(8,9),(8,10),(8,11),(8,12),(8,13),(8,14),(8,15),(9,10),(9,11),(9,12),(9,13),(9,14),(9,15),(10,11),(10,12),(10,13),(10,14),(11,12),(11,13),(11,14),(11,15),(12,13),(12,14),(12,15),(13,15),(14,15)],16)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,6} + 1
[[1,3,3,4]]
=> ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
=> ([(0,10),(1,8),(1,14),(2,9),(2,13),(3,4),(3,11),(4,12),(5,7),(5,11),(5,13),(6,11),(6,13),(6,14),(7,12),(7,15),(8,10),(8,15),(9,10),(9,15),(11,12),(12,14),(13,15),(14,15)],16)
=> ([(0,3),(0,4),(0,8),(0,9),(0,10),(0,11),(0,12),(0,13),(0,14),(0,15),(1,2),(1,4),(1,5),(1,6),(1,7),(1,9),(1,10),(1,11),(1,13),(1,14),(1,15),(2,3),(2,4),(2,5),(2,6),(2,7),(2,10),(2,12),(2,13),(2,14),(2,15),(3,5),(3,6),(3,8),(3,9),(3,10),(3,11),(3,12),(3,14),(3,15),(4,5),(4,6),(4,7),(4,10),(4,11),(4,12),(4,13),(4,15),(5,6),(5,7),(5,8),(5,9),(5,11),(5,13),(5,14),(5,15),(6,7),(6,8),(6,9),(6,12),(6,13),(6,14),(6,15),(7,8),(7,10),(7,11),(7,12),(7,13),(7,14),(7,15),(8,9),(8,10),(8,11),(8,12),(8,13),(8,14),(8,15),(9,10),(9,11),(9,12),(9,13),(9,14),(9,15),(10,11),(10,12),(10,13),(10,14),(11,12),(11,13),(11,14),(11,15),(12,13),(12,14),(12,15),(13,15),(14,15)],16)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,6} + 1
[[1,3,4,4]]
=> ([(0,1),(1,4),(1,5),(2,14),(3,13),(4,6),(4,17),(5,7),(5,17),(6,15),(7,16),(8,11),(8,12),(10,18),(11,3),(11,18),(12,2),(12,18),(13,9),(14,9),(15,10),(15,11),(16,10),(16,12),(17,8),(17,15),(17,16),(18,13),(18,14)],19)
=> ([(0,12),(1,10),(1,11),(2,11),(2,14),(3,10),(3,13),(4,8),(4,15),(5,9),(5,16),(6,15),(6,16),(6,17),(7,13),(7,14),(7,18),(8,12),(8,18),(9,12),(9,18),(10,17),(11,17),(13,15),(13,17),(14,16),(14,17),(15,18),(16,18)],19)
=> ([(0,1),(0,2),(0,3),(0,6),(0,8),(0,9),(0,12),(0,13),(0,14),(0,15),(0,16),(0,17),(0,18),(1,4),(1,5),(1,7),(1,10),(1,11),(1,12),(1,13),(1,14),(1,15),(1,16),(1,17),(1,18),(2,3),(2,5),(2,6),(2,8),(2,9),(2,10),(2,11),(2,12),(2,13),(2,14),(2,16),(2,17),(2,18),(3,4),(3,6),(3,8),(3,9),(3,10),(3,11),(3,12),(3,13),(3,14),(3,15),(3,17),(3,18),(4,5),(4,7),(4,8),(4,9),(4,10),(4,11),(4,12),(4,13),(4,15),(4,16),(4,17),(4,18),(5,7),(5,8),(5,9),(5,10),(5,11),(5,12),(5,14),(5,15),(5,16),(5,17),(5,18),(6,7),(6,8),(6,9),(6,10),(6,11),(6,12),(6,13),(6,14),(6,15),(6,16),(6,17),(6,18),(7,8),(7,9),(7,10),(7,11),(7,12),(7,13),(7,14),(7,15),(7,16),(7,17),(7,18),(8,9),(8,10),(8,11),(8,13),(8,14),(8,15),(8,17),(8,18),(9,10),(9,11),(9,13),(9,14),(9,16),(9,17),(9,18),(10,11),(10,12),(10,13),(10,15),(10,16),(10,18),(11,12),(11,14),(11,15),(11,16),(11,18),(12,13),(12,14),(12,15),(12,16),(12,17),(13,14),(13,15),(13,16),(13,17),(13,18),(14,15),(14,16),(14,17),(14,18),(15,16),(15,17),(15,18),(16,17),(16,18),(17,18)],19)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,6} + 1
[[1,4,4,4]]
=> ([(0,9),(2,16),(2,17),(3,13),(4,12),(5,10),(6,11),(7,5),(7,15),(8,6),(8,15),(9,7),(9,8),(10,14),(10,16),(11,14),(11,17),(12,18),(13,18),(14,19),(15,2),(15,10),(15,11),(16,4),(16,19),(17,3),(17,19),(18,1),(19,12),(19,13)],20)
=> ([(0,13),(1,12),(2,9),(2,15),(3,8),(3,14),(4,10),(4,16),(5,11),(5,17),(6,16),(6,17),(6,18),(7,14),(7,15),(7,19),(8,12),(8,18),(9,12),(9,18),(10,13),(10,19),(11,13),(11,19),(14,16),(14,18),(15,17),(15,18),(16,19),(17,19)],20)
=> ([(0,1),(0,4),(0,5),(0,7),(0,10),(0,11),(0,12),(0,13),(0,14),(0,15),(0,16),(0,17),(0,18),(0,19),(1,2),(1,3),(1,6),(1,8),(1,9),(1,12),(1,13),(1,14),(1,15),(1,16),(1,17),(1,18),(1,19),(2,3),(2,5),(2,6),(2,8),(2,9),(2,10),(2,11),(2,12),(2,13),(2,14),(2,15),(2,17),(2,18),(2,19),(3,4),(3,6),(3,8),(3,9),(3,10),(3,11),(3,12),(3,13),(3,14),(3,15),(3,16),(3,18),(3,19),(4,5),(4,7),(4,8),(4,9),(4,10),(4,11),(4,12),(4,13),(4,14),(4,16),(4,17),(4,18),(4,19),(5,7),(5,8),(5,9),(5,10),(5,11),(5,12),(5,13),(5,15),(5,16),(5,17),(5,18),(5,19),(6,7),(6,8),(6,9),(6,10),(6,11),(6,12),(6,13),(6,14),(6,15),(6,16),(6,17),(6,18),(6,19),(7,8),(7,9),(7,10),(7,11),(7,12),(7,13),(7,14),(7,15),(7,16),(7,17),(7,18),(7,19),(8,9),(8,10),(8,11),(8,12),(8,14),(8,15),(8,16),(8,18),(8,19),(9,10),(9,11),(9,12),(9,14),(9,15),(9,17),(9,18),(9,19),(10,11),(10,13),(10,14),(10,16),(10,17),(10,18),(10,19),(11,13),(11,15),(11,16),(11,17),(11,18),(11,19),(12,13),(12,14),(12,15),(12,16),(12,17),(12,18),(13,14),(13,15),(13,16),(13,17),(13,19),(14,15),(14,16),(14,17),(14,18),(14,19),(15,16),(15,17),(15,18),(15,19),(16,17),(16,18),(16,19),(17,18),(17,19),(18,19)],20)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,6} + 1
[[2,2,2,4]]
=> ([(0,6),(1,9),(2,8),(3,5),(3,7),(4,1),(4,7),(5,2),(5,10),(6,3),(6,4),(7,9),(7,10),(8,12),(9,11),(10,8),(10,11),(11,12)],13)
=> ([(0,10),(1,6),(1,8),(2,5),(2,9),(3,7),(3,9),(4,7),(4,10),(4,12),(5,6),(5,11),(6,12),(7,11),(8,10),(8,12),(9,11),(11,12)],13)
=> ([(0,3),(0,5),(0,7),(0,8),(0,9),(0,10),(0,11),(0,12),(1,2),(1,4),(1,6),(1,8),(1,9),(1,10),(1,11),(1,12),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(2,11),(2,12),(3,4),(3,5),(3,6),(3,7),(3,9),(3,10),(3,12),(4,5),(4,6),(4,8),(4,10),(4,11),(4,12),(5,7),(5,8),(5,9),(5,10),(5,12),(6,7),(6,8),(6,9),(6,11),(6,12),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,11),(9,10),(9,11),(9,12),(10,11),(10,12),(11,12)],13)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,6} + 1
[[2,2,3,4]]
=> ([(0,1),(1,2),(1,3),(2,4),(2,13),(3,6),(3,13),(4,15),(5,14),(6,5),(6,16),(7,10),(7,12),(8,18),(9,18),(10,17),(11,9),(11,17),(12,8),(12,17),(13,7),(13,15),(13,16),(14,8),(14,9),(15,10),(15,11),(16,11),(16,12),(16,14),(17,18)],19)
=> ([(0,11),(1,10),(1,16),(2,9),(2,15),(3,9),(3,11),(3,17),(4,6),(4,7),(4,14),(5,8),(5,14),(5,16),(6,12),(6,15),(7,13),(7,15),(8,12),(8,17),(9,18),(10,11),(10,17),(12,14),(12,18),(13,14),(13,16),(13,18),(15,18),(16,17),(17,18)],19)
=> ([(0,2),(0,6),(0,7),(0,8),(0,10),(0,11),(0,12),(0,13),(0,14),(0,15),(0,16),(0,17),(0,18),(1,3),(1,4),(1,5),(1,6),(1,9),(1,11),(1,12),(1,13),(1,14),(1,15),(1,16),(1,17),(1,18),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(2,10),(2,12),(2,13),(2,14),(2,15),(2,17),(2,18),(3,4),(3,5),(3,7),(3,8),(3,9),(3,10),(3,11),(3,13),(3,14),(3,15),(3,16),(3,17),(3,18),(4,5),(4,7),(4,8),(4,9),(4,11),(4,12),(4,13),(4,14),(4,16),(4,17),(4,18),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(5,13),(5,14),(5,16),(5,18),(6,7),(6,8),(6,9),(6,10),(6,12),(6,13),(6,15),(6,16),(6,17),(6,18),(7,8),(7,9),(7,10),(7,11),(7,12),(7,14),(7,15),(7,17),(7,18),(8,10),(8,11),(8,12),(8,14),(8,15),(8,16),(8,17),(8,18),(9,10),(9,11),(9,12),(9,13),(9,14),(9,15),(9,16),(9,18),(10,12),(10,13),(10,14),(10,15),(10,16),(10,17),(10,18),(11,12),(11,13),(11,14),(11,15),(11,16),(11,17),(11,18),(12,13),(12,15),(12,16),(12,17),(12,18),(13,14),(13,15),(13,16),(13,17),(14,16),(14,17),(14,18),(15,16),(15,17),(15,18),(16,17),(16,18),(17,18)],19)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,6} + 1
[[2,2,4,4]]
=> ([(0,1),(1,3),(1,4),(2,15),(3,6),(3,18),(4,5),(4,18),(5,17),(6,7),(6,19),(7,16),(8,12),(8,14),(10,21),(11,21),(12,2),(12,20),(13,11),(13,20),(14,10),(14,20),(15,9),(16,10),(16,11),(17,12),(17,13),(18,8),(18,17),(18,19),(19,13),(19,14),(19,16),(20,15),(20,21),(21,9)],22)
=> ([(0,12),(1,10),(1,16),(2,11),(2,15),(3,11),(3,18),(4,9),(4,17),(5,10),(5,12),(5,19),(6,13),(6,18),(6,19),(7,14),(7,15),(7,16),(8,13),(8,15),(8,16),(9,12),(9,19),(10,20),(11,21),(13,20),(13,21),(14,17),(14,20),(14,21),(15,21),(16,20),(17,18),(17,19),(18,21),(19,20)],22)
=> ([(0,2),(0,4),(0,7),(0,8),(0,10),(0,11),(0,12),(0,13),(0,14),(0,15),(0,16),(0,17),(0,18),(0,19),(0,20),(0,21),(1,2),(1,3),(1,5),(1,6),(1,7),(1,8),(1,9),(1,13),(1,14),(1,15),(1,16),(1,17),(1,18),(1,19),(1,20),(1,21),(2,4),(2,5),(2,9),(2,10),(2,11),(2,12),(2,13),(2,15),(2,16),(2,17),(2,18),(2,19),(2,20),(2,21),(3,5),(3,6),(3,7),(3,8),(3,9),(3,10),(3,11),(3,12),(3,14),(3,15),(3,16),(3,17),(3,18),(3,19),(3,20),(3,21),(4,5),(4,6),(4,8),(4,9),(4,10),(4,11),(4,12),(4,13),(4,14),(4,15),(4,16),(4,18),(4,19),(4,20),(4,21),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(5,12),(5,14),(5,15),(5,17),(5,18),(5,20),(5,21),(6,7),(6,8),(6,9),(6,11),(6,12),(6,13),(6,14),(6,15),(6,17),(6,18),(6,19),(6,20),(6,21),(7,8),(7,9),(7,11),(7,12),(7,13),(7,14),(7,15),(7,16),(7,17),(7,19),(7,20),(7,21),(8,9),(8,10),(8,11),(8,12),(8,14),(8,15),(8,17),(8,18),(8,19),(8,21),(9,10),(9,11),(9,13),(9,14),(9,15),(9,16),(9,17),(9,18),(9,20),(9,21),(10,11),(10,12),(10,13),(10,14),(10,15),(10,16),(10,17),(10,18),(10,19),(10,20),(10,21),(11,12),(11,13),(11,14),(11,16),(11,18),(11,19),(11,20),(11,21),(12,13),(12,14),(12,16),(12,17),(12,18),(12,19),(12,20),(12,21),(13,14),(13,15),(13,16),(13,17),(13,18),(13,19),(13,20),(13,21),(14,15),(14,16),(14,17),(14,19),(14,21),(15,16),(15,17),(15,18),(15,19),(15,20),(16,17),(16,18),(16,19),(16,20),(16,21),(17,18),(17,19),(17,20),(17,21),(18,19),(18,20),(18,21),(19,20),(19,21),(20,21)],22)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,6} + 1
[[2,3,3,4]]
=> ([(0,1),(1,2),(1,3),(2,4),(2,16),(3,6),(3,16),(4,18),(5,17),(6,5),(6,19),(7,9),(7,11),(8,10),(8,14),(9,21),(10,22),(11,21),(12,20),(13,12),(13,22),(14,7),(14,15),(14,22),(15,9),(15,20),(16,8),(16,18),(16,19),(17,12),(17,15),(18,10),(18,13),(19,13),(19,14),(19,17),(20,21),(22,11),(22,20)],23)
=> ([(0,13),(1,12),(1,17),(2,11),(2,18),(3,4),(3,8),(3,22),(4,9),(4,21),(5,6),(5,17),(5,21),(6,19),(6,22),(7,11),(7,13),(7,19),(8,9),(8,15),(9,16),(10,14),(10,16),(10,18),(11,20),(12,13),(12,19),(14,17),(14,20),(14,21),(15,16),(15,18),(15,22),(16,21),(17,19),(18,20),(19,20),(20,22),(21,22)],23)
=> ([(0,2),(0,5),(0,7),(0,8),(0,10),(0,11),(0,12),(0,13),(0,14),(0,15),(0,16),(0,17),(0,18),(0,19),(0,20),(0,21),(0,22),(1,3),(1,4),(1,6),(1,7),(1,8),(1,9),(1,12),(1,13),(1,14),(1,15),(1,16),(1,17),(1,18),(1,19),(1,20),(1,21),(1,22),(2,3),(2,4),(2,5),(2,8),(2,9),(2,10),(2,11),(2,12),(2,13),(2,14),(2,17),(2,18),(2,19),(2,20),(2,21),(2,22),(3,4),(3,5),(3,6),(3,7),(3,9),(3,11),(3,12),(3,13),(3,14),(3,15),(3,16),(3,17),(3,20),(3,21),(3,22),(4,6),(4,7),(4,8),(4,9),(4,10),(4,11),(4,13),(4,14),(4,15),(4,16),(4,17),(4,18),(4,19),(4,21),(4,22),(5,6),(5,8),(5,9),(5,10),(5,11),(5,12),(5,14),(5,15),(5,16),(5,17),(5,18),(5,19),(5,20),(5,21),(5,22),(6,7),(6,9),(6,10),(6,11),(6,13),(6,14),(6,15),(6,16),(6,17),(6,18),(6,19),(6,20),(6,21),(6,22),(7,8),(7,9),(7,10),(7,11),(7,13),(7,14),(7,15),(7,16),(7,18),(7,19),(7,20),(7,21),(7,22),(8,9),(8,10),(8,11),(8,12),(8,13),(8,14),(8,16),(8,17),(8,18),(8,19),(8,20),(8,22),(9,10),(9,11),(9,12),(9,13),(9,14),(9,15),(9,16),(9,17),(9,18),(9,21),(9,22),(10,11),(10,12),(10,13),(10,14),(10,16),(10,17),(10,18),(10,19),(10,20),(10,21),(10,22),(11,12),(11,14),(11,15),(11,17),(11,18),(11,19),(11,20),(11,21),(11,22),(12,13),(12,14),(12,15),(12,16),(12,17),(12,18),(12,19),(12,20),(12,21),(12,22),(13,14),(13,15),(13,16),(13,18),(13,19),(13,20),(13,21),(13,22),(14,15),(14,16),(14,17),(14,20),(14,21),(15,16),(15,17),(15,18),(15,19),(15,20),(15,21),(15,22),(16,18),(16,19),(16,20),(16,21),(16,22),(17,18),(17,19),(17,20),(17,21),(17,22),(18,19),(18,20),(18,22),(19,20),(19,21),(19,22),(20,21),(20,22),(21,22)],23)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,6} + 1
[[2,3,4,4]]
=> ([(0,1),(1,3),(1,4),(2,21),(3,6),(3,22),(4,5),(4,22),(5,20),(6,7),(6,23),(7,19),(8,13),(8,18),(9,14),(9,17),(10,26),(11,26),(12,27),(13,24),(14,2),(14,25),(15,13),(15,27),(16,12),(16,25),(17,8),(17,15),(17,25),(18,10),(18,24),(19,12),(19,15),(20,14),(20,16),(21,10),(21,11),(22,9),(22,20),(22,23),(23,16),(23,17),(23,19),(24,26),(25,18),(25,21),(25,27),(27,11),(27,24)],28)
=> ([(0,14),(1,13),(1,22),(2,19),(2,20),(3,12),(3,21),(4,13),(4,14),(4,23),(5,7),(5,19),(5,24),(6,20),(6,23),(6,26),(7,8),(7,18),(8,17),(8,19),(9,15),(9,22),(9,24),(10,11),(10,17),(10,26),(11,16),(11,18),(12,14),(12,23),(13,25),(15,21),(15,25),(15,27),(16,22),(16,24),(16,26),(17,18),(17,27),(18,24),(19,27),(20,21),(20,27),(21,23),(22,25),(23,25),(24,27),(25,26),(26,27)],28)
=> ([(0,3),(0,4),(0,9),(0,10),(0,11),(0,12),(0,13),(0,14),(0,15),(0,16),(0,17),(0,18),(0,19),(0,20),(0,21),(0,22),(0,23),(0,24),(0,25),(0,26),(0,27),(1,2),(1,4),(1,5),(1,6),(1,7),(1,8),(1,10),(1,11),(1,12),(1,14),(1,15),(1,17),(1,18),(1,19),(1,20),(1,21),(1,22),(1,23),(1,24),(1,25),(1,26),(1,27),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(2,11),(2,12),(2,13),(2,16),(2,17),(2,18),(2,19),(2,20),(2,21),(2,22),(2,23),(2,24),(2,25),(2,26),(2,27),(3,6),(3,7),(3,8),(3,9),(3,10),(3,11),(3,13),(3,14),(3,15),(3,16),(3,17),(3,19),(3,20),(3,21),(3,22),(3,23),(3,24),(3,25),(3,26),(3,27),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(4,12),(4,14),(4,15),(4,16),(4,17),(4,18),(4,20),(4,22),(4,23),(4,24),(4,25),(4,26),(4,27),(5,6),(5,7),(5,8),(5,9),(5,10),(5,12),(5,13),(5,15),(5,16),(5,17),(5,18),(5,19),(5,20),(5,21),(5,22),(5,23),(5,24),(5,25),(5,26),(5,27),(6,7),(6,8),(6,9),(6,10),(6,12),(6,14),(6,15),(6,16),(6,17),(6,18),(6,19),(6,20),(6,21),(6,22),(6,23),(6,25),(6,26),(6,27),(7,8),(7,9),(7,11),(7,12),(7,13),(7,14),(7,15),(7,18),(7,19),(7,20),(7,21),(7,22),(7,23),(7,24),(7,25),(7,26),(7,27),(8,9),(8,10),(8,11),(8,12),(8,13),(8,14),(8,16),(8,18),(8,19),(8,20),(8,21),(8,22),(8,23),(8,25),(8,26),(8,27),(9,10),(9,11),(9,13),(9,14),(9,15),(9,16),(9,17),(9,18),(9,19),(9,20),(9,21),(9,23),(9,24),(9,25),(9,26),(9,27),(10,11),(10,12),(10,13),(10,14),(10,15),(10,16),(10,17),(10,18),(10,19),(10,20),(10,21),(10,24),(10,25),(10,26),(10,27),(11,12),(11,13),(11,14),(11,15),(11,16),(11,17),(11,18),(11,19),(11,20),(11,21),(11,22),(11,23),(11,24),(11,26),(11,27),(12,13),(12,15),(12,16),(12,17),(12,18),(12,19),(12,20),(12,21),(12,22),(12,23),(12,24),(12,25),(12,27),(13,14),(13,15),(13,16),(13,17),(13,18),(13,19),(13,20),(13,21),(13,22),(13,23),(13,24),(13,25),(13,26),(13,27),(14,15),(14,16),(14,17),(14,18),(14,19),(14,20),(14,21),(14,22),(14,23),(14,24),(14,25),(14,26),(14,27),(15,16),(15,17),(15,18),(15,19),(15,20),(15,21),(15,22),(15,24),(15,25),(15,26),(15,27),(16,17),(16,18),(16,19),(16,20),(16,21),(16,23),(16,24),(16,25),(16,26),(16,27),(17,18),(17,19),(17,20),(17,21),(17,22),(17,24),(17,25),(17,26),(17,27),(18,19),(18,20),(18,22),(18,23),(18,24),(18,25),(18,27),(19,21),(19,22),(19,23),(19,24),(19,26),(19,27),(20,22),(20,23),(20,24),(20,25),(20,26),(21,22),(21,23),(21,24),(21,25),(21,26),(21,27),(22,23),(22,24),(22,25),(22,26),(22,27),(23,24),(23,25),(23,26),(23,27),(24,25),(24,26),(24,27),(25,26),(25,27),(26,27)],28)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,6} + 1
[[2,4,4,4]]
=> ([(0,1),(1,4),(1,5),(2,23),(3,16),(4,7),(4,24),(5,6),(5,24),(6,22),(7,8),(7,25),(8,21),(9,13),(9,20),(10,15),(10,19),(11,28),(12,29),(13,26),(14,3),(14,28),(15,2),(15,27),(17,13),(17,29),(18,12),(18,27),(19,9),(19,17),(19,27),(20,14),(20,26),(21,12),(21,17),(22,15),(22,18),(23,11),(23,14),(24,10),(24,22),(24,25),(25,18),(25,19),(25,21),(26,28),(27,20),(27,23),(27,29),(28,16),(29,11),(29,26)],30)
=> ([(0,14),(1,13),(1,22),(2,23),(2,24),(3,12),(3,21),(4,5),(4,15),(5,19),(6,20),(6,22),(6,28),(7,24),(7,26),(7,27),(8,19),(8,23),(8,28),(9,13),(9,14),(9,27),(10,11),(10,17),(10,18),(11,16),(11,26),(12,14),(12,27),(13,25),(15,16),(15,19),(15,23),(16,18),(16,29),(17,22),(17,26),(17,28),(18,19),(18,28),(20,21),(20,25),(20,29),(21,24),(21,27),(22,25),(23,29),(24,29),(25,26),(25,27),(26,29),(28,29)],30)
=> ([(0,3),(0,4),(0,9),(0,10),(0,11),(0,12),(0,13),(0,14),(0,15),(0,16),(0,17),(0,18),(0,19),(0,20),(0,21),(0,22),(0,23),(0,24),(0,25),(0,26),(0,27),(0,28),(0,29),(1,2),(1,4),(1,5),(1,6),(1,7),(1,8),(1,10),(1,11),(1,12),(1,13),(1,14),(1,17),(1,18),(1,19),(1,20),(1,21),(1,22),(1,23),(1,24),(1,25),(1,26),(1,27),(1,28),(1,29),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(2,11),(2,12),(2,13),(2,14),(2,15),(2,16),(2,19),(2,20),(2,21),(2,22),(2,23),(2,24),(2,25),(2,26),(2,27),(2,28),(2,29),(3,5),(3,6),(3,8),(3,9),(3,10),(3,12),(3,13),(3,14),(3,15),(3,16),(3,17),(3,18),(3,19),(3,20),(3,22),(3,23),(3,24),(3,25),(3,26),(3,27),(3,28),(3,29),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(4,11),(4,13),(4,14),(4,15),(4,17),(4,18),(4,19),(4,21),(4,23),(4,24),(4,25),(4,26),(4,27),(4,28),(4,29),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(5,13),(5,14),(5,15),(5,17),(5,18),(5,19),(5,20),(5,21),(5,22),(5,23),(5,24),(5,25),(5,27),(5,28),(5,29),(6,7),(6,8),(6,9),(6,10),(6,11),(6,12),(6,14),(6,15),(6,16),(6,18),(6,19),(6,20),(6,21),(6,22),(6,23),(6,24),(6,25),(6,27),(6,28),(6,29),(7,8),(7,9),(7,10),(7,11),(7,13),(7,14),(7,15),(7,16),(7,17),(7,19),(7,20),(7,21),(7,22),(7,23),(7,24),(7,25),(7,26),(7,27),(7,28),(7,29),(8,9),(8,11),(8,12),(8,14),(8,16),(8,17),(8,18),(8,19),(8,20),(8,21),(8,22),(8,23),(8,24),(8,25),(8,26),(8,27),(8,28),(8,29),(9,10),(9,12),(9,13),(9,14),(9,15),(9,16),(9,17),(9,18),(9,19),(9,20),(9,21),(9,22),(9,24),(9,25),(9,26),(9,27),(9,28),(9,29),(10,11),(10,12),(10,13),(10,15),(10,16),(10,17),(10,18),(10,19),(10,20),(10,21),(10,22),(10,24),(10,25),(10,26),(10,27),(10,28),(10,29),(11,12),(11,13),(11,14),(11,15),(11,16),(11,17),(11,19),(11,20),(11,21),(11,22),(11,23),(11,24),(11,25),(11,26),(11,28),(11,29),(12,13),(12,14),(12,15),(12,16),(12,17),(12,18),(12,19),(12,20),(12,21),(12,22),(12,23),(12,24),(12,25),(12,26),(12,27),(12,29),(13,15),(13,16),(13,17),(13,18),(13,19),(13,20),(13,21),(13,22),(13,23),(13,24),(13,26),(13,27),(13,28),(13,29),(14,15),(14,16),(14,18),(14,19),(14,20),(14,21),(14,22),(14,23),(14,25),(14,26),(14,27),(14,28),(14,29),(15,16),(15,17),(15,18),(15,19),(15,20),(15,21),(15,22),(15,24),(15,25),(15,26),(15,27),(15,28),(15,29),(16,17),(16,18),(16,19),(16,20),(16,21),(16,22),(16,23),(16,24),(16,25),(16,26),(16,27),(16,28),(16,29),(17,18),(17,19),(17,20),(17,21),(17,22),(17,23),(17,24),(17,25),(17,26),(17,27),(17,28),(17,29),(18,19),(18,20),(18,21),(18,22),(18,23),(18,24),(18,25),(18,26),(18,27),(18,28),(18,29),(19,21),(19,23),(19,24),(19,25),(19,26),(19,27),(19,28),(20,22),(20,23),(20,24),(20,25),(20,26),(20,27),(20,28),(20,29),(21,22),(21,23),(21,24),(21,25),(21,26),(21,28),(21,29),(22,23),(22,24),(22,25),(22,26),(22,27),(22,29),(23,24),(23,25),(23,26),(23,27),(23,28),(23,29),(24,26),(24,27),(24,28),(24,29),(25,26),(25,27),(25,28),(25,29),(26,27),(26,28),(26,29),(27,28),(27,29),(28,29)],30)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,6} + 1
[[3,3,3,4]]
=> ([(0,1),(1,3),(1,4),(2,14),(3,6),(3,20),(4,5),(4,20),(5,19),(6,7),(6,21),(7,18),(8,12),(8,13),(9,11),(9,17),(10,22),(11,24),(12,23),(13,2),(13,23),(15,13),(15,22),(16,10),(16,24),(17,8),(17,15),(17,24),(18,10),(18,15),(19,11),(19,16),(20,9),(20,19),(20,21),(21,16),(21,17),(21,18),(22,23),(23,14),(24,12),(24,22)],25)
=> ([(0,13),(1,12),(1,19),(2,11),(2,20),(3,4),(3,17),(4,18),(5,6),(5,19),(5,23),(6,21),(6,24),(7,11),(7,13),(7,21),(8,9),(8,17),(8,24),(9,18),(9,23),(10,14),(10,16),(10,20),(11,22),(12,13),(12,21),(14,19),(14,22),(14,23),(15,16),(15,17),(15,20),(15,24),(16,18),(16,23),(17,18),(19,21),(20,22),(21,22),(22,24),(23,24)],25)
=> ([(0,3),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(0,12),(0,14),(0,15),(0,16),(0,17),(0,18),(0,19),(0,20),(0,21),(0,22),(0,23),(0,24),(1,2),(1,4),(1,7),(1,8),(1,9),(1,10),(1,11),(1,13),(1,14),(1,15),(1,16),(1,17),(1,18),(1,19),(1,20),(1,21),(1,22),(1,23),(1,24),(2,3),(2,4),(2,6),(2,7),(2,9),(2,10),(2,11),(2,12),(2,13),(2,14),(2,15),(2,16),(2,17),(2,20),(2,21),(2,22),(2,23),(2,24),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(3,11),(3,12),(3,14),(3,16),(3,18),(3,19),(3,20),(3,21),(3,22),(3,23),(3,24),(4,5),(4,7),(4,10),(4,11),(4,12),(4,13),(4,14),(4,15),(4,16),(4,17),(4,18),(4,19),(4,20),(4,21),(4,22),(4,23),(4,24),(5,6),(5,7),(5,8),(5,9),(5,11),(5,12),(5,13),(5,15),(5,16),(5,17),(5,18),(5,19),(5,20),(5,21),(5,22),(5,23),(5,24),(6,7),(6,8),(6,9),(6,10),(6,11),(6,12),(6,13),(6,15),(6,16),(6,17),(6,18),(6,19),(6,20),(6,22),(6,23),(6,24),(7,10),(7,11),(7,12),(7,13),(7,14),(7,15),(7,16),(7,17),(7,19),(7,20),(7,21),(7,23),(7,24),(8,9),(8,10),(8,11),(8,12),(8,13),(8,15),(8,16),(8,17),(8,18),(8,19),(8,20),(8,21),(8,22),(8,23),(8,24),(9,10),(9,12),(9,13),(9,14),(9,15),(9,16),(9,17),(9,18),(9,19),(9,21),(9,22),(9,23),(9,24),(10,11),(10,12),(10,13),(10,14),(10,15),(10,16),(10,17),(10,18),(10,20),(10,21),(10,22),(10,24),(11,12),(11,13),(11,14),(11,15),(11,16),(11,17),(11,19),(11,20),(11,21),(11,22),(11,23),(11,24),(12,13),(12,14),(12,15),(12,16),(12,18),(12,19),(12,20),(12,22),(12,23),(12,24),(13,14),(13,15),(13,16),(13,17),(13,18),(13,20),(13,21),(13,22),(13,23),(13,24),(14,15),(14,16),(14,17),(14,18),(14,19),(14,20),(14,21),(14,22),(14,23),(14,24),(15,17),(15,18),(15,19),(15,20),(15,21),(15,22),(15,24),(16,18),(16,19),(16,20),(16,21),(16,22),(16,23),(17,18),(17,19),(17,20),(17,21),(17,22),(17,23),(17,24),(18,19),(18,20),(18,21),(18,22),(18,23),(18,24),(19,20),(19,21),(19,22),(19,23),(19,24),(20,21),(20,23),(20,24),(21,22),(21,23),(21,24),(22,23),(22,24),(23,24)],25)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,6} + 1
[[3,3,4,4]]
=> ([(0,1),(1,4),(1,5),(2,24),(3,21),(4,7),(4,25),(5,6),(5,25),(6,23),(7,8),(7,26),(8,22),(9,16),(9,20),(10,15),(10,19),(11,29),(12,29),(14,30),(15,2),(15,28),(16,3),(16,27),(17,16),(17,30),(18,14),(18,28),(19,9),(19,17),(19,28),(20,12),(20,27),(21,13),(22,14),(22,17),(23,15),(23,18),(24,11),(24,12),(25,10),(25,23),(25,26),(26,18),(26,19),(26,22),(27,21),(27,29),(28,20),(28,24),(28,30),(29,13),(30,11),(30,27)],31)
=> ([(0,15),(1,13),(1,24),(2,14),(2,19),(3,14),(3,21),(4,12),(4,20),(5,22),(5,23),(6,12),(6,15),(6,25),(7,17),(7,20),(7,26),(8,16),(8,19),(8,27),(9,23),(9,25),(9,27),(10,16),(10,21),(10,22),(11,21),(11,22),(11,26),(12,28),(13,15),(13,25),(14,29),(16,29),(16,30),(17,24),(17,28),(17,30),(18,19),(18,20),(18,26),(18,27),(19,29),(20,28),(21,29),(22,30),(23,24),(23,30),(24,25),(25,28),(26,29),(26,30),(27,28),(27,30)],31)
=> ([(0,3),(0,4),(0,5),(0,10),(0,11),(0,12),(0,13),(0,14),(0,15),(0,16),(0,17),(0,18),(0,19),(0,20),(0,21),(0,22),(0,23),(0,24),(0,25),(0,26),(0,27),(0,28),(0,29),(0,30),(1,2),(1,3),(1,4),(1,6),(1,7),(1,8),(1,9),(1,10),(1,11),(1,13),(1,14),(1,15),(1,16),(1,17),(1,18),(1,19),(1,20),(1,21),(1,24),(1,25),(1,26),(1,27),(1,28),(1,29),(1,30),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(2,10),(2,11),(2,13),(2,14),(2,15),(2,16),(2,18),(2,20),(2,21),(2,22),(2,23),(2,24),(2,25),(2,26),(2,27),(2,28),(2,29),(2,30),(3,5),(3,7),(3,8),(3,9),(3,11),(3,12),(3,13),(3,14),(3,15),(3,17),(3,18),(3,19),(3,20),(3,21),(3,22),(3,23),(3,24),(3,25),(3,26),(3,27),(3,28),(3,29),(3,30),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(4,12),(4,13),(4,14),(4,16),(4,19),(4,20),(4,21),(4,22),(4,23),(4,24),(4,25),(4,26),(4,27),(4,28),(4,29),(4,30),(5,6),(5,7),(5,8),(5,10),(5,11),(5,12),(5,15),(5,16),(5,17),(5,18),(5,19),(5,21),(5,22),(5,23),(5,24),(5,25),(5,26),(5,27),(5,28),(5,29),(5,30),(6,7),(6,8),(6,9),(6,10),(6,12),(6,13),(6,14),(6,15),(6,16),(6,17),(6,18),(6,19),(6,20),(6,21),(6,22),(6,24),(6,25),(6,26),(6,27),(6,28),(6,29),(6,30),(7,8),(7,9),(7,10),(7,12),(7,13),(7,14),(7,15),(7,16),(7,18),(7,19),(7,20),(7,21),(7,22),(7,23),(7,24),(7,25),(7,27),(7,28),(7,29),(7,30),(8,9),(8,10),(8,11),(8,12),(8,13),(8,14),(8,15),(8,16),(8,17),(8,18),(8,19),(8,20),(8,21),(8,23),(8,25),(8,27),(8,28),(8,29),(8,30),(9,10),(9,11),(9,12),(9,13),(9,14),(9,15),(9,16),(9,17),(9,18),(9,20),(9,21),(9,22),(9,23),(9,25),(9,26),(9,27),(9,28),(9,29),(9,30),(10,11),(10,13),(10,14),(10,15),(10,16),(10,17),(10,18),(10,19),(10,20),(10,21),(10,22),(10,24),(10,25),(10,26),(10,28),(10,29),(10,30),(11,12),(11,13),(11,14),(11,15),(11,16),(11,17),(11,18),(11,19),(11,20),(11,21),(11,22),(11,23),(11,24),(11,25),(11,26),(11,27),(11,29),(11,30),(12,14),(12,15),(12,16),(12,17),(12,18),(12,19),(12,20),(12,21),(12,22),(12,23),(12,24),(12,25),(12,26),(12,27),(12,28),(12,29),(12,30),(13,14),(13,15),(13,16),(13,17),(13,18),(13,20),(13,21),(13,22),(13,23),(13,24),(13,26),(13,27),(13,28),(13,29),(13,30),(14,15),(14,16),(14,17),(14,18),(14,19),(14,20),(14,21),(14,23),(14,25),(14,26),(14,27),(14,28),(14,30),(15,16),(15,17),(15,18),(15,19),(15,20),(15,22),(15,23),(15,24),(15,25),(15,26),(15,27),(15,29),(15,30),(16,17),(16,19),(16,20),(16,21),(16,22),(16,23),(16,24),(16,25),(16,26),(16,28),(16,29),(16,30),(17,18),(17,19),(17,20),(17,21),(17,22),(17,23),(17,24),(17,25),(17,26),(17,27),(17,28),(17,29),(17,30),(18,19),(18,20),(18,22),(18,23),(18,24),(18,25),(18,26),(18,27),(18,28),(18,29),(18,30),(19,20),(19,21),(19,22),(19,23),(19,24),(19,25),(19,26),(19,27),(19,28),(19,29),(19,30),(20,21),(20,22),(20,23),(20,24),(20,26),(20,27),(20,28),(20,30),(21,22),(21,23),(21,24),(21,25),(21,26),(21,27),(21,28),(21,29),(22,23),(22,24),(22,25),(22,26),(22,27),(22,28),(22,29),(22,30),(23,24),(23,25),(23,26),(23,27),(23,28),(23,29),(23,30),(24,25),(24,26),(24,27),(24,28),(24,29),(24,30),(25,26),(25,27),(25,28),(25,29),(25,30),(26,27),(26,28),(26,29),(26,30),(27,28),(27,29),(27,30),(28,29),(28,30),(29,30)],31)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,6} + 1
[[3,4,4,4]]
=> ([(0,1),(1,5),(1,6),(2,24),(3,27),(4,23),(5,8),(5,28),(6,9),(6,28),(7,26),(8,7),(8,29),(9,25),(10,16),(10,22),(11,17),(11,21),(13,30),(14,33),(15,4),(15,33),(16,2),(16,32),(17,3),(17,31),(18,16),(18,30),(19,12),(20,13),(20,31),(21,10),(21,18),(21,31),(22,15),(22,32),(23,12),(24,19),(25,17),(25,20),(26,13),(26,18),(27,14),(27,15),(28,11),(28,25),(28,29),(29,20),(29,21),(29,26),(30,14),(30,32),(31,22),(31,27),(31,30),(32,24),(32,33),(33,19),(33,23)],34)
=> ([(0,17),(1,12),(1,16),(2,13),(2,24),(3,15),(3,21),(4,14),(4,22),(5,16),(5,23),(6,25),(6,26),(7,14),(7,17),(7,31),(8,25),(8,29),(8,32),(9,26),(9,30),(9,31),(10,19),(10,22),(10,29),(11,18),(11,21),(11,30),(12,15),(12,32),(13,17),(13,31),(14,28),(15,27),(16,32),(18,23),(18,27),(18,33),(19,24),(19,28),(19,33),(20,21),(20,22),(20,29),(20,30),(21,27),(22,28),(23,25),(23,32),(24,26),(24,31),(25,33),(26,33),(27,29),(27,32),(28,30),(28,31),(29,33),(30,33)],34)
=> ([(0,3),(0,4),(0,5),(0,6),(0,11),(0,12),(0,13),(0,14),(0,15),(0,16),(0,17),(0,18),(0,19),(0,20),(0,21),(0,22),(0,23),(0,24),(0,25),(0,26),(0,27),(0,28),(0,29),(0,30),(0,31),(0,32),(0,33),(1,2),(1,3),(1,5),(1,6),(1,7),(1,8),(1,9),(1,10),(1,11),(1,12),(1,13),(1,14),(1,16),(1,17),(1,18),(1,19),(1,20),(1,22),(1,23),(1,24),(1,26),(1,27),(1,28),(1,29),(1,30),(1,31),(1,32),(1,33),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(2,10),(2,11),(2,12),(2,13),(2,14),(2,16),(2,17),(2,18),(2,19),(2,21),(2,22),(2,23),(2,25),(2,26),(2,27),(2,28),(2,29),(2,30),(2,31),(2,32),(2,33),(3,4),(3,5),(3,7),(3,9),(3,10),(3,11),(3,12),(3,13),(3,15),(3,16),(3,17),(3,18),(3,20),(3,21),(3,22),(3,23),(3,24),(3,25),(3,26),(3,27),(3,28),(3,29),(3,30),(3,31),(3,32),(3,33),(4,6),(4,8),(4,9),(4,10),(4,11),(4,12),(4,14),(4,15),(4,16),(4,17),(4,19),(4,20),(4,21),(4,22),(4,23),(4,24),(4,25),(4,26),(4,27),(4,28),(4,29),(4,30),(4,31),(4,32),(4,33),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(5,13),(5,14),(5,15),(5,17),(5,18),(5,19),(5,20),(5,23),(5,24),(5,25),(5,26),(5,27),(5,28),(5,29),(5,30),(5,31),(5,32),(5,33),(6,7),(6,8),(6,9),(6,10),(6,12),(6,13),(6,14),(6,15),(6,16),(6,18),(6,19),(6,21),(6,22),(6,24),(6,25),(6,26),(6,27),(6,28),(6,29),(6,30),(6,31),(6,32),(6,33),(7,8),(7,9),(7,10),(7,11),(7,13),(7,14),(7,15),(7,16),(7,17),(7,18),(7,19),(7,20),(7,21),(7,22),(7,23),(7,25),(7,26),(7,27),(7,28),(7,29),(7,30),(7,31),(7,32),(7,33),(8,9),(8,10),(8,12),(8,13),(8,14),(8,15),(8,16),(8,17),(8,18),(8,19),(8,20),(8,21),(8,22),(8,23),(8,24),(8,26),(8,27),(8,28),(8,29),(8,30),(8,31),(8,32),(8,33),(9,10),(9,11),(9,13),(9,14),(9,15),(9,16),(9,17),(9,18),(9,19),(9,20),(9,22),(9,23),(9,24),(9,25),(9,26),(9,27),(9,28),(9,29),(9,30),(9,32),(9,33),(10,12),(10,13),(10,14),(10,15),(10,16),(10,17),(10,18),(10,19),(10,21),(10,22),(10,23),(10,24),(10,25),(10,26),(10,27),(10,28),(10,29),(10,30),(10,32),(10,33),(11,12),(11,13),(11,14),(11,15),(11,16),(11,17),(11,18),(11,19),(11,20),(11,21),(11,22),(11,23),(11,24),(11,25),(11,26),(11,27),(11,28),(11,29),(11,31),(11,32),(11,33),(12,13),(12,14),(12,15),(12,16),(12,17),(12,18),(12,19),(12,20),(12,21),(12,22),(12,23),(12,24),(12,25),(12,26),(12,27),(12,28),(12,30),(12,31),(12,32),(12,33),(13,14),(13,16),(13,17),(13,18),(13,19),(13,20),(13,21),(13,22),(13,23),(13,25),(13,26),(13,27),(13,29),(13,30),(13,31),(13,32),(13,33),(14,16),(14,17),(14,18),(14,19),(14,20),(14,21),(14,22),(14,23),(14,24),(14,26),(14,28),(14,29),(14,30),(14,31),(14,32),(14,33),(15,16),(15,17),(15,18),(15,19),(15,20),(15,21),(15,22),(15,23),(15,24),(15,25),(15,26),(15,27),(15,28),(15,29),(15,30),(15,31),(15,32),(15,33),(16,17),(16,18),(16,19),(16,20),(16,21),(16,22),(16,23),(16,24),(16,25),(16,27),(16,28),(16,30),(16,31),(16,32),(16,33),(17,18),(17,19),(17,20),(17,21),(17,22),(17,23),(17,24),(17,25),(17,26),(17,27),(17,28),(17,29),(17,31),(17,33),(18,19),(18,20),(18,21),(18,23),(18,24),(18,25),(18,26),(18,27),(18,29),(18,30),(18,31),(18,32),(18,33),(19,20),(19,21),(19,22),(19,24),(19,25),(19,26),(19,28),(19,29),(19,30),(19,31),(19,32),(19,33),(20,21),(20,22),(20,23),(20,24),(20,25),(20,26),(20,27),(20,28),(20,29),(20,30),(20,31),(20,32),(20,33),(21,22),(21,23),(21,24),(21,25),(21,26),(21,27),(21,28),(21,29),(21,30),(21,31),(21,32),(21,33),(22,23),(22,24),(22,25),(22,27),(22,28),(22,29),(22,30),(22,31),(22,32),(22,33),(23,24),(23,25),(23,26),(23,27),(23,28),(23,29),(23,30),(23,31),(23,33),(24,25),(24,26),(24,27),(24,28),(24,29),(24,30),(24,31),(24,32),(24,33),(25,26),(25,27),(25,28),(25,29),(25,30),(25,31),(25,32),(25,33),(26,27),(26,28),(26,29),(26,30),(26,31),(26,32),(27,28),(27,29),(27,30),(27,31),(27,32),(27,33),(28,29),(28,30),(28,31),(28,32),(28,33),(29,30),(29,31),(29,32),(29,33),(30,31),(30,32),(30,33),(31,32),(31,33),(32,33)],34)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,6} + 1
[[4,4,4,4]]
=> ([(0,2),(2,6),(2,7),(3,25),(4,28),(5,24),(6,9),(6,29),(7,10),(7,29),(8,27),(9,8),(9,30),(10,26),(11,17),(11,23),(12,18),(12,22),(13,31),(14,34),(15,1),(16,5),(16,34),(17,3),(17,33),(18,4),(18,32),(19,15),(20,17),(20,31),(21,13),(21,32),(22,11),(22,20),(22,32),(23,16),(23,33),(24,15),(25,19),(26,18),(26,21),(27,13),(27,20),(28,14),(28,16),(29,12),(29,26),(29,30),(30,21),(30,22),(30,27),(31,14),(31,33),(32,23),(32,28),(32,31),(33,25),(33,34),(34,19),(34,24)],35)
=> ([(0,18),(1,17),(2,15),(2,22),(3,16),(3,23),(4,13),(4,24),(5,14),(5,25),(6,26),(6,27),(7,15),(7,17),(7,32),(8,16),(8,18),(8,33),(9,26),(9,30),(9,32),(10,27),(10,31),(10,33),(11,19),(11,22),(11,31),(12,20),(12,23),(12,30),(13,17),(13,32),(14,18),(14,33),(15,28),(16,29),(19,24),(19,28),(19,34),(20,25),(20,29),(20,34),(21,22),(21,23),(21,30),(21,31),(22,28),(23,29),(24,26),(24,32),(25,27),(25,33),(26,34),(27,34),(28,30),(28,32),(29,31),(29,33),(30,34),(31,34)],35)
=> ([(0,3),(0,4),(0,5),(0,6),(0,11),(0,12),(0,13),(0,14),(0,15),(0,16),(0,17),(0,18),(0,19),(0,20),(0,21),(0,22),(0,23),(0,24),(0,25),(0,26),(0,27),(0,28),(0,29),(0,30),(0,31),(0,32),(0,33),(0,34),(1,2),(1,4),(1,5),(1,6),(1,7),(1,8),(1,9),(1,10),(1,11),(1,12),(1,13),(1,14),(1,16),(1,17),(1,18),(1,19),(1,21),(1,22),(1,23),(1,25),(1,26),(1,27),(1,28),(1,29),(1,30),(1,31),(1,32),(1,33),(1,34),(2,3),(2,5),(2,6),(2,7),(2,8),(2,9),(2,10),(2,11),(2,12),(2,13),(2,14),(2,16),(2,17),(2,18),(2,19),(2,20),(2,22),(2,23),(2,24),(2,26),(2,27),(2,28),(2,29),(2,30),(2,31),(2,32),(2,33),(2,34),(3,4),(3,5),(3,7),(3,9),(3,10),(3,11),(3,12),(3,13),(3,15),(3,16),(3,17),(3,18),(3,20),(3,21),(3,22),(3,23),(3,24),(3,25),(3,26),(3,27),(3,28),(3,29),(3,30),(3,31),(3,32),(3,33),(3,34),(4,6),(4,8),(4,9),(4,10),(4,11),(4,12),(4,14),(4,15),(4,16),(4,17),(4,19),(4,20),(4,21),(4,22),(4,23),(4,24),(4,25),(4,26),(4,27),(4,28),(4,29),(4,30),(4,31),(4,32),(4,33),(4,34),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(5,13),(5,14),(5,15),(5,16),(5,18),(5,19),(5,20),(5,22),(5,24),(5,25),(5,26),(5,27),(5,28),(5,29),(5,30),(5,31),(5,32),(5,33),(5,34),(6,7),(6,8),(6,9),(6,10),(6,12),(6,13),(6,14),(6,15),(6,17),(6,18),(6,19),(6,21),(6,23),(6,24),(6,25),(6,26),(6,27),(6,28),(6,29),(6,30),(6,31),(6,32),(6,33),(6,34),(7,8),(7,9),(7,10),(7,11),(7,13),(7,14),(7,15),(7,16),(7,17),(7,18),(7,19),(7,20),(7,21),(7,22),(7,23),(7,25),(7,26),(7,27),(7,28),(7,29),(7,30),(7,31),(7,32),(7,33),(7,34),(8,9),(8,10),(8,12),(8,13),(8,14),(8,15),(8,16),(8,17),(8,18),(8,19),(8,20),(8,21),(8,22),(8,23),(8,24),(8,26),(8,27),(8,28),(8,29),(8,30),(8,31),(8,32),(8,33),(8,34),(9,10),(9,11),(9,13),(9,14),(9,15),(9,16),(9,17),(9,18),(9,19),(9,20),(9,22),(9,23),(9,24),(9,25),(9,26),(9,27),(9,28),(9,29),(9,30),(9,31),(9,33),(9,34),(10,12),(10,13),(10,14),(10,15),(10,16),(10,17),(10,18),(10,19),(10,21),(10,22),(10,23),(10,24),(10,25),(10,26),(10,27),(10,28),(10,29),(10,30),(10,31),(10,33),(10,34),(11,12),(11,13),(11,14),(11,15),(11,16),(11,17),(11,18),(11,19),(11,20),(11,21),(11,22),(11,23),(11,24),(11,25),(11,26),(11,27),(11,28),(11,29),(11,30),(11,32),(11,33),(11,34),(12,13),(12,14),(12,15),(12,16),(12,17),(12,18),(12,19),(12,20),(12,21),(12,22),(12,23),(12,24),(12,25),(12,26),(12,27),(12,28),(12,29),(12,31),(12,32),(12,33),(12,34),(13,14),(13,16),(13,17),(13,18),(13,19),(13,20),(13,21),(13,22),(13,23),(13,25),(13,26),(13,27),(13,28),(13,30),(13,31),(13,32),(13,33),(13,34),(14,16),(14,17),(14,18),(14,19),(14,20),(14,21),(14,22),(14,23),(14,24),(14,26),(14,27),(14,29),(14,30),(14,31),(14,32),(14,33),(14,34),(15,16),(15,17),(15,18),(15,19),(15,20),(15,21),(15,22),(15,23),(15,24),(15,25),(15,26),(15,27),(15,28),(15,29),(15,30),(15,31),(15,32),(15,33),(15,34),(16,17),(16,18),(16,19),(16,20),(16,21),(16,22),(16,23),(16,24),(16,25),(16,26),(16,28),(16,29),(16,30),(16,32),(16,33),(16,34),(17,18),(17,19),(17,20),(17,21),(17,22),(17,23),(17,24),(17,25),(17,27),(17,28),(17,29),(17,31),(17,32),(17,33),(17,34),(18,19),(18,20),(18,21),(18,22),(18,24),(18,25),(18,26),(18,27),(18,28),(18,30),(18,31),(18,32),(18,33),(18,34),(19,20),(19,21),(19,23),(19,24),(19,25),(19,26),(19,27),(19,29),(19,30),(19,31),(19,32),(19,33),(19,34),(20,21),(20,22),(20,23),(20,24),(20,25),(20,26),(20,27),(20,28),(20,29),(20,30),(20,31),(20,32),(20,33),(20,34),(21,22),(21,23),(21,24),(21,25),(21,26),(21,27),(21,28),(21,29),(21,30),(21,31),(21,32),(21,33),(21,34),(22,23),(22,24),(22,25),(22,26),(22,28),(22,29),(22,30),(22,31),(22,32),(22,33),(22,34),(23,24),(23,25),(23,27),(23,28),(23,29),(23,30),(23,31),(23,32),(23,33),(23,34),(24,25),(24,26),(24,27),(24,28),(24,29),(24,30),(24,31),(24,32),(24,33),(24,34),(25,26),(25,27),(25,28),(25,29),(25,30),(25,31),(25,32),(25,33),(25,34),(26,27),(26,28),(26,29),(26,30),(26,31),(26,32),(26,33),(27,28),(27,29),(27,30),(27,31),(27,32),(27,34),(28,29),(28,30),(28,31),(28,32),(28,33),(28,34),(29,30),(29,31),(29,32),(29,33),(29,34),(30,31),(30,32),(30,33),(30,34),(31,32),(31,33),(31,34),(32,33),(32,34),(33,34)],35)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,6} + 1
[[1,1,4],[4]]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5)],7)
=> ([(0,1),(0,4),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(5,6)],7)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,6} + 1
[[1,2,2],[4]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(0,6),(1,2),(1,4),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
=> ([(0,1),(0,4),(0,6),(1,3),(1,4),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,6} + 1
[[1,2,4],[2]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(0,6),(1,2),(1,4),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
=> ([(0,1),(0,4),(0,6),(1,3),(1,4),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,6} + 1
[[1,2,3],[4]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ([(0,7),(1,3),(1,8),(2,7),(2,8),(3,5),(4,5),(4,6),(5,8),(6,7),(6,8)],9)
=> ([(0,3),(0,6),(0,7),(0,8),(1,2),(1,4),(1,5),(1,7),(1,8),(2,3),(2,4),(2,5),(2,8),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(4,8),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,6} + 1
[[1,2,4],[3]]
=> ([(0,5),(0,6),(1,8),(2,9),(3,8),(3,9),(4,1),(5,4),(6,7),(7,2),(7,3),(8,10),(9,10)],11)
=> ([(0,1),(0,2),(1,4),(2,3),(3,8),(4,9),(5,7),(5,8),(6,7),(6,9),(7,10),(8,10),(9,10)],11)
=> ([(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(1,2),(1,3),(1,6),(1,7),(1,8),(1,9),(1,10),(2,3),(2,5),(2,7),(2,8),(2,9),(2,10),(3,4),(3,6),(3,8),(3,9),(3,10),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,10),(7,9),(7,10),(8,9)],11)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,6} + 1
[[1,3,4],[2]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ([(0,7),(1,3),(1,8),(2,7),(2,8),(3,5),(4,5),(4,6),(5,8),(6,7),(6,8)],9)
=> ([(0,3),(0,6),(0,7),(0,8),(1,2),(1,4),(1,5),(1,7),(1,8),(2,3),(2,4),(2,5),(2,8),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(4,8),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,6} + 1
[[1,2,4],[4]]
=> ([(0,6),(0,7),(1,9),(2,12),(3,9),(3,12),(4,10),(5,1),(6,5),(7,8),(8,2),(8,3),(9,11),(11,10),(12,4),(12,11)],13)
=> ([(0,1),(0,2),(1,4),(2,6),(3,5),(3,12),(4,8),(5,9),(6,10),(7,8),(7,12),(8,11),(9,10),(9,12),(10,11),(11,12)],13)
=> ([(0,3),(0,4),(0,7),(0,8),(0,9),(0,10),(0,11),(0,12),(1,2),(1,5),(1,6),(1,7),(1,8),(1,9),(1,10),(1,11),(1,12),(2,4),(2,5),(2,6),(2,8),(2,9),(2,10),(2,11),(2,12),(3,4),(3,5),(3,6),(3,7),(3,9),(3,10),(3,11),(3,12),(4,6),(4,7),(4,8),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(5,12),(6,8),(6,9),(6,10),(6,11),(6,12),(7,8),(7,9),(7,10),(7,11),(7,12),(8,9),(8,11),(8,12),(9,10),(9,12),(10,11)],13)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,6} + 1
[[1,4,4],[2]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ([(0,8),(1,7),(2,5),(2,6),(3,7),(3,9),(4,8),(4,9),(5,7),(5,9),(6,8),(6,9)],10)
=> ([(0,3),(0,4),(0,7),(0,8),(0,9),(1,2),(1,4),(1,5),(1,6),(1,8),(1,9),(2,3),(2,5),(2,6),(2,8),(2,9),(3,4),(3,5),(3,7),(3,8),(4,6),(4,7),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,6} + 1
[[1,3,3],[4]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ([(0,8),(1,7),(2,5),(2,6),(3,7),(3,9),(4,8),(4,9),(5,7),(5,9),(6,8),(6,9)],10)
=> ([(0,3),(0,4),(0,7),(0,8),(0,9),(1,2),(1,4),(1,5),(1,6),(1,8),(1,9),(2,3),(2,5),(2,6),(2,8),(2,9),(3,4),(3,5),(3,7),(3,8),(4,6),(4,7),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,6} + 1
[[1,3,4],[3]]
=> ([(0,6),(1,7),(2,8),(3,9),(4,3),(4,7),(5,2),(5,10),(6,1),(6,4),(7,5),(7,9),(9,10),(10,8)],11)
=> ([(0,9),(1,7),(1,8),(2,9),(2,10),(3,4),(3,5),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(8,10)],11)
=> ([(0,4),(0,5),(0,7),(0,8),(0,9),(0,10),(1,2),(1,3),(1,5),(1,6),(1,7),(1,8),(1,10),(2,3),(2,5),(2,6),(2,7),(2,9),(2,10),(3,4),(3,6),(3,7),(3,8),(3,10),(4,5),(4,6),(4,8),(4,9),(4,10),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(6,10),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,6} + 1
[[1,3,4],[4]]
=> ([(0,9),(0,11),(1,18),(2,17),(3,19),(4,13),(4,19),(5,12),(5,13),(6,16),(7,14),(8,5),(8,18),(9,10),(10,3),(10,4),(11,1),(11,8),(12,17),(13,15),(15,16),(16,14),(17,7),(18,2),(18,12),(19,6),(19,15)],20)
=> ([(0,15),(0,18),(1,14),(1,18),(2,16),(2,19),(3,17),(3,19),(4,6),(4,14),(5,7),(5,15),(6,16),(7,17),(8,9),(8,12),(8,13),(9,10),(9,11),(10,14),(10,18),(11,15),(11,18),(12,16),(12,19),(13,17),(13,19)],20)
=> ([(0,1),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(0,11),(0,14),(0,15),(0,16),(0,17),(0,18),(0,19),(1,2),(1,3),(1,6),(1,7),(1,8),(1,9),(1,10),(1,11),(1,12),(1,13),(1,16),(1,17),(1,18),(1,19),(2,3),(2,4),(2,5),(2,6),(2,7),(2,9),(2,10),(2,12),(2,13),(2,14),(2,15),(2,16),(2,17),(2,18),(2,19),(3,4),(3,5),(3,6),(3,7),(3,8),(3,10),(3,12),(3,13),(3,14),(3,15),(3,16),(3,17),(3,18),(3,19),(4,5),(4,6),(4,8),(4,9),(4,11),(4,12),(4,13),(4,14),(4,15),(4,16),(4,17),(4,18),(4,19),(5,7),(5,8),(5,9),(5,11),(5,12),(5,13),(5,14),(5,15),(5,16),(5,17),(5,18),(5,19),(6,7),(6,8),(6,9),(6,10),(6,11),(6,12),(6,13),(6,14),(6,16),(6,17),(6,18),(7,8),(7,9),(7,10),(7,11),(7,12),(7,13),(7,15),(7,16),(7,17),(7,19),(8,9),(8,10),(8,11),(8,12),(8,14),(8,15),(8,16),(8,18),(8,19),(9,10),(9,11),(9,13),(9,14),(9,15),(9,17),(9,18),(9,19),(10,12),(10,13),(10,14),(10,15),(10,16),(10,17),(10,18),(10,19),(11,12),(11,13),(11,14),(11,15),(11,16),(11,17),(11,18),(11,19),(12,13),(12,14),(12,15),(12,16),(12,17),(12,18),(12,19),(13,14),(13,15),(13,16),(13,17),(13,18),(13,19),(14,15),(14,16),(14,17),(14,18),(14,19),(15,16),(15,17),(15,18),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,6} + 1
[[1,4,4],[3]]
=> ([(0,7),(1,10),(2,11),(3,8),(4,9),(5,2),(5,9),(6,3),(6,12),(7,4),(7,5),(8,10),(9,6),(9,11),(11,12),(12,1),(12,8)],13)
=> ([(0,10),(1,3),(1,11),(2,10),(2,12),(3,6),(4,6),(4,8),(5,7),(5,9),(6,11),(7,10),(7,12),(8,11),(8,12),(9,11),(9,12)],13)
=> ([(0,1),(0,5),(0,6),(0,8),(0,9),(0,10),(0,11),(0,12),(1,4),(1,6),(1,7),(1,9),(1,10),(1,11),(1,12),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(2,10),(2,12),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(3,11),(3,12),(4,5),(4,6),(4,7),(4,8),(4,11),(4,12),(5,7),(5,8),(5,9),(5,10),(5,12),(6,7),(6,9),(6,10),(6,11),(7,8),(7,10),(7,11),(7,12),(8,9),(8,10),(8,11),(8,12),(9,10),(9,11),(9,12),(10,11),(10,12),(11,12)],13)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,6} + 1
[[1,4,4],[4]]
=> ([(0,8),(2,13),(3,11),(4,9),(5,10),(6,3),(6,10),(7,4),(7,12),(8,5),(8,6),(9,13),(10,7),(10,11),(11,12),(12,2),(12,9),(13,1)],14)
=> ([(0,11),(1,10),(2,6),(2,8),(3,7),(3,9),(4,10),(4,12),(5,11),(5,13),(6,10),(6,12),(7,11),(7,13),(8,12),(8,13),(9,12),(9,13)],14)
=> ([(0,1),(0,5),(0,6),(0,7),(0,9),(0,10),(0,11),(0,12),(0,13),(1,4),(1,6),(1,7),(1,8),(1,10),(1,11),(1,12),(1,13),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(2,10),(2,12),(2,13),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(3,11),(3,12),(3,13),(4,5),(4,6),(4,8),(4,9),(4,10),(4,12),(4,13),(5,7),(5,8),(5,9),(5,11),(5,12),(5,13),(6,7),(6,8),(6,10),(6,11),(6,12),(7,9),(7,10),(7,11),(7,13),(8,9),(8,10),(8,11),(8,12),(8,13),(9,10),(9,11),(9,12),(9,13),(10,11),(10,12),(10,13),(11,12),(11,13),(12,13)],14)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,6} + 1
[[2,2,2],[4]]
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> ([(0,8),(1,4),(1,8),(2,3),(2,6),(3,7),(4,5),(4,6),(5,7),(5,8),(6,7)],9)
=> ([(0,1),(0,2),(0,3),(0,5),(0,6),(1,2),(1,5),(1,6),(1,8),(2,5),(2,7),(2,8),(3,4),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,6} + 1
[[2,2,3],[4]]
=> ([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
=> ([(0,9),(1,7),(1,8),(2,7),(2,11),(3,6),(3,8),(4,9),(4,11),(5,6),(5,9),(5,11),(6,10),(7,10),(8,10),(10,11)],12)
=> ([(0,2),(0,4),(0,5),(0,6),(0,9),(0,10),(0,11),(1,3),(1,6),(1,7),(1,8),(1,9),(1,10),(1,11),(2,3),(2,4),(2,5),(2,6),(2,7),(2,10),(2,11),(3,4),(3,7),(3,8),(3,9),(3,10),(3,11),(4,5),(4,6),(4,7),(4,8),(4,11),(5,6),(5,7),(5,8),(5,9),(5,11),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,6} + 1
[[2,2,4],[3]]
=> ([(0,6),(0,7),(1,11),(2,9),(3,9),(3,10),(4,5),(5,1),(5,10),(6,4),(7,8),(8,2),(8,3),(9,12),(10,11),(10,12),(11,13),(12,13)],14)
=> ([(0,1),(0,2),(1,3),(2,8),(3,9),(4,7),(4,8),(5,7),(5,10),(6,9),(6,11),(7,13),(8,13),(9,12),(10,11),(10,13),(11,12),(12,13)],14)
=> ([(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(0,11),(0,12),(0,13),(1,2),(1,3),(1,4),(1,7),(1,8),(1,9),(1,10),(1,11),(1,12),(1,13),(2,3),(2,4),(2,6),(2,8),(2,9),(2,10),(2,11),(2,12),(2,13),(3,4),(3,5),(3,6),(3,9),(3,10),(3,11),(3,12),(3,13),(4,5),(4,6),(4,7),(4,10),(4,11),(4,12),(4,13),(5,6),(5,7),(5,8),(5,9),(5,11),(5,12),(5,13),(6,7),(6,8),(6,9),(6,12),(6,13),(7,8),(7,9),(7,10),(7,11),(7,12),(7,13),(8,9),(8,10),(8,11),(8,12),(8,13),(9,10),(9,11),(9,13),(10,11),(10,12),(10,13),(11,12)],14)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,6} + 1
Description
The order of the largest clique of the graph.
A clique in a graph $G$ is a subset $U \subseteq V(G)$ such that any pair of vertices in $U$ are adjacent. I.e. the subgraph induced by $U$ is a complete graph.
Matching statistic: St000098
Values
[[1,2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 1 = 0 + 1
[[2,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2 = 1 + 1
[[1],[2]]
=> ([],1)
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[1,1,2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 1 = 0 + 1
[[1,2,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2 = 1 + 1
[[2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[[1,1],[2]]
=> ([],1)
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[1,2],[2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 1 = 0 + 1
[[1,1,3]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2 = 1 + 1
[[1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
[[1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 3 + 1
[[2,2,3]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(0,6),(1,2),(1,4),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
=> ([(0,1),(0,4),(0,6),(1,3),(1,4),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,0,0,0} + 1
[[2,3,3]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ([(0,7),(1,3),(1,8),(2,7),(2,8),(3,5),(4,5),(4,6),(5,8),(6,7),(6,8)],9)
=> ([(0,3),(0,6),(0,7),(0,8),(1,2),(1,4),(1,5),(1,7),(1,8),(2,3),(2,4),(2,5),(2,8),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(4,8),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {0,0,0,0} + 1
[[3,3,3]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ([(0,8),(1,7),(2,5),(2,6),(3,7),(3,9),(4,8),(4,9),(5,7),(5,9),(6,8),(6,9)],10)
=> ([(0,3),(0,4),(0,7),(0,8),(0,9),(1,2),(1,4),(1,5),(1,6),(1,8),(1,9),(2,3),(2,5),(2,6),(2,8),(2,9),(3,4),(3,5),(3,7),(3,8),(4,6),(4,7),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? ∊ {0,0,0,0} + 1
[[1,1],[3]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 1 = 0 + 1
[[1,2],[3]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2 = 1 + 1
[[1,3],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2 = 1 + 1
[[1,3],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[[2,2],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[[2,3],[3]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ([(0,6),(0,7),(1,2),(1,3),(2,5),(3,4),(4,6),(5,7)],8)
=> ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,4),(1,5),(1,6),(1,7),(2,3),(2,5),(2,6),(2,7),(3,4),(3,5),(3,7),(4,6),(4,7),(5,6)],8)
=> ? ∊ {0,0,0,0} + 1
[[1],[2],[3]]
=> ([],1)
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[1,1,1,2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 1 = 0 + 1
[[1,1,2,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2 = 1 + 1
[[1,2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[[2,2,2,2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[[1,1,1],[2]]
=> ([],1)
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[1,1,2],[2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 1 = 0 + 1
[[1,2,2],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2 = 1 + 1
[[1,1],[2,2]]
=> ([],1)
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[1,1,1,3]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2 = 1 + 1
[[1,1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
[[1,1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 3 + 1
[[1,2,2,3]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(0,6),(1,2),(1,4),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
=> ([(0,1),(0,4),(0,6),(1,3),(1,4),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,0,0,0,0,0,1,2,2,2,2} + 1
[[1,2,3,3]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ([(0,7),(1,3),(1,8),(2,7),(2,8),(3,5),(4,5),(4,6),(5,8),(6,7),(6,8)],9)
=> ([(0,3),(0,6),(0,7),(0,8),(1,2),(1,4),(1,5),(1,7),(1,8),(2,3),(2,4),(2,5),(2,8),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(4,8),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {0,0,0,0,0,0,1,2,2,2,2} + 1
[[1,3,3,3]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ([(0,8),(1,7),(2,5),(2,6),(3,7),(3,9),(4,8),(4,9),(5,7),(5,9),(6,8),(6,9)],10)
=> ([(0,3),(0,4),(0,7),(0,8),(0,9),(1,2),(1,4),(1,5),(1,6),(1,8),(1,9),(2,3),(2,5),(2,6),(2,8),(2,9),(3,4),(3,5),(3,7),(3,8),(4,6),(4,7),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? ∊ {0,0,0,0,0,0,1,2,2,2,2} + 1
[[2,2,2,3]]
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> ([(0,8),(1,4),(1,8),(2,3),(2,6),(3,7),(4,5),(4,6),(5,7),(5,8),(6,7)],9)
=> ([(0,1),(0,2),(0,3),(0,5),(0,6),(1,2),(1,5),(1,6),(1,8),(2,5),(2,7),(2,8),(3,4),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {0,0,0,0,0,0,1,2,2,2,2} + 1
[[2,2,3,3]]
=> ([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
=> ([(0,9),(1,7),(1,8),(2,7),(2,11),(3,6),(3,8),(4,9),(4,11),(5,6),(5,9),(5,11),(6,10),(7,10),(8,10),(10,11)],12)
=> ([(0,2),(0,4),(0,5),(0,6),(0,9),(0,10),(0,11),(1,3),(1,6),(1,7),(1,8),(1,9),(1,10),(1,11),(2,3),(2,4),(2,5),(2,6),(2,7),(2,10),(2,11),(3,4),(3,7),(3,8),(3,9),(3,10),(3,11),(4,5),(4,6),(4,7),(4,8),(4,11),(5,6),(5,7),(5,8),(5,9),(5,11),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? ∊ {0,0,0,0,0,0,1,2,2,2,2} + 1
[[2,3,3,3]]
=> ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
=> ([(0,10),(1,4),(1,12),(2,10),(2,11),(3,11),(3,12),(4,8),(5,6),(5,7),(6,9),(6,13),(7,8),(7,13),(8,12),(9,10),(9,11),(11,13),(12,13)],14)
=> ([(0,3),(0,4),(0,5),(0,8),(0,9),(0,10),(0,11),(0,12),(0,13),(1,2),(1,3),(1,5),(1,6),(1,7),(1,9),(1,11),(1,12),(1,13),(2,4),(2,5),(2,6),(2,7),(2,10),(2,11),(2,12),(2,13),(3,4),(3,5),(3,6),(3,8),(3,9),(3,10),(3,12),(3,13),(4,7),(4,8),(4,9),(4,10),(4,11),(4,12),(4,13),(5,6),(5,7),(5,8),(5,9),(5,11),(5,12),(6,7),(6,8),(6,9),(6,10),(6,11),(6,13),(7,8),(7,9),(7,10),(7,11),(7,13),(8,9),(8,10),(8,11),(8,12),(8,13),(9,10),(9,12),(9,13),(10,11),(10,12),(10,13),(11,12),(11,13),(12,13)],14)
=> ? ∊ {0,0,0,0,0,0,1,2,2,2,2} + 1
[[3,3,3,3]]
=> ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
=> ([(0,11),(1,10),(2,8),(2,9),(3,10),(3,13),(4,11),(4,14),(5,13),(5,14),(6,8),(6,10),(6,13),(7,9),(7,11),(7,14),(8,12),(9,12),(12,13),(12,14)],15)
=> ([(0,3),(0,4),(0,5),(0,6),(0,9),(0,10),(0,11),(0,12),(0,13),(0,14),(1,2),(1,4),(1,5),(1,6),(1,7),(1,8),(1,11),(1,12),(1,13),(1,14),(2,3),(2,5),(2,6),(2,7),(2,8),(2,10),(2,12),(2,13),(2,14),(3,4),(3,5),(3,7),(3,9),(3,10),(3,11),(3,12),(3,13),(3,14),(4,6),(4,8),(4,9),(4,10),(4,11),(4,12),(4,13),(4,14),(5,6),(5,7),(5,8),(5,9),(5,10),(5,12),(5,13),(6,7),(6,8),(6,9),(6,11),(6,12),(6,14),(7,8),(7,9),(7,10),(7,11),(7,13),(7,14),(8,9),(8,10),(8,11),(8,13),(8,14),(9,10),(9,11),(9,12),(9,13),(9,14),(10,11),(10,12),(10,13),(10,14),(11,12),(11,13),(11,14),(12,13),(12,14),(13,14)],15)
=> ? ∊ {0,0,0,0,0,0,1,2,2,2,2} + 1
[[1,1,1],[3]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 1 = 0 + 1
[[1,1,2],[3]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2 = 1 + 1
[[1,1,3],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2 = 1 + 1
[[1,1,3],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[[1,2,2],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[[1,2,3],[2]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
[[1,2,3],[3]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ([(0,6),(0,7),(1,2),(1,3),(2,5),(3,4),(4,6),(5,7)],8)
=> ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,4),(1,5),(1,6),(1,7),(2,3),(2,5),(2,6),(2,7),(3,4),(3,5),(3,7),(4,6),(4,7),(5,6)],8)
=> ? ∊ {0,0,0,0,0,0,1,2,2,2,2} + 1
[[1,3,3],[2]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 3 + 1
[[1,3,3],[3]]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5)],7)
=> ([(0,1),(0,4),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(5,6)],7)
=> ? ∊ {0,0,0,0,0,0,1,2,2,2,2} + 1
[[2,2,2],[3]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[[2,2,3],[3]]
=> ([(0,6),(0,7),(1,9),(2,8),(3,5),(4,2),(5,1),(5,8),(6,3),(7,4),(8,9)],10)
=> ([(0,1),(0,3),(1,2),(2,4),(3,5),(4,8),(5,9),(6,7),(6,8),(7,9),(8,9)],10)
=> ([(0,3),(0,5),(0,6),(0,7),(0,8),(0,9),(1,2),(1,4),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,9),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,8),(7,9),(8,9)],10)
=> ? ∊ {0,0,0,0,0,0,1,2,2,2,2} + 1
[[2,3,3],[3]]
=> ([(0,9),(0,10),(1,11),(2,14),(3,12),(4,13),(5,4),(5,11),(6,5),(7,3),(8,1),(8,14),(9,6),(10,2),(10,8),(11,13),(13,12),(14,7)],15)
=> ([(0,4),(0,5),(1,2),(1,3),(2,10),(3,9),(4,11),(5,12),(6,9),(6,11),(7,10),(7,12),(8,13),(8,14),(9,13),(10,14),(11,13),(12,14)],15)
=> ([(0,1),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(0,11),(0,13),(0,14),(1,2),(1,3),(1,6),(1,7),(1,8),(1,9),(1,10),(1,11),(1,13),(1,14),(2,3),(2,4),(2,5),(2,6),(2,8),(2,9),(2,11),(2,12),(2,13),(2,14),(3,4),(3,5),(3,6),(3,8),(3,9),(3,10),(3,12),(3,13),(3,14),(4,5),(4,7),(4,8),(4,10),(4,11),(4,12),(4,13),(4,14),(5,7),(5,9),(5,10),(5,11),(5,12),(5,13),(5,14),(6,7),(6,8),(6,9),(6,10),(6,11),(6,12),(6,13),(6,14),(7,8),(7,9),(7,10),(7,11),(7,12),(7,13),(7,14),(8,9),(8,10),(8,11),(8,12),(8,13),(9,10),(9,11),(9,12),(9,14),(10,11),(10,12),(10,13),(11,12),(11,14),(12,13),(12,14),(13,14)],15)
=> ? ∊ {0,0,0,0,0,0,1,2,2,2,2} + 1
[[1,1],[2,3]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 1 = 0 + 1
[[1,1],[3,3]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2 = 1 + 1
[[1,2],[2,3]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2 = 1 + 1
[[1,2],[3,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
[[2,2],[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 3 + 1
[[1,1],[2],[3]]
=> ([],1)
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[1,2],[2],[3]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 1 = 0 + 1
[[1,3],[2],[3]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2 = 1 + 1
[[1,1,1,1,2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 1 = 0 + 1
[[1,1,1,2,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2 = 1 + 1
[[1,1,2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[[1,2,2,2,2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[[2,2,2,2,2]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,1),(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[[1,1,1,1],[2]]
=> ([],1)
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[1,1,2,4]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(0,6),(1,2),(1,4),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
=> ([(0,1),(0,4),(0,6),(1,3),(1,4),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,6} + 1
[[1,1,3,4]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ([(0,7),(1,3),(1,8),(2,7),(2,8),(3,5),(4,5),(4,6),(5,8),(6,7),(6,8)],9)
=> ([(0,3),(0,6),(0,7),(0,8),(1,2),(1,4),(1,5),(1,7),(1,8),(2,3),(2,4),(2,5),(2,8),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(4,8),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,6} + 1
[[1,1,4,4]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ([(0,8),(1,7),(2,5),(2,6),(3,7),(3,9),(4,8),(4,9),(5,7),(5,9),(6,8),(6,9)],10)
=> ([(0,3),(0,4),(0,7),(0,8),(0,9),(1,2),(1,4),(1,5),(1,6),(1,8),(1,9),(2,3),(2,5),(2,6),(2,8),(2,9),(3,4),(3,5),(3,7),(3,8),(4,6),(4,7),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,6} + 1
[[1,2,2,4]]
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> ([(0,8),(1,5),(1,7),(2,4),(2,6),(3,6),(3,7),(4,8),(4,9),(5,8),(5,9),(6,9),(7,9)],10)
=> ([(0,5),(0,6),(0,7),(0,8),(0,9),(1,2),(1,3),(1,4),(1,5),(1,7),(1,9),(2,3),(2,4),(2,5),(2,6),(2,9),(3,4),(3,6),(3,8),(3,9),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,6} + 1
[[1,2,3,4]]
=> ([(0,6),(1,9),(1,10),(2,8),(3,7),(4,3),(4,12),(5,2),(5,12),(6,4),(6,5),(7,9),(7,11),(8,10),(8,11),(9,13),(10,13),(11,13),(12,1),(12,7),(12,8)],14)
=> ([(0,10),(1,9),(1,12),(2,8),(2,11),(3,5),(3,6),(3,7),(4,5),(4,6),(4,13),(5,11),(6,12),(7,11),(7,12),(8,10),(8,13),(9,10),(9,13),(11,13),(12,13)],14)
=> ([(0,6),(0,7),(0,8),(0,9),(0,10),(0,11),(0,12),(0,13),(1,2),(1,3),(1,4),(1,5),(1,8),(1,9),(1,10),(1,12),(1,13),(2,3),(2,4),(2,5),(2,7),(2,9),(2,10),(2,11),(2,13),(3,4),(3,5),(3,6),(3,7),(3,8),(3,10),(3,11),(3,13),(4,5),(4,6),(4,7),(4,8),(4,10),(4,12),(4,13),(5,6),(5,9),(5,10),(5,11),(5,12),(5,13),(6,7),(6,8),(6,9),(6,11),(6,12),(6,13),(7,8),(7,9),(7,11),(7,12),(7,13),(8,9),(8,11),(8,12),(8,13),(9,10),(9,11),(9,12),(10,11),(10,12),(10,13),(11,12),(11,13),(12,13)],14)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,6} + 1
[[1,2,4,4]]
=> ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
=> ([(0,10),(1,8),(1,14),(2,9),(2,13),(3,4),(3,11),(4,12),(5,7),(5,11),(5,13),(6,11),(6,13),(6,14),(7,12),(7,15),(8,10),(8,15),(9,10),(9,15),(11,12),(12,14),(13,15),(14,15)],16)
=> ([(0,3),(0,4),(0,8),(0,9),(0,10),(0,11),(0,12),(0,13),(0,14),(0,15),(1,2),(1,4),(1,5),(1,6),(1,7),(1,9),(1,10),(1,11),(1,13),(1,14),(1,15),(2,3),(2,4),(2,5),(2,6),(2,7),(2,10),(2,12),(2,13),(2,14),(2,15),(3,5),(3,6),(3,8),(3,9),(3,10),(3,11),(3,12),(3,14),(3,15),(4,5),(4,6),(4,7),(4,10),(4,11),(4,12),(4,13),(4,15),(5,6),(5,7),(5,8),(5,9),(5,11),(5,13),(5,14),(5,15),(6,7),(6,8),(6,9),(6,12),(6,13),(6,14),(6,15),(7,8),(7,10),(7,11),(7,12),(7,13),(7,14),(7,15),(8,9),(8,10),(8,11),(8,12),(8,13),(8,14),(8,15),(9,10),(9,11),(9,12),(9,13),(9,14),(9,15),(10,11),(10,12),(10,13),(10,14),(11,12),(11,13),(11,14),(11,15),(12,13),(12,14),(12,15),(13,15),(14,15)],16)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,6} + 1
[[1,3,3,4]]
=> ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
=> ([(0,10),(1,8),(1,14),(2,9),(2,13),(3,4),(3,11),(4,12),(5,7),(5,11),(5,13),(6,11),(6,13),(6,14),(7,12),(7,15),(8,10),(8,15),(9,10),(9,15),(11,12),(12,14),(13,15),(14,15)],16)
=> ([(0,3),(0,4),(0,8),(0,9),(0,10),(0,11),(0,12),(0,13),(0,14),(0,15),(1,2),(1,4),(1,5),(1,6),(1,7),(1,9),(1,10),(1,11),(1,13),(1,14),(1,15),(2,3),(2,4),(2,5),(2,6),(2,7),(2,10),(2,12),(2,13),(2,14),(2,15),(3,5),(3,6),(3,8),(3,9),(3,10),(3,11),(3,12),(3,14),(3,15),(4,5),(4,6),(4,7),(4,10),(4,11),(4,12),(4,13),(4,15),(5,6),(5,7),(5,8),(5,9),(5,11),(5,13),(5,14),(5,15),(6,7),(6,8),(6,9),(6,12),(6,13),(6,14),(6,15),(7,8),(7,10),(7,11),(7,12),(7,13),(7,14),(7,15),(8,9),(8,10),(8,11),(8,12),(8,13),(8,14),(8,15),(9,10),(9,11),(9,12),(9,13),(9,14),(9,15),(10,11),(10,12),(10,13),(10,14),(11,12),(11,13),(11,14),(11,15),(12,13),(12,14),(12,15),(13,15),(14,15)],16)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,6} + 1
[[1,3,4,4]]
=> ([(0,1),(1,4),(1,5),(2,14),(3,13),(4,6),(4,17),(5,7),(5,17),(6,15),(7,16),(8,11),(8,12),(10,18),(11,3),(11,18),(12,2),(12,18),(13,9),(14,9),(15,10),(15,11),(16,10),(16,12),(17,8),(17,15),(17,16),(18,13),(18,14)],19)
=> ([(0,12),(1,10),(1,11),(2,11),(2,14),(3,10),(3,13),(4,8),(4,15),(5,9),(5,16),(6,15),(6,16),(6,17),(7,13),(7,14),(7,18),(8,12),(8,18),(9,12),(9,18),(10,17),(11,17),(13,15),(13,17),(14,16),(14,17),(15,18),(16,18)],19)
=> ([(0,1),(0,2),(0,3),(0,6),(0,8),(0,9),(0,12),(0,13),(0,14),(0,15),(0,16),(0,17),(0,18),(1,4),(1,5),(1,7),(1,10),(1,11),(1,12),(1,13),(1,14),(1,15),(1,16),(1,17),(1,18),(2,3),(2,5),(2,6),(2,8),(2,9),(2,10),(2,11),(2,12),(2,13),(2,14),(2,16),(2,17),(2,18),(3,4),(3,6),(3,8),(3,9),(3,10),(3,11),(3,12),(3,13),(3,14),(3,15),(3,17),(3,18),(4,5),(4,7),(4,8),(4,9),(4,10),(4,11),(4,12),(4,13),(4,15),(4,16),(4,17),(4,18),(5,7),(5,8),(5,9),(5,10),(5,11),(5,12),(5,14),(5,15),(5,16),(5,17),(5,18),(6,7),(6,8),(6,9),(6,10),(6,11),(6,12),(6,13),(6,14),(6,15),(6,16),(6,17),(6,18),(7,8),(7,9),(7,10),(7,11),(7,12),(7,13),(7,14),(7,15),(7,16),(7,17),(7,18),(8,9),(8,10),(8,11),(8,13),(8,14),(8,15),(8,17),(8,18),(9,10),(9,11),(9,13),(9,14),(9,16),(9,17),(9,18),(10,11),(10,12),(10,13),(10,15),(10,16),(10,18),(11,12),(11,14),(11,15),(11,16),(11,18),(12,13),(12,14),(12,15),(12,16),(12,17),(13,14),(13,15),(13,16),(13,17),(13,18),(14,15),(14,16),(14,17),(14,18),(15,16),(15,17),(15,18),(16,17),(16,18),(17,18)],19)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,6} + 1
[[1,4,4,4]]
=> ([(0,9),(2,16),(2,17),(3,13),(4,12),(5,10),(6,11),(7,5),(7,15),(8,6),(8,15),(9,7),(9,8),(10,14),(10,16),(11,14),(11,17),(12,18),(13,18),(14,19),(15,2),(15,10),(15,11),(16,4),(16,19),(17,3),(17,19),(18,1),(19,12),(19,13)],20)
=> ([(0,13),(1,12),(2,9),(2,15),(3,8),(3,14),(4,10),(4,16),(5,11),(5,17),(6,16),(6,17),(6,18),(7,14),(7,15),(7,19),(8,12),(8,18),(9,12),(9,18),(10,13),(10,19),(11,13),(11,19),(14,16),(14,18),(15,17),(15,18),(16,19),(17,19)],20)
=> ([(0,1),(0,4),(0,5),(0,7),(0,10),(0,11),(0,12),(0,13),(0,14),(0,15),(0,16),(0,17),(0,18),(0,19),(1,2),(1,3),(1,6),(1,8),(1,9),(1,12),(1,13),(1,14),(1,15),(1,16),(1,17),(1,18),(1,19),(2,3),(2,5),(2,6),(2,8),(2,9),(2,10),(2,11),(2,12),(2,13),(2,14),(2,15),(2,17),(2,18),(2,19),(3,4),(3,6),(3,8),(3,9),(3,10),(3,11),(3,12),(3,13),(3,14),(3,15),(3,16),(3,18),(3,19),(4,5),(4,7),(4,8),(4,9),(4,10),(4,11),(4,12),(4,13),(4,14),(4,16),(4,17),(4,18),(4,19),(5,7),(5,8),(5,9),(5,10),(5,11),(5,12),(5,13),(5,15),(5,16),(5,17),(5,18),(5,19),(6,7),(6,8),(6,9),(6,10),(6,11),(6,12),(6,13),(6,14),(6,15),(6,16),(6,17),(6,18),(6,19),(7,8),(7,9),(7,10),(7,11),(7,12),(7,13),(7,14),(7,15),(7,16),(7,17),(7,18),(7,19),(8,9),(8,10),(8,11),(8,12),(8,14),(8,15),(8,16),(8,18),(8,19),(9,10),(9,11),(9,12),(9,14),(9,15),(9,17),(9,18),(9,19),(10,11),(10,13),(10,14),(10,16),(10,17),(10,18),(10,19),(11,13),(11,15),(11,16),(11,17),(11,18),(11,19),(12,13),(12,14),(12,15),(12,16),(12,17),(12,18),(13,14),(13,15),(13,16),(13,17),(13,19),(14,15),(14,16),(14,17),(14,18),(14,19),(15,16),(15,17),(15,18),(15,19),(16,17),(16,18),(16,19),(17,18),(17,19),(18,19)],20)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,6} + 1
[[2,2,2,4]]
=> ([(0,6),(1,9),(2,8),(3,5),(3,7),(4,1),(4,7),(5,2),(5,10),(6,3),(6,4),(7,9),(7,10),(8,12),(9,11),(10,8),(10,11),(11,12)],13)
=> ([(0,10),(1,6),(1,8),(2,5),(2,9),(3,7),(3,9),(4,7),(4,10),(4,12),(5,6),(5,11),(6,12),(7,11),(8,10),(8,12),(9,11),(11,12)],13)
=> ([(0,3),(0,5),(0,7),(0,8),(0,9),(0,10),(0,11),(0,12),(1,2),(1,4),(1,6),(1,8),(1,9),(1,10),(1,11),(1,12),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(2,11),(2,12),(3,4),(3,5),(3,6),(3,7),(3,9),(3,10),(3,12),(4,5),(4,6),(4,8),(4,10),(4,11),(4,12),(5,7),(5,8),(5,9),(5,10),(5,12),(6,7),(6,8),(6,9),(6,11),(6,12),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,11),(9,10),(9,11),(9,12),(10,11),(10,12),(11,12)],13)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,6} + 1
[[2,2,3,4]]
=> ([(0,1),(1,2),(1,3),(2,4),(2,13),(3,6),(3,13),(4,15),(5,14),(6,5),(6,16),(7,10),(7,12),(8,18),(9,18),(10,17),(11,9),(11,17),(12,8),(12,17),(13,7),(13,15),(13,16),(14,8),(14,9),(15,10),(15,11),(16,11),(16,12),(16,14),(17,18)],19)
=> ([(0,11),(1,10),(1,16),(2,9),(2,15),(3,9),(3,11),(3,17),(4,6),(4,7),(4,14),(5,8),(5,14),(5,16),(6,12),(6,15),(7,13),(7,15),(8,12),(8,17),(9,18),(10,11),(10,17),(12,14),(12,18),(13,14),(13,16),(13,18),(15,18),(16,17),(17,18)],19)
=> ([(0,2),(0,6),(0,7),(0,8),(0,10),(0,11),(0,12),(0,13),(0,14),(0,15),(0,16),(0,17),(0,18),(1,3),(1,4),(1,5),(1,6),(1,9),(1,11),(1,12),(1,13),(1,14),(1,15),(1,16),(1,17),(1,18),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(2,10),(2,12),(2,13),(2,14),(2,15),(2,17),(2,18),(3,4),(3,5),(3,7),(3,8),(3,9),(3,10),(3,11),(3,13),(3,14),(3,15),(3,16),(3,17),(3,18),(4,5),(4,7),(4,8),(4,9),(4,11),(4,12),(4,13),(4,14),(4,16),(4,17),(4,18),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(5,13),(5,14),(5,16),(5,18),(6,7),(6,8),(6,9),(6,10),(6,12),(6,13),(6,15),(6,16),(6,17),(6,18),(7,8),(7,9),(7,10),(7,11),(7,12),(7,14),(7,15),(7,17),(7,18),(8,10),(8,11),(8,12),(8,14),(8,15),(8,16),(8,17),(8,18),(9,10),(9,11),(9,12),(9,13),(9,14),(9,15),(9,16),(9,18),(10,12),(10,13),(10,14),(10,15),(10,16),(10,17),(10,18),(11,12),(11,13),(11,14),(11,15),(11,16),(11,17),(11,18),(12,13),(12,15),(12,16),(12,17),(12,18),(13,14),(13,15),(13,16),(13,17),(14,16),(14,17),(14,18),(15,16),(15,17),(15,18),(16,17),(16,18),(17,18)],19)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,6} + 1
[[2,2,4,4]]
=> ([(0,1),(1,3),(1,4),(2,15),(3,6),(3,18),(4,5),(4,18),(5,17),(6,7),(6,19),(7,16),(8,12),(8,14),(10,21),(11,21),(12,2),(12,20),(13,11),(13,20),(14,10),(14,20),(15,9),(16,10),(16,11),(17,12),(17,13),(18,8),(18,17),(18,19),(19,13),(19,14),(19,16),(20,15),(20,21),(21,9)],22)
=> ([(0,12),(1,10),(1,16),(2,11),(2,15),(3,11),(3,18),(4,9),(4,17),(5,10),(5,12),(5,19),(6,13),(6,18),(6,19),(7,14),(7,15),(7,16),(8,13),(8,15),(8,16),(9,12),(9,19),(10,20),(11,21),(13,20),(13,21),(14,17),(14,20),(14,21),(15,21),(16,20),(17,18),(17,19),(18,21),(19,20)],22)
=> ([(0,2),(0,4),(0,7),(0,8),(0,10),(0,11),(0,12),(0,13),(0,14),(0,15),(0,16),(0,17),(0,18),(0,19),(0,20),(0,21),(1,2),(1,3),(1,5),(1,6),(1,7),(1,8),(1,9),(1,13),(1,14),(1,15),(1,16),(1,17),(1,18),(1,19),(1,20),(1,21),(2,4),(2,5),(2,9),(2,10),(2,11),(2,12),(2,13),(2,15),(2,16),(2,17),(2,18),(2,19),(2,20),(2,21),(3,5),(3,6),(3,7),(3,8),(3,9),(3,10),(3,11),(3,12),(3,14),(3,15),(3,16),(3,17),(3,18),(3,19),(3,20),(3,21),(4,5),(4,6),(4,8),(4,9),(4,10),(4,11),(4,12),(4,13),(4,14),(4,15),(4,16),(4,18),(4,19),(4,20),(4,21),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(5,12),(5,14),(5,15),(5,17),(5,18),(5,20),(5,21),(6,7),(6,8),(6,9),(6,11),(6,12),(6,13),(6,14),(6,15),(6,17),(6,18),(6,19),(6,20),(6,21),(7,8),(7,9),(7,11),(7,12),(7,13),(7,14),(7,15),(7,16),(7,17),(7,19),(7,20),(7,21),(8,9),(8,10),(8,11),(8,12),(8,14),(8,15),(8,17),(8,18),(8,19),(8,21),(9,10),(9,11),(9,13),(9,14),(9,15),(9,16),(9,17),(9,18),(9,20),(9,21),(10,11),(10,12),(10,13),(10,14),(10,15),(10,16),(10,17),(10,18),(10,19),(10,20),(10,21),(11,12),(11,13),(11,14),(11,16),(11,18),(11,19),(11,20),(11,21),(12,13),(12,14),(12,16),(12,17),(12,18),(12,19),(12,20),(12,21),(13,14),(13,15),(13,16),(13,17),(13,18),(13,19),(13,20),(13,21),(14,15),(14,16),(14,17),(14,19),(14,21),(15,16),(15,17),(15,18),(15,19),(15,20),(16,17),(16,18),(16,19),(16,20),(16,21),(17,18),(17,19),(17,20),(17,21),(18,19),(18,20),(18,21),(19,20),(19,21),(20,21)],22)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,6} + 1
[[2,3,3,4]]
=> ([(0,1),(1,2),(1,3),(2,4),(2,16),(3,6),(3,16),(4,18),(5,17),(6,5),(6,19),(7,9),(7,11),(8,10),(8,14),(9,21),(10,22),(11,21),(12,20),(13,12),(13,22),(14,7),(14,15),(14,22),(15,9),(15,20),(16,8),(16,18),(16,19),(17,12),(17,15),(18,10),(18,13),(19,13),(19,14),(19,17),(20,21),(22,11),(22,20)],23)
=> ([(0,13),(1,12),(1,17),(2,11),(2,18),(3,4),(3,8),(3,22),(4,9),(4,21),(5,6),(5,17),(5,21),(6,19),(6,22),(7,11),(7,13),(7,19),(8,9),(8,15),(9,16),(10,14),(10,16),(10,18),(11,20),(12,13),(12,19),(14,17),(14,20),(14,21),(15,16),(15,18),(15,22),(16,21),(17,19),(18,20),(19,20),(20,22),(21,22)],23)
=> ([(0,2),(0,5),(0,7),(0,8),(0,10),(0,11),(0,12),(0,13),(0,14),(0,15),(0,16),(0,17),(0,18),(0,19),(0,20),(0,21),(0,22),(1,3),(1,4),(1,6),(1,7),(1,8),(1,9),(1,12),(1,13),(1,14),(1,15),(1,16),(1,17),(1,18),(1,19),(1,20),(1,21),(1,22),(2,3),(2,4),(2,5),(2,8),(2,9),(2,10),(2,11),(2,12),(2,13),(2,14),(2,17),(2,18),(2,19),(2,20),(2,21),(2,22),(3,4),(3,5),(3,6),(3,7),(3,9),(3,11),(3,12),(3,13),(3,14),(3,15),(3,16),(3,17),(3,20),(3,21),(3,22),(4,6),(4,7),(4,8),(4,9),(4,10),(4,11),(4,13),(4,14),(4,15),(4,16),(4,17),(4,18),(4,19),(4,21),(4,22),(5,6),(5,8),(5,9),(5,10),(5,11),(5,12),(5,14),(5,15),(5,16),(5,17),(5,18),(5,19),(5,20),(5,21),(5,22),(6,7),(6,9),(6,10),(6,11),(6,13),(6,14),(6,15),(6,16),(6,17),(6,18),(6,19),(6,20),(6,21),(6,22),(7,8),(7,9),(7,10),(7,11),(7,13),(7,14),(7,15),(7,16),(7,18),(7,19),(7,20),(7,21),(7,22),(8,9),(8,10),(8,11),(8,12),(8,13),(8,14),(8,16),(8,17),(8,18),(8,19),(8,20),(8,22),(9,10),(9,11),(9,12),(9,13),(9,14),(9,15),(9,16),(9,17),(9,18),(9,21),(9,22),(10,11),(10,12),(10,13),(10,14),(10,16),(10,17),(10,18),(10,19),(10,20),(10,21),(10,22),(11,12),(11,14),(11,15),(11,17),(11,18),(11,19),(11,20),(11,21),(11,22),(12,13),(12,14),(12,15),(12,16),(12,17),(12,18),(12,19),(12,20),(12,21),(12,22),(13,14),(13,15),(13,16),(13,18),(13,19),(13,20),(13,21),(13,22),(14,15),(14,16),(14,17),(14,20),(14,21),(15,16),(15,17),(15,18),(15,19),(15,20),(15,21),(15,22),(16,18),(16,19),(16,20),(16,21),(16,22),(17,18),(17,19),(17,20),(17,21),(17,22),(18,19),(18,20),(18,22),(19,20),(19,21),(19,22),(20,21),(20,22),(21,22)],23)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,6} + 1
[[2,3,4,4]]
=> ([(0,1),(1,3),(1,4),(2,21),(3,6),(3,22),(4,5),(4,22),(5,20),(6,7),(6,23),(7,19),(8,13),(8,18),(9,14),(9,17),(10,26),(11,26),(12,27),(13,24),(14,2),(14,25),(15,13),(15,27),(16,12),(16,25),(17,8),(17,15),(17,25),(18,10),(18,24),(19,12),(19,15),(20,14),(20,16),(21,10),(21,11),(22,9),(22,20),(22,23),(23,16),(23,17),(23,19),(24,26),(25,18),(25,21),(25,27),(27,11),(27,24)],28)
=> ([(0,14),(1,13),(1,22),(2,19),(2,20),(3,12),(3,21),(4,13),(4,14),(4,23),(5,7),(5,19),(5,24),(6,20),(6,23),(6,26),(7,8),(7,18),(8,17),(8,19),(9,15),(9,22),(9,24),(10,11),(10,17),(10,26),(11,16),(11,18),(12,14),(12,23),(13,25),(15,21),(15,25),(15,27),(16,22),(16,24),(16,26),(17,18),(17,27),(18,24),(19,27),(20,21),(20,27),(21,23),(22,25),(23,25),(24,27),(25,26),(26,27)],28)
=> ([(0,3),(0,4),(0,9),(0,10),(0,11),(0,12),(0,13),(0,14),(0,15),(0,16),(0,17),(0,18),(0,19),(0,20),(0,21),(0,22),(0,23),(0,24),(0,25),(0,26),(0,27),(1,2),(1,4),(1,5),(1,6),(1,7),(1,8),(1,10),(1,11),(1,12),(1,14),(1,15),(1,17),(1,18),(1,19),(1,20),(1,21),(1,22),(1,23),(1,24),(1,25),(1,26),(1,27),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(2,11),(2,12),(2,13),(2,16),(2,17),(2,18),(2,19),(2,20),(2,21),(2,22),(2,23),(2,24),(2,25),(2,26),(2,27),(3,6),(3,7),(3,8),(3,9),(3,10),(3,11),(3,13),(3,14),(3,15),(3,16),(3,17),(3,19),(3,20),(3,21),(3,22),(3,23),(3,24),(3,25),(3,26),(3,27),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(4,12),(4,14),(4,15),(4,16),(4,17),(4,18),(4,20),(4,22),(4,23),(4,24),(4,25),(4,26),(4,27),(5,6),(5,7),(5,8),(5,9),(5,10),(5,12),(5,13),(5,15),(5,16),(5,17),(5,18),(5,19),(5,20),(5,21),(5,22),(5,23),(5,24),(5,25),(5,26),(5,27),(6,7),(6,8),(6,9),(6,10),(6,12),(6,14),(6,15),(6,16),(6,17),(6,18),(6,19),(6,20),(6,21),(6,22),(6,23),(6,25),(6,26),(6,27),(7,8),(7,9),(7,11),(7,12),(7,13),(7,14),(7,15),(7,18),(7,19),(7,20),(7,21),(7,22),(7,23),(7,24),(7,25),(7,26),(7,27),(8,9),(8,10),(8,11),(8,12),(8,13),(8,14),(8,16),(8,18),(8,19),(8,20),(8,21),(8,22),(8,23),(8,25),(8,26),(8,27),(9,10),(9,11),(9,13),(9,14),(9,15),(9,16),(9,17),(9,18),(9,19),(9,20),(9,21),(9,23),(9,24),(9,25),(9,26),(9,27),(10,11),(10,12),(10,13),(10,14),(10,15),(10,16),(10,17),(10,18),(10,19),(10,20),(10,21),(10,24),(10,25),(10,26),(10,27),(11,12),(11,13),(11,14),(11,15),(11,16),(11,17),(11,18),(11,19),(11,20),(11,21),(11,22),(11,23),(11,24),(11,26),(11,27),(12,13),(12,15),(12,16),(12,17),(12,18),(12,19),(12,20),(12,21),(12,22),(12,23),(12,24),(12,25),(12,27),(13,14),(13,15),(13,16),(13,17),(13,18),(13,19),(13,20),(13,21),(13,22),(13,23),(13,24),(13,25),(13,26),(13,27),(14,15),(14,16),(14,17),(14,18),(14,19),(14,20),(14,21),(14,22),(14,23),(14,24),(14,25),(14,26),(14,27),(15,16),(15,17),(15,18),(15,19),(15,20),(15,21),(15,22),(15,24),(15,25),(15,26),(15,27),(16,17),(16,18),(16,19),(16,20),(16,21),(16,23),(16,24),(16,25),(16,26),(16,27),(17,18),(17,19),(17,20),(17,21),(17,22),(17,24),(17,25),(17,26),(17,27),(18,19),(18,20),(18,22),(18,23),(18,24),(18,25),(18,27),(19,21),(19,22),(19,23),(19,24),(19,26),(19,27),(20,22),(20,23),(20,24),(20,25),(20,26),(21,22),(21,23),(21,24),(21,25),(21,26),(21,27),(22,23),(22,24),(22,25),(22,26),(22,27),(23,24),(23,25),(23,26),(23,27),(24,25),(24,26),(24,27),(25,26),(25,27),(26,27)],28)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,6} + 1
[[2,4,4,4]]
=> ([(0,1),(1,4),(1,5),(2,23),(3,16),(4,7),(4,24),(5,6),(5,24),(6,22),(7,8),(7,25),(8,21),(9,13),(9,20),(10,15),(10,19),(11,28),(12,29),(13,26),(14,3),(14,28),(15,2),(15,27),(17,13),(17,29),(18,12),(18,27),(19,9),(19,17),(19,27),(20,14),(20,26),(21,12),(21,17),(22,15),(22,18),(23,11),(23,14),(24,10),(24,22),(24,25),(25,18),(25,19),(25,21),(26,28),(27,20),(27,23),(27,29),(28,16),(29,11),(29,26)],30)
=> ([(0,14),(1,13),(1,22),(2,23),(2,24),(3,12),(3,21),(4,5),(4,15),(5,19),(6,20),(6,22),(6,28),(7,24),(7,26),(7,27),(8,19),(8,23),(8,28),(9,13),(9,14),(9,27),(10,11),(10,17),(10,18),(11,16),(11,26),(12,14),(12,27),(13,25),(15,16),(15,19),(15,23),(16,18),(16,29),(17,22),(17,26),(17,28),(18,19),(18,28),(20,21),(20,25),(20,29),(21,24),(21,27),(22,25),(23,29),(24,29),(25,26),(25,27),(26,29),(28,29)],30)
=> ([(0,3),(0,4),(0,9),(0,10),(0,11),(0,12),(0,13),(0,14),(0,15),(0,16),(0,17),(0,18),(0,19),(0,20),(0,21),(0,22),(0,23),(0,24),(0,25),(0,26),(0,27),(0,28),(0,29),(1,2),(1,4),(1,5),(1,6),(1,7),(1,8),(1,10),(1,11),(1,12),(1,13),(1,14),(1,17),(1,18),(1,19),(1,20),(1,21),(1,22),(1,23),(1,24),(1,25),(1,26),(1,27),(1,28),(1,29),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(2,11),(2,12),(2,13),(2,14),(2,15),(2,16),(2,19),(2,20),(2,21),(2,22),(2,23),(2,24),(2,25),(2,26),(2,27),(2,28),(2,29),(3,5),(3,6),(3,8),(3,9),(3,10),(3,12),(3,13),(3,14),(3,15),(3,16),(3,17),(3,18),(3,19),(3,20),(3,22),(3,23),(3,24),(3,25),(3,26),(3,27),(3,28),(3,29),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(4,11),(4,13),(4,14),(4,15),(4,17),(4,18),(4,19),(4,21),(4,23),(4,24),(4,25),(4,26),(4,27),(4,28),(4,29),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(5,13),(5,14),(5,15),(5,17),(5,18),(5,19),(5,20),(5,21),(5,22),(5,23),(5,24),(5,25),(5,27),(5,28),(5,29),(6,7),(6,8),(6,9),(6,10),(6,11),(6,12),(6,14),(6,15),(6,16),(6,18),(6,19),(6,20),(6,21),(6,22),(6,23),(6,24),(6,25),(6,27),(6,28),(6,29),(7,8),(7,9),(7,10),(7,11),(7,13),(7,14),(7,15),(7,16),(7,17),(7,19),(7,20),(7,21),(7,22),(7,23),(7,24),(7,25),(7,26),(7,27),(7,28),(7,29),(8,9),(8,11),(8,12),(8,14),(8,16),(8,17),(8,18),(8,19),(8,20),(8,21),(8,22),(8,23),(8,24),(8,25),(8,26),(8,27),(8,28),(8,29),(9,10),(9,12),(9,13),(9,14),(9,15),(9,16),(9,17),(9,18),(9,19),(9,20),(9,21),(9,22),(9,24),(9,25),(9,26),(9,27),(9,28),(9,29),(10,11),(10,12),(10,13),(10,15),(10,16),(10,17),(10,18),(10,19),(10,20),(10,21),(10,22),(10,24),(10,25),(10,26),(10,27),(10,28),(10,29),(11,12),(11,13),(11,14),(11,15),(11,16),(11,17),(11,19),(11,20),(11,21),(11,22),(11,23),(11,24),(11,25),(11,26),(11,28),(11,29),(12,13),(12,14),(12,15),(12,16),(12,17),(12,18),(12,19),(12,20),(12,21),(12,22),(12,23),(12,24),(12,25),(12,26),(12,27),(12,29),(13,15),(13,16),(13,17),(13,18),(13,19),(13,20),(13,21),(13,22),(13,23),(13,24),(13,26),(13,27),(13,28),(13,29),(14,15),(14,16),(14,18),(14,19),(14,20),(14,21),(14,22),(14,23),(14,25),(14,26),(14,27),(14,28),(14,29),(15,16),(15,17),(15,18),(15,19),(15,20),(15,21),(15,22),(15,24),(15,25),(15,26),(15,27),(15,28),(15,29),(16,17),(16,18),(16,19),(16,20),(16,21),(16,22),(16,23),(16,24),(16,25),(16,26),(16,27),(16,28),(16,29),(17,18),(17,19),(17,20),(17,21),(17,22),(17,23),(17,24),(17,25),(17,26),(17,27),(17,28),(17,29),(18,19),(18,20),(18,21),(18,22),(18,23),(18,24),(18,25),(18,26),(18,27),(18,28),(18,29),(19,21),(19,23),(19,24),(19,25),(19,26),(19,27),(19,28),(20,22),(20,23),(20,24),(20,25),(20,26),(20,27),(20,28),(20,29),(21,22),(21,23),(21,24),(21,25),(21,26),(21,28),(21,29),(22,23),(22,24),(22,25),(22,26),(22,27),(22,29),(23,24),(23,25),(23,26),(23,27),(23,28),(23,29),(24,26),(24,27),(24,28),(24,29),(25,26),(25,27),(25,28),(25,29),(26,27),(26,28),(26,29),(27,28),(27,29),(28,29)],30)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,6} + 1
[[3,3,3,4]]
=> ([(0,1),(1,3),(1,4),(2,14),(3,6),(3,20),(4,5),(4,20),(5,19),(6,7),(6,21),(7,18),(8,12),(8,13),(9,11),(9,17),(10,22),(11,24),(12,23),(13,2),(13,23),(15,13),(15,22),(16,10),(16,24),(17,8),(17,15),(17,24),(18,10),(18,15),(19,11),(19,16),(20,9),(20,19),(20,21),(21,16),(21,17),(21,18),(22,23),(23,14),(24,12),(24,22)],25)
=> ([(0,13),(1,12),(1,19),(2,11),(2,20),(3,4),(3,17),(4,18),(5,6),(5,19),(5,23),(6,21),(6,24),(7,11),(7,13),(7,21),(8,9),(8,17),(8,24),(9,18),(9,23),(10,14),(10,16),(10,20),(11,22),(12,13),(12,21),(14,19),(14,22),(14,23),(15,16),(15,17),(15,20),(15,24),(16,18),(16,23),(17,18),(19,21),(20,22),(21,22),(22,24),(23,24)],25)
=> ([(0,3),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(0,12),(0,14),(0,15),(0,16),(0,17),(0,18),(0,19),(0,20),(0,21),(0,22),(0,23),(0,24),(1,2),(1,4),(1,7),(1,8),(1,9),(1,10),(1,11),(1,13),(1,14),(1,15),(1,16),(1,17),(1,18),(1,19),(1,20),(1,21),(1,22),(1,23),(1,24),(2,3),(2,4),(2,6),(2,7),(2,9),(2,10),(2,11),(2,12),(2,13),(2,14),(2,15),(2,16),(2,17),(2,20),(2,21),(2,22),(2,23),(2,24),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(3,11),(3,12),(3,14),(3,16),(3,18),(3,19),(3,20),(3,21),(3,22),(3,23),(3,24),(4,5),(4,7),(4,10),(4,11),(4,12),(4,13),(4,14),(4,15),(4,16),(4,17),(4,18),(4,19),(4,20),(4,21),(4,22),(4,23),(4,24),(5,6),(5,7),(5,8),(5,9),(5,11),(5,12),(5,13),(5,15),(5,16),(5,17),(5,18),(5,19),(5,20),(5,21),(5,22),(5,23),(5,24),(6,7),(6,8),(6,9),(6,10),(6,11),(6,12),(6,13),(6,15),(6,16),(6,17),(6,18),(6,19),(6,20),(6,22),(6,23),(6,24),(7,10),(7,11),(7,12),(7,13),(7,14),(7,15),(7,16),(7,17),(7,19),(7,20),(7,21),(7,23),(7,24),(8,9),(8,10),(8,11),(8,12),(8,13),(8,15),(8,16),(8,17),(8,18),(8,19),(8,20),(8,21),(8,22),(8,23),(8,24),(9,10),(9,12),(9,13),(9,14),(9,15),(9,16),(9,17),(9,18),(9,19),(9,21),(9,22),(9,23),(9,24),(10,11),(10,12),(10,13),(10,14),(10,15),(10,16),(10,17),(10,18),(10,20),(10,21),(10,22),(10,24),(11,12),(11,13),(11,14),(11,15),(11,16),(11,17),(11,19),(11,20),(11,21),(11,22),(11,23),(11,24),(12,13),(12,14),(12,15),(12,16),(12,18),(12,19),(12,20),(12,22),(12,23),(12,24),(13,14),(13,15),(13,16),(13,17),(13,18),(13,20),(13,21),(13,22),(13,23),(13,24),(14,15),(14,16),(14,17),(14,18),(14,19),(14,20),(14,21),(14,22),(14,23),(14,24),(15,17),(15,18),(15,19),(15,20),(15,21),(15,22),(15,24),(16,18),(16,19),(16,20),(16,21),(16,22),(16,23),(17,18),(17,19),(17,20),(17,21),(17,22),(17,23),(17,24),(18,19),(18,20),(18,21),(18,22),(18,23),(18,24),(19,20),(19,21),(19,22),(19,23),(19,24),(20,21),(20,23),(20,24),(21,22),(21,23),(21,24),(22,23),(22,24),(23,24)],25)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,6} + 1
[[3,3,4,4]]
=> ([(0,1),(1,4),(1,5),(2,24),(3,21),(4,7),(4,25),(5,6),(5,25),(6,23),(7,8),(7,26),(8,22),(9,16),(9,20),(10,15),(10,19),(11,29),(12,29),(14,30),(15,2),(15,28),(16,3),(16,27),(17,16),(17,30),(18,14),(18,28),(19,9),(19,17),(19,28),(20,12),(20,27),(21,13),(22,14),(22,17),(23,15),(23,18),(24,11),(24,12),(25,10),(25,23),(25,26),(26,18),(26,19),(26,22),(27,21),(27,29),(28,20),(28,24),(28,30),(29,13),(30,11),(30,27)],31)
=> ([(0,15),(1,13),(1,24),(2,14),(2,19),(3,14),(3,21),(4,12),(4,20),(5,22),(5,23),(6,12),(6,15),(6,25),(7,17),(7,20),(7,26),(8,16),(8,19),(8,27),(9,23),(9,25),(9,27),(10,16),(10,21),(10,22),(11,21),(11,22),(11,26),(12,28),(13,15),(13,25),(14,29),(16,29),(16,30),(17,24),(17,28),(17,30),(18,19),(18,20),(18,26),(18,27),(19,29),(20,28),(21,29),(22,30),(23,24),(23,30),(24,25),(25,28),(26,29),(26,30),(27,28),(27,30)],31)
=> ([(0,3),(0,4),(0,5),(0,10),(0,11),(0,12),(0,13),(0,14),(0,15),(0,16),(0,17),(0,18),(0,19),(0,20),(0,21),(0,22),(0,23),(0,24),(0,25),(0,26),(0,27),(0,28),(0,29),(0,30),(1,2),(1,3),(1,4),(1,6),(1,7),(1,8),(1,9),(1,10),(1,11),(1,13),(1,14),(1,15),(1,16),(1,17),(1,18),(1,19),(1,20),(1,21),(1,24),(1,25),(1,26),(1,27),(1,28),(1,29),(1,30),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(2,10),(2,11),(2,13),(2,14),(2,15),(2,16),(2,18),(2,20),(2,21),(2,22),(2,23),(2,24),(2,25),(2,26),(2,27),(2,28),(2,29),(2,30),(3,5),(3,7),(3,8),(3,9),(3,11),(3,12),(3,13),(3,14),(3,15),(3,17),(3,18),(3,19),(3,20),(3,21),(3,22),(3,23),(3,24),(3,25),(3,26),(3,27),(3,28),(3,29),(3,30),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(4,12),(4,13),(4,14),(4,16),(4,19),(4,20),(4,21),(4,22),(4,23),(4,24),(4,25),(4,26),(4,27),(4,28),(4,29),(4,30),(5,6),(5,7),(5,8),(5,10),(5,11),(5,12),(5,15),(5,16),(5,17),(5,18),(5,19),(5,21),(5,22),(5,23),(5,24),(5,25),(5,26),(5,27),(5,28),(5,29),(5,30),(6,7),(6,8),(6,9),(6,10),(6,12),(6,13),(6,14),(6,15),(6,16),(6,17),(6,18),(6,19),(6,20),(6,21),(6,22),(6,24),(6,25),(6,26),(6,27),(6,28),(6,29),(6,30),(7,8),(7,9),(7,10),(7,12),(7,13),(7,14),(7,15),(7,16),(7,18),(7,19),(7,20),(7,21),(7,22),(7,23),(7,24),(7,25),(7,27),(7,28),(7,29),(7,30),(8,9),(8,10),(8,11),(8,12),(8,13),(8,14),(8,15),(8,16),(8,17),(8,18),(8,19),(8,20),(8,21),(8,23),(8,25),(8,27),(8,28),(8,29),(8,30),(9,10),(9,11),(9,12),(9,13),(9,14),(9,15),(9,16),(9,17),(9,18),(9,20),(9,21),(9,22),(9,23),(9,25),(9,26),(9,27),(9,28),(9,29),(9,30),(10,11),(10,13),(10,14),(10,15),(10,16),(10,17),(10,18),(10,19),(10,20),(10,21),(10,22),(10,24),(10,25),(10,26),(10,28),(10,29),(10,30),(11,12),(11,13),(11,14),(11,15),(11,16),(11,17),(11,18),(11,19),(11,20),(11,21),(11,22),(11,23),(11,24),(11,25),(11,26),(11,27),(11,29),(11,30),(12,14),(12,15),(12,16),(12,17),(12,18),(12,19),(12,20),(12,21),(12,22),(12,23),(12,24),(12,25),(12,26),(12,27),(12,28),(12,29),(12,30),(13,14),(13,15),(13,16),(13,17),(13,18),(13,20),(13,21),(13,22),(13,23),(13,24),(13,26),(13,27),(13,28),(13,29),(13,30),(14,15),(14,16),(14,17),(14,18),(14,19),(14,20),(14,21),(14,23),(14,25),(14,26),(14,27),(14,28),(14,30),(15,16),(15,17),(15,18),(15,19),(15,20),(15,22),(15,23),(15,24),(15,25),(15,26),(15,27),(15,29),(15,30),(16,17),(16,19),(16,20),(16,21),(16,22),(16,23),(16,24),(16,25),(16,26),(16,28),(16,29),(16,30),(17,18),(17,19),(17,20),(17,21),(17,22),(17,23),(17,24),(17,25),(17,26),(17,27),(17,28),(17,29),(17,30),(18,19),(18,20),(18,22),(18,23),(18,24),(18,25),(18,26),(18,27),(18,28),(18,29),(18,30),(19,20),(19,21),(19,22),(19,23),(19,24),(19,25),(19,26),(19,27),(19,28),(19,29),(19,30),(20,21),(20,22),(20,23),(20,24),(20,26),(20,27),(20,28),(20,30),(21,22),(21,23),(21,24),(21,25),(21,26),(21,27),(21,28),(21,29),(22,23),(22,24),(22,25),(22,26),(22,27),(22,28),(22,29),(22,30),(23,24),(23,25),(23,26),(23,27),(23,28),(23,29),(23,30),(24,25),(24,26),(24,27),(24,28),(24,29),(24,30),(25,26),(25,27),(25,28),(25,29),(25,30),(26,27),(26,28),(26,29),(26,30),(27,28),(27,29),(27,30),(28,29),(28,30),(29,30)],31)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,6} + 1
[[3,4,4,4]]
=> ([(0,1),(1,5),(1,6),(2,24),(3,27),(4,23),(5,8),(5,28),(6,9),(6,28),(7,26),(8,7),(8,29),(9,25),(10,16),(10,22),(11,17),(11,21),(13,30),(14,33),(15,4),(15,33),(16,2),(16,32),(17,3),(17,31),(18,16),(18,30),(19,12),(20,13),(20,31),(21,10),(21,18),(21,31),(22,15),(22,32),(23,12),(24,19),(25,17),(25,20),(26,13),(26,18),(27,14),(27,15),(28,11),(28,25),(28,29),(29,20),(29,21),(29,26),(30,14),(30,32),(31,22),(31,27),(31,30),(32,24),(32,33),(33,19),(33,23)],34)
=> ([(0,17),(1,12),(1,16),(2,13),(2,24),(3,15),(3,21),(4,14),(4,22),(5,16),(5,23),(6,25),(6,26),(7,14),(7,17),(7,31),(8,25),(8,29),(8,32),(9,26),(9,30),(9,31),(10,19),(10,22),(10,29),(11,18),(11,21),(11,30),(12,15),(12,32),(13,17),(13,31),(14,28),(15,27),(16,32),(18,23),(18,27),(18,33),(19,24),(19,28),(19,33),(20,21),(20,22),(20,29),(20,30),(21,27),(22,28),(23,25),(23,32),(24,26),(24,31),(25,33),(26,33),(27,29),(27,32),(28,30),(28,31),(29,33),(30,33)],34)
=> ([(0,3),(0,4),(0,5),(0,6),(0,11),(0,12),(0,13),(0,14),(0,15),(0,16),(0,17),(0,18),(0,19),(0,20),(0,21),(0,22),(0,23),(0,24),(0,25),(0,26),(0,27),(0,28),(0,29),(0,30),(0,31),(0,32),(0,33),(1,2),(1,3),(1,5),(1,6),(1,7),(1,8),(1,9),(1,10),(1,11),(1,12),(1,13),(1,14),(1,16),(1,17),(1,18),(1,19),(1,20),(1,22),(1,23),(1,24),(1,26),(1,27),(1,28),(1,29),(1,30),(1,31),(1,32),(1,33),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(2,10),(2,11),(2,12),(2,13),(2,14),(2,16),(2,17),(2,18),(2,19),(2,21),(2,22),(2,23),(2,25),(2,26),(2,27),(2,28),(2,29),(2,30),(2,31),(2,32),(2,33),(3,4),(3,5),(3,7),(3,9),(3,10),(3,11),(3,12),(3,13),(3,15),(3,16),(3,17),(3,18),(3,20),(3,21),(3,22),(3,23),(3,24),(3,25),(3,26),(3,27),(3,28),(3,29),(3,30),(3,31),(3,32),(3,33),(4,6),(4,8),(4,9),(4,10),(4,11),(4,12),(4,14),(4,15),(4,16),(4,17),(4,19),(4,20),(4,21),(4,22),(4,23),(4,24),(4,25),(4,26),(4,27),(4,28),(4,29),(4,30),(4,31),(4,32),(4,33),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(5,13),(5,14),(5,15),(5,17),(5,18),(5,19),(5,20),(5,23),(5,24),(5,25),(5,26),(5,27),(5,28),(5,29),(5,30),(5,31),(5,32),(5,33),(6,7),(6,8),(6,9),(6,10),(6,12),(6,13),(6,14),(6,15),(6,16),(6,18),(6,19),(6,21),(6,22),(6,24),(6,25),(6,26),(6,27),(6,28),(6,29),(6,30),(6,31),(6,32),(6,33),(7,8),(7,9),(7,10),(7,11),(7,13),(7,14),(7,15),(7,16),(7,17),(7,18),(7,19),(7,20),(7,21),(7,22),(7,23),(7,25),(7,26),(7,27),(7,28),(7,29),(7,30),(7,31),(7,32),(7,33),(8,9),(8,10),(8,12),(8,13),(8,14),(8,15),(8,16),(8,17),(8,18),(8,19),(8,20),(8,21),(8,22),(8,23),(8,24),(8,26),(8,27),(8,28),(8,29),(8,30),(8,31),(8,32),(8,33),(9,10),(9,11),(9,13),(9,14),(9,15),(9,16),(9,17),(9,18),(9,19),(9,20),(9,22),(9,23),(9,24),(9,25),(9,26),(9,27),(9,28),(9,29),(9,30),(9,32),(9,33),(10,12),(10,13),(10,14),(10,15),(10,16),(10,17),(10,18),(10,19),(10,21),(10,22),(10,23),(10,24),(10,25),(10,26),(10,27),(10,28),(10,29),(10,30),(10,32),(10,33),(11,12),(11,13),(11,14),(11,15),(11,16),(11,17),(11,18),(11,19),(11,20),(11,21),(11,22),(11,23),(11,24),(11,25),(11,26),(11,27),(11,28),(11,29),(11,31),(11,32),(11,33),(12,13),(12,14),(12,15),(12,16),(12,17),(12,18),(12,19),(12,20),(12,21),(12,22),(12,23),(12,24),(12,25),(12,26),(12,27),(12,28),(12,30),(12,31),(12,32),(12,33),(13,14),(13,16),(13,17),(13,18),(13,19),(13,20),(13,21),(13,22),(13,23),(13,25),(13,26),(13,27),(13,29),(13,30),(13,31),(13,32),(13,33),(14,16),(14,17),(14,18),(14,19),(14,20),(14,21),(14,22),(14,23),(14,24),(14,26),(14,28),(14,29),(14,30),(14,31),(14,32),(14,33),(15,16),(15,17),(15,18),(15,19),(15,20),(15,21),(15,22),(15,23),(15,24),(15,25),(15,26),(15,27),(15,28),(15,29),(15,30),(15,31),(15,32),(15,33),(16,17),(16,18),(16,19),(16,20),(16,21),(16,22),(16,23),(16,24),(16,25),(16,27),(16,28),(16,30),(16,31),(16,32),(16,33),(17,18),(17,19),(17,20),(17,21),(17,22),(17,23),(17,24),(17,25),(17,26),(17,27),(17,28),(17,29),(17,31),(17,33),(18,19),(18,20),(18,21),(18,23),(18,24),(18,25),(18,26),(18,27),(18,29),(18,30),(18,31),(18,32),(18,33),(19,20),(19,21),(19,22),(19,24),(19,25),(19,26),(19,28),(19,29),(19,30),(19,31),(19,32),(19,33),(20,21),(20,22),(20,23),(20,24),(20,25),(20,26),(20,27),(20,28),(20,29),(20,30),(20,31),(20,32),(20,33),(21,22),(21,23),(21,24),(21,25),(21,26),(21,27),(21,28),(21,29),(21,30),(21,31),(21,32),(21,33),(22,23),(22,24),(22,25),(22,27),(22,28),(22,29),(22,30),(22,31),(22,32),(22,33),(23,24),(23,25),(23,26),(23,27),(23,28),(23,29),(23,30),(23,31),(23,33),(24,25),(24,26),(24,27),(24,28),(24,29),(24,30),(24,31),(24,32),(24,33),(25,26),(25,27),(25,28),(25,29),(25,30),(25,31),(25,32),(25,33),(26,27),(26,28),(26,29),(26,30),(26,31),(26,32),(27,28),(27,29),(27,30),(27,31),(27,32),(27,33),(28,29),(28,30),(28,31),(28,32),(28,33),(29,30),(29,31),(29,32),(29,33),(30,31),(30,32),(30,33),(31,32),(31,33),(32,33)],34)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,6} + 1
[[4,4,4,4]]
=> ([(0,2),(2,6),(2,7),(3,25),(4,28),(5,24),(6,9),(6,29),(7,10),(7,29),(8,27),(9,8),(9,30),(10,26),(11,17),(11,23),(12,18),(12,22),(13,31),(14,34),(15,1),(16,5),(16,34),(17,3),(17,33),(18,4),(18,32),(19,15),(20,17),(20,31),(21,13),(21,32),(22,11),(22,20),(22,32),(23,16),(23,33),(24,15),(25,19),(26,18),(26,21),(27,13),(27,20),(28,14),(28,16),(29,12),(29,26),(29,30),(30,21),(30,22),(30,27),(31,14),(31,33),(32,23),(32,28),(32,31),(33,25),(33,34),(34,19),(34,24)],35)
=> ([(0,18),(1,17),(2,15),(2,22),(3,16),(3,23),(4,13),(4,24),(5,14),(5,25),(6,26),(6,27),(7,15),(7,17),(7,32),(8,16),(8,18),(8,33),(9,26),(9,30),(9,32),(10,27),(10,31),(10,33),(11,19),(11,22),(11,31),(12,20),(12,23),(12,30),(13,17),(13,32),(14,18),(14,33),(15,28),(16,29),(19,24),(19,28),(19,34),(20,25),(20,29),(20,34),(21,22),(21,23),(21,30),(21,31),(22,28),(23,29),(24,26),(24,32),(25,27),(25,33),(26,34),(27,34),(28,30),(28,32),(29,31),(29,33),(30,34),(31,34)],35)
=> ([(0,3),(0,4),(0,5),(0,6),(0,11),(0,12),(0,13),(0,14),(0,15),(0,16),(0,17),(0,18),(0,19),(0,20),(0,21),(0,22),(0,23),(0,24),(0,25),(0,26),(0,27),(0,28),(0,29),(0,30),(0,31),(0,32),(0,33),(0,34),(1,2),(1,4),(1,5),(1,6),(1,7),(1,8),(1,9),(1,10),(1,11),(1,12),(1,13),(1,14),(1,16),(1,17),(1,18),(1,19),(1,21),(1,22),(1,23),(1,25),(1,26),(1,27),(1,28),(1,29),(1,30),(1,31),(1,32),(1,33),(1,34),(2,3),(2,5),(2,6),(2,7),(2,8),(2,9),(2,10),(2,11),(2,12),(2,13),(2,14),(2,16),(2,17),(2,18),(2,19),(2,20),(2,22),(2,23),(2,24),(2,26),(2,27),(2,28),(2,29),(2,30),(2,31),(2,32),(2,33),(2,34),(3,4),(3,5),(3,7),(3,9),(3,10),(3,11),(3,12),(3,13),(3,15),(3,16),(3,17),(3,18),(3,20),(3,21),(3,22),(3,23),(3,24),(3,25),(3,26),(3,27),(3,28),(3,29),(3,30),(3,31),(3,32),(3,33),(3,34),(4,6),(4,8),(4,9),(4,10),(4,11),(4,12),(4,14),(4,15),(4,16),(4,17),(4,19),(4,20),(4,21),(4,22),(4,23),(4,24),(4,25),(4,26),(4,27),(4,28),(4,29),(4,30),(4,31),(4,32),(4,33),(4,34),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(5,13),(5,14),(5,15),(5,16),(5,18),(5,19),(5,20),(5,22),(5,24),(5,25),(5,26),(5,27),(5,28),(5,29),(5,30),(5,31),(5,32),(5,33),(5,34),(6,7),(6,8),(6,9),(6,10),(6,12),(6,13),(6,14),(6,15),(6,17),(6,18),(6,19),(6,21),(6,23),(6,24),(6,25),(6,26),(6,27),(6,28),(6,29),(6,30),(6,31),(6,32),(6,33),(6,34),(7,8),(7,9),(7,10),(7,11),(7,13),(7,14),(7,15),(7,16),(7,17),(7,18),(7,19),(7,20),(7,21),(7,22),(7,23),(7,25),(7,26),(7,27),(7,28),(7,29),(7,30),(7,31),(7,32),(7,33),(7,34),(8,9),(8,10),(8,12),(8,13),(8,14),(8,15),(8,16),(8,17),(8,18),(8,19),(8,20),(8,21),(8,22),(8,23),(8,24),(8,26),(8,27),(8,28),(8,29),(8,30),(8,31),(8,32),(8,33),(8,34),(9,10),(9,11),(9,13),(9,14),(9,15),(9,16),(9,17),(9,18),(9,19),(9,20),(9,22),(9,23),(9,24),(9,25),(9,26),(9,27),(9,28),(9,29),(9,30),(9,31),(9,33),(9,34),(10,12),(10,13),(10,14),(10,15),(10,16),(10,17),(10,18),(10,19),(10,21),(10,22),(10,23),(10,24),(10,25),(10,26),(10,27),(10,28),(10,29),(10,30),(10,31),(10,33),(10,34),(11,12),(11,13),(11,14),(11,15),(11,16),(11,17),(11,18),(11,19),(11,20),(11,21),(11,22),(11,23),(11,24),(11,25),(11,26),(11,27),(11,28),(11,29),(11,30),(11,32),(11,33),(11,34),(12,13),(12,14),(12,15),(12,16),(12,17),(12,18),(12,19),(12,20),(12,21),(12,22),(12,23),(12,24),(12,25),(12,26),(12,27),(12,28),(12,29),(12,31),(12,32),(12,33),(12,34),(13,14),(13,16),(13,17),(13,18),(13,19),(13,20),(13,21),(13,22),(13,23),(13,25),(13,26),(13,27),(13,28),(13,30),(13,31),(13,32),(13,33),(13,34),(14,16),(14,17),(14,18),(14,19),(14,20),(14,21),(14,22),(14,23),(14,24),(14,26),(14,27),(14,29),(14,30),(14,31),(14,32),(14,33),(14,34),(15,16),(15,17),(15,18),(15,19),(15,20),(15,21),(15,22),(15,23),(15,24),(15,25),(15,26),(15,27),(15,28),(15,29),(15,30),(15,31),(15,32),(15,33),(15,34),(16,17),(16,18),(16,19),(16,20),(16,21),(16,22),(16,23),(16,24),(16,25),(16,26),(16,28),(16,29),(16,30),(16,32),(16,33),(16,34),(17,18),(17,19),(17,20),(17,21),(17,22),(17,23),(17,24),(17,25),(17,27),(17,28),(17,29),(17,31),(17,32),(17,33),(17,34),(18,19),(18,20),(18,21),(18,22),(18,24),(18,25),(18,26),(18,27),(18,28),(18,30),(18,31),(18,32),(18,33),(18,34),(19,20),(19,21),(19,23),(19,24),(19,25),(19,26),(19,27),(19,29),(19,30),(19,31),(19,32),(19,33),(19,34),(20,21),(20,22),(20,23),(20,24),(20,25),(20,26),(20,27),(20,28),(20,29),(20,30),(20,31),(20,32),(20,33),(20,34),(21,22),(21,23),(21,24),(21,25),(21,26),(21,27),(21,28),(21,29),(21,30),(21,31),(21,32),(21,33),(21,34),(22,23),(22,24),(22,25),(22,26),(22,28),(22,29),(22,30),(22,31),(22,32),(22,33),(22,34),(23,24),(23,25),(23,27),(23,28),(23,29),(23,30),(23,31),(23,32),(23,33),(23,34),(24,25),(24,26),(24,27),(24,28),(24,29),(24,30),(24,31),(24,32),(24,33),(24,34),(25,26),(25,27),(25,28),(25,29),(25,30),(25,31),(25,32),(25,33),(25,34),(26,27),(26,28),(26,29),(26,30),(26,31),(26,32),(26,33),(27,28),(27,29),(27,30),(27,31),(27,32),(27,34),(28,29),(28,30),(28,31),(28,32),(28,33),(28,34),(29,30),(29,31),(29,32),(29,33),(29,34),(30,31),(30,32),(30,33),(30,34),(31,32),(31,33),(31,34),(32,33),(32,34),(33,34)],35)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,6} + 1
[[1,1,4],[4]]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5)],7)
=> ([(0,1),(0,4),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(5,6)],7)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,6} + 1
[[1,2,2],[4]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(0,6),(1,2),(1,4),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
=> ([(0,1),(0,4),(0,6),(1,3),(1,4),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,6} + 1
[[1,2,4],[2]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(0,6),(1,2),(1,4),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
=> ([(0,1),(0,4),(0,6),(1,3),(1,4),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,6} + 1
[[1,2,3],[4]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ([(0,7),(1,3),(1,8),(2,7),(2,8),(3,5),(4,5),(4,6),(5,8),(6,7),(6,8)],9)
=> ([(0,3),(0,6),(0,7),(0,8),(1,2),(1,4),(1,5),(1,7),(1,8),(2,3),(2,4),(2,5),(2,8),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(4,8),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,6} + 1
[[1,2,4],[3]]
=> ([(0,5),(0,6),(1,8),(2,9),(3,8),(3,9),(4,1),(5,4),(6,7),(7,2),(7,3),(8,10),(9,10)],11)
=> ([(0,1),(0,2),(1,4),(2,3),(3,8),(4,9),(5,7),(5,8),(6,7),(6,9),(7,10),(8,10),(9,10)],11)
=> ([(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(1,2),(1,3),(1,6),(1,7),(1,8),(1,9),(1,10),(2,3),(2,5),(2,7),(2,8),(2,9),(2,10),(3,4),(3,6),(3,8),(3,9),(3,10),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,10),(7,9),(7,10),(8,9)],11)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,6} + 1
[[1,3,4],[2]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ([(0,7),(1,3),(1,8),(2,7),(2,8),(3,5),(4,5),(4,6),(5,8),(6,7),(6,8)],9)
=> ([(0,3),(0,6),(0,7),(0,8),(1,2),(1,4),(1,5),(1,7),(1,8),(2,3),(2,4),(2,5),(2,8),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(4,8),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,6} + 1
[[1,2,4],[4]]
=> ([(0,6),(0,7),(1,9),(2,12),(3,9),(3,12),(4,10),(5,1),(6,5),(7,8),(8,2),(8,3),(9,11),(11,10),(12,4),(12,11)],13)
=> ([(0,1),(0,2),(1,4),(2,6),(3,5),(3,12),(4,8),(5,9),(6,10),(7,8),(7,12),(8,11),(9,10),(9,12),(10,11),(11,12)],13)
=> ([(0,3),(0,4),(0,7),(0,8),(0,9),(0,10),(0,11),(0,12),(1,2),(1,5),(1,6),(1,7),(1,8),(1,9),(1,10),(1,11),(1,12),(2,4),(2,5),(2,6),(2,8),(2,9),(2,10),(2,11),(2,12),(3,4),(3,5),(3,6),(3,7),(3,9),(3,10),(3,11),(3,12),(4,6),(4,7),(4,8),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(5,12),(6,8),(6,9),(6,10),(6,11),(6,12),(7,8),(7,9),(7,10),(7,11),(7,12),(8,9),(8,11),(8,12),(9,10),(9,12),(10,11)],13)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,6} + 1
[[1,4,4],[2]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ([(0,8),(1,7),(2,5),(2,6),(3,7),(3,9),(4,8),(4,9),(5,7),(5,9),(6,8),(6,9)],10)
=> ([(0,3),(0,4),(0,7),(0,8),(0,9),(1,2),(1,4),(1,5),(1,6),(1,8),(1,9),(2,3),(2,5),(2,6),(2,8),(2,9),(3,4),(3,5),(3,7),(3,8),(4,6),(4,7),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,6} + 1
[[1,3,3],[4]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ([(0,8),(1,7),(2,5),(2,6),(3,7),(3,9),(4,8),(4,9),(5,7),(5,9),(6,8),(6,9)],10)
=> ([(0,3),(0,4),(0,7),(0,8),(0,9),(1,2),(1,4),(1,5),(1,6),(1,8),(1,9),(2,3),(2,5),(2,6),(2,8),(2,9),(3,4),(3,5),(3,7),(3,8),(4,6),(4,7),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,6} + 1
[[1,3,4],[3]]
=> ([(0,6),(1,7),(2,8),(3,9),(4,3),(4,7),(5,2),(5,10),(6,1),(6,4),(7,5),(7,9),(9,10),(10,8)],11)
=> ([(0,9),(1,7),(1,8),(2,9),(2,10),(3,4),(3,5),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(8,10)],11)
=> ([(0,4),(0,5),(0,7),(0,8),(0,9),(0,10),(1,2),(1,3),(1,5),(1,6),(1,7),(1,8),(1,10),(2,3),(2,5),(2,6),(2,7),(2,9),(2,10),(3,4),(3,6),(3,7),(3,8),(3,10),(4,5),(4,6),(4,8),(4,9),(4,10),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(6,10),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,6} + 1
[[1,3,4],[4]]
=> ([(0,9),(0,11),(1,18),(2,17),(3,19),(4,13),(4,19),(5,12),(5,13),(6,16),(7,14),(8,5),(8,18),(9,10),(10,3),(10,4),(11,1),(11,8),(12,17),(13,15),(15,16),(16,14),(17,7),(18,2),(18,12),(19,6),(19,15)],20)
=> ([(0,15),(0,18),(1,14),(1,18),(2,16),(2,19),(3,17),(3,19),(4,6),(4,14),(5,7),(5,15),(6,16),(7,17),(8,9),(8,12),(8,13),(9,10),(9,11),(10,14),(10,18),(11,15),(11,18),(12,16),(12,19),(13,17),(13,19)],20)
=> ([(0,1),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(0,11),(0,14),(0,15),(0,16),(0,17),(0,18),(0,19),(1,2),(1,3),(1,6),(1,7),(1,8),(1,9),(1,10),(1,11),(1,12),(1,13),(1,16),(1,17),(1,18),(1,19),(2,3),(2,4),(2,5),(2,6),(2,7),(2,9),(2,10),(2,12),(2,13),(2,14),(2,15),(2,16),(2,17),(2,18),(2,19),(3,4),(3,5),(3,6),(3,7),(3,8),(3,10),(3,12),(3,13),(3,14),(3,15),(3,16),(3,17),(3,18),(3,19),(4,5),(4,6),(4,8),(4,9),(4,11),(4,12),(4,13),(4,14),(4,15),(4,16),(4,17),(4,18),(4,19),(5,7),(5,8),(5,9),(5,11),(5,12),(5,13),(5,14),(5,15),(5,16),(5,17),(5,18),(5,19),(6,7),(6,8),(6,9),(6,10),(6,11),(6,12),(6,13),(6,14),(6,16),(6,17),(6,18),(7,8),(7,9),(7,10),(7,11),(7,12),(7,13),(7,15),(7,16),(7,17),(7,19),(8,9),(8,10),(8,11),(8,12),(8,14),(8,15),(8,16),(8,18),(8,19),(9,10),(9,11),(9,13),(9,14),(9,15),(9,17),(9,18),(9,19),(10,12),(10,13),(10,14),(10,15),(10,16),(10,17),(10,18),(10,19),(11,12),(11,13),(11,14),(11,15),(11,16),(11,17),(11,18),(11,19),(12,13),(12,14),(12,15),(12,16),(12,17),(12,18),(12,19),(13,14),(13,15),(13,16),(13,17),(13,18),(13,19),(14,15),(14,16),(14,17),(14,18),(14,19),(15,16),(15,17),(15,18),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,6} + 1
[[1,4,4],[3]]
=> ([(0,7),(1,10),(2,11),(3,8),(4,9),(5,2),(5,9),(6,3),(6,12),(7,4),(7,5),(8,10),(9,6),(9,11),(11,12),(12,1),(12,8)],13)
=> ([(0,10),(1,3),(1,11),(2,10),(2,12),(3,6),(4,6),(4,8),(5,7),(5,9),(6,11),(7,10),(7,12),(8,11),(8,12),(9,11),(9,12)],13)
=> ([(0,1),(0,5),(0,6),(0,8),(0,9),(0,10),(0,11),(0,12),(1,4),(1,6),(1,7),(1,9),(1,10),(1,11),(1,12),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(2,10),(2,12),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(3,11),(3,12),(4,5),(4,6),(4,7),(4,8),(4,11),(4,12),(5,7),(5,8),(5,9),(5,10),(5,12),(6,7),(6,9),(6,10),(6,11),(7,8),(7,10),(7,11),(7,12),(8,9),(8,10),(8,11),(8,12),(9,10),(9,11),(9,12),(10,11),(10,12),(11,12)],13)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,6} + 1
[[1,4,4],[4]]
=> ([(0,8),(2,13),(3,11),(4,9),(5,10),(6,3),(6,10),(7,4),(7,12),(8,5),(8,6),(9,13),(10,7),(10,11),(11,12),(12,2),(12,9),(13,1)],14)
=> ([(0,11),(1,10),(2,6),(2,8),(3,7),(3,9),(4,10),(4,12),(5,11),(5,13),(6,10),(6,12),(7,11),(7,13),(8,12),(8,13),(9,12),(9,13)],14)
=> ([(0,1),(0,5),(0,6),(0,7),(0,9),(0,10),(0,11),(0,12),(0,13),(1,4),(1,6),(1,7),(1,8),(1,10),(1,11),(1,12),(1,13),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(2,10),(2,12),(2,13),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(3,11),(3,12),(3,13),(4,5),(4,6),(4,8),(4,9),(4,10),(4,12),(4,13),(5,7),(5,8),(5,9),(5,11),(5,12),(5,13),(6,7),(6,8),(6,10),(6,11),(6,12),(7,9),(7,10),(7,11),(7,13),(8,9),(8,10),(8,11),(8,12),(8,13),(9,10),(9,11),(9,12),(9,13),(10,11),(10,12),(10,13),(11,12),(11,13),(12,13)],14)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,6} + 1
[[2,2,2],[4]]
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> ([(0,8),(1,4),(1,8),(2,3),(2,6),(3,7),(4,5),(4,6),(5,7),(5,8),(6,7)],9)
=> ([(0,1),(0,2),(0,3),(0,5),(0,6),(1,2),(1,5),(1,6),(1,8),(2,5),(2,7),(2,8),(3,4),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,6} + 1
[[2,2,3],[4]]
=> ([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
=> ([(0,9),(1,7),(1,8),(2,7),(2,11),(3,6),(3,8),(4,9),(4,11),(5,6),(5,9),(5,11),(6,10),(7,10),(8,10),(10,11)],12)
=> ([(0,2),(0,4),(0,5),(0,6),(0,9),(0,10),(0,11),(1,3),(1,6),(1,7),(1,8),(1,9),(1,10),(1,11),(2,3),(2,4),(2,5),(2,6),(2,7),(2,10),(2,11),(3,4),(3,7),(3,8),(3,9),(3,10),(3,11),(4,5),(4,6),(4,7),(4,8),(4,11),(5,6),(5,7),(5,8),(5,9),(5,11),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,6} + 1
[[2,2,4],[3]]
=> ([(0,6),(0,7),(1,11),(2,9),(3,9),(3,10),(4,5),(5,1),(5,10),(6,4),(7,8),(8,2),(8,3),(9,12),(10,11),(10,12),(11,13),(12,13)],14)
=> ([(0,1),(0,2),(1,3),(2,8),(3,9),(4,7),(4,8),(5,7),(5,10),(6,9),(6,11),(7,13),(8,13),(9,12),(10,11),(10,13),(11,12),(12,13)],14)
=> ([(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(0,11),(0,12),(0,13),(1,2),(1,3),(1,4),(1,7),(1,8),(1,9),(1,10),(1,11),(1,12),(1,13),(2,3),(2,4),(2,6),(2,8),(2,9),(2,10),(2,11),(2,12),(2,13),(3,4),(3,5),(3,6),(3,9),(3,10),(3,11),(3,12),(3,13),(4,5),(4,6),(4,7),(4,10),(4,11),(4,12),(4,13),(5,6),(5,7),(5,8),(5,9),(5,11),(5,12),(5,13),(6,7),(6,8),(6,9),(6,12),(6,13),(7,8),(7,9),(7,10),(7,11),(7,12),(7,13),(8,9),(8,10),(8,11),(8,12),(8,13),(9,10),(9,11),(9,13),(10,11),(10,12),(10,13),(11,12)],14)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,5,6} + 1
Description
The chromatic number of a graph.
The minimal number of colors needed to color the vertices of the graph such that no two vertices which share an edge have the same color.
Matching statistic: St000510
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00307: Posets —promotion cycle type⟶ Integer partitions
St000510: Integer partitions ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 18%
Mp00307: Posets —promotion cycle type⟶ Integer partitions
St000510: Integer partitions ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 18%
Values
[[1,2]]
=> ([(0,1)],2)
=> [1]
=> ? ∊ {0,0,1}
[[2,2]]
=> ([(0,2),(2,1)],3)
=> [1]
=> ? ∊ {0,0,1}
[[1],[2]]
=> ([],1)
=> [1]
=> ? ∊ {0,0,1}
[[1,1,2]]
=> ([(0,1)],2)
=> [1]
=> ? ∊ {0,0,0,1,1}
[[1,2,2]]
=> ([(0,2),(2,1)],3)
=> [1]
=> ? ∊ {0,0,0,1,1}
[[2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> [1]
=> ? ∊ {0,0,0,1,1}
[[1,1],[2]]
=> ([],1)
=> [1]
=> ? ∊ {0,0,0,1,1}
[[1,2],[2]]
=> ([(0,1)],2)
=> [1]
=> ? ∊ {0,0,0,1,1}
[[1,1,3]]
=> ([(0,2),(2,1)],3)
=> [1]
=> ? ∊ {0,0,0,1,1,1,2,3}
[[1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [2]
=> 1
[[1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [2]
=> 1
[[2,2,3]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [3,2]
=> 0
[[2,3,3]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> [10,2]
=> 0
[[3,3,3]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> [10,2]
=> 0
[[1,1],[3]]
=> ([(0,1)],2)
=> [1]
=> ? ∊ {0,0,0,1,1,1,2,3}
[[1,2],[3]]
=> ([(0,2),(2,1)],3)
=> [1]
=> ? ∊ {0,0,0,1,1,1,2,3}
[[1,3],[2]]
=> ([(0,2),(2,1)],3)
=> [1]
=> ? ∊ {0,0,0,1,1,1,2,3}
[[1,3],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> [1]
=> ? ∊ {0,0,0,1,1,1,2,3}
[[2,2],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> [1]
=> ? ∊ {0,0,0,1,1,1,2,3}
[[2,3],[3]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> [6,6,6,2]
=> ? ∊ {0,0,0,1,1,1,2,3}
[[1],[2],[3]]
=> ([],1)
=> [1]
=> ? ∊ {0,0,0,1,1,1,2,3}
[[1,1,1,2]]
=> ([(0,1)],2)
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,2}
[[1,1,2,2]]
=> ([(0,2),(2,1)],3)
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,2}
[[1,2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,2}
[[2,2,2,2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,2}
[[1,1,1],[2]]
=> ([],1)
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,2}
[[1,1,2],[2]]
=> ([(0,1)],2)
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,2}
[[1,2,2],[2]]
=> ([(0,2),(2,1)],3)
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,2}
[[1,1],[2,2]]
=> ([],1)
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,2}
[[1,1,1,3]]
=> ([(0,2),(2,1)],3)
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,3,3,3}
[[1,1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [2]
=> 1
[[1,1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [2]
=> 1
[[1,2,2,3]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [3,2]
=> 0
[[1,2,3,3]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> [10,2]
=> 0
[[1,3,3,3]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> [10,2]
=> 0
[[2,2,2,3]]
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> [8,4,2]
=> ? ∊ {0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,3,3,3}
[[2,2,3,3]]
=> ([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
=> [24,24,24,24,14]
=> ? ∊ {0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,3,3,3}
[[2,3,3,3]]
=> ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
=> [15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,5,5,3,3]
=> ? ∊ {0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,3,3,3}
[[3,3,3,3]]
=> ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
=> [15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,5,5,3,3]
=> ? ∊ {0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,3,3,3}
[[1,1,1],[3]]
=> ([(0,1)],2)
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,3,3,3}
[[1,1,2],[3]]
=> ([(0,2),(2,1)],3)
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,3,3,3}
[[1,1,3],[2]]
=> ([(0,2),(2,1)],3)
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,3,3,3}
[[1,1,3],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,3,3,3}
[[1,2,2],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,3,3,3}
[[1,2,3],[2]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [2]
=> 1
[[1,2,3],[3]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> [6,6,6,2]
=> ? ∊ {0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,3,3,3}
[[1,3,3],[2]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [2]
=> 1
[[1,3,3],[3]]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> [2]
=> 1
[[2,2,2],[3]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,3,3,3}
[[2,2,3],[3]]
=> ([(0,6),(0,7),(1,9),(2,8),(3,5),(4,2),(5,1),(5,8),(6,3),(7,4),(8,9)],10)
=> [26,13,7,7,2]
=> ? ∊ {0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,3,3,3}
[[2,3,3],[3]]
=> ([(0,9),(0,10),(1,11),(2,14),(3,12),(4,13),(5,4),(5,11),(6,5),(7,3),(8,1),(8,14),(9,6),(10,2),(10,8),(11,13),(13,12),(14,7)],15)
=> ?
=> ? ∊ {0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,3,3,3}
[[1,1],[2,3]]
=> ([(0,1)],2)
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,3,3,3}
[[1,1],[3,3]]
=> ([(0,2),(2,1)],3)
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,3,3,3}
[[1,2],[2,3]]
=> ([(0,2),(2,1)],3)
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,3,3,3}
[[1,2],[3,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [2]
=> 1
[[2,2],[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [2]
=> 1
[[1,1],[2],[3]]
=> ([],1)
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,3,3,3}
[[1,2],[2],[3]]
=> ([(0,1)],2)
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,3,3,3}
[[1,3],[2],[3]]
=> ([(0,2),(2,1)],3)
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,3,3,3}
[[1,1,1,1,2]]
=> ([(0,1)],2)
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2}
[[1,1,1,2,2]]
=> ([(0,2),(2,1)],3)
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2}
[[1,1,2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2}
[[1,2,2,2,2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2}
[[2,2,2,2,2]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2}
[[1,1,1,1],[2]]
=> ([],1)
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,1,2,2}
[[1,1,2,4]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [3,2]
=> 0
[[1,1,3,4]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> [10,2]
=> 0
[[1,1,4,4]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> [10,2]
=> 0
[[1,1,2],[4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [2]
=> 1
[[1,1,3],[4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [2]
=> 1
[[1,1,4],[3]]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> [2]
=> 1
[[1,1,4],[4]]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> [2]
=> 1
[[1,2,2],[4]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [3,2]
=> 0
[[1,2,4],[2]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [3,2]
=> 0
[[1,2,3],[4]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> [10,2]
=> 0
[[1,3,4],[2]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> [10,2]
=> 0
[[1,4,4],[2]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> [10,2]
=> 0
[[1,3,3],[4]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> [10,2]
=> 0
[[1,1],[3,4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [2]
=> 1
[[1,1],[4,4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [2]
=> 1
[[1,2],[2,4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [2]
=> 1
[[1,3],[2,4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [2]
=> 1
[[1,3],[3,4]]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> [2]
=> 1
[[1,2],[2],[4]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> [2]
=> 1
[[1,3],[2],[4]]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [2]
=> 1
[[1,3],[3],[4]]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> [2]
=> 1
[[1,1,1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [2]
=> 1
[[1,1,1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [2]
=> 1
[[1,1,2,2,3]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [3,2]
=> 0
[[1,1,2,3,3]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> [10,2]
=> 0
[[1,1,3,3,3]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> [10,2]
=> 0
[[1,1,2,3],[2]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [2]
=> 1
[[1,1,3,3],[2]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [2]
=> 1
[[1,1,3,3],[3]]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> [2]
=> 1
[[1,2,2,3],[2]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> [3,2]
=> 0
[[1,2,3,3],[2]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> [10,2]
=> 0
[[1,3,3,3],[2]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> [10,2]
=> 0
[[1,3,3,3],[3]]
=> ([(0,7),(2,9),(3,10),(4,8),(5,4),(5,10),(6,1),(7,3),(7,5),(8,9),(9,6),(10,2),(10,8)],11)
=> [10,2]
=> 0
[[1,1,2],[3,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [2]
=> 1
[[1,2,2],[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [2]
=> 1
Description
The number of invariant oriented cycles when acting with a permutation of given cycle type.
The following 62 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000928The sum of the coefficients of the character polynomial of an integer partition. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001644The dimension of a graph. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000260The radius of a connected graph. St000456The monochromatic index of a connected graph. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001568The smallest positive integer that does not appear twice in the partition. St001877Number of indecomposable injective modules with projective dimension 2. St001118The acyclic chromatic index of a graph. St001624The breadth of a lattice. St000455The second largest eigenvalue of a graph if it is integral. St000264The girth of a graph, which is not a tree. St000284The Plancherel distribution on integer partitions. St000478Another weight of a partition according to Alladi. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000567The sum of the products of all pairs of parts. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000668The least common multiple of the parts of the partition. St000681The Grundy value of Chomp on Ferrers diagrams. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000706The product of the factorials of the multiplicities of an integer partition. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000929The constant term of the character polynomial of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St000934The 2-degree of an integer partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St000940The number of characters of the symmetric group whose value on the partition is zero. St000941The number of characters of the symmetric group whose value on the partition is even. St000993The multiplicity of the largest part of an integer partition. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001128The exponens consonantiae of a partition. St000379The number of Hamiltonian cycles in a graph. St001281The normalized isoperimetric number of a graph. St001592The maximal number of simple paths between any two different vertices of a graph. St000259The diameter of a connected graph. St000718The largest Laplacian eigenvalue of a graph if it is integral. St000713The dimension of the irreducible representation of Sp(4) labelled by an integer partition. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St001098The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for vertex labelled trees. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset.
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