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Your data matches 32 different statistics following compositions of up to 3 maps.
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Matching statistic: St000560
St000560: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> 0
{{1},{2}}
=> 0
{{1,2,3}}
=> 0
{{1,2},{3}}
=> 0
{{1,3},{2}}
=> 0
{{1},{2,3}}
=> 0
{{1},{2},{3}}
=> 0
{{1,2,3,4}}
=> 0
{{1,2,3},{4}}
=> 0
{{1,2,4},{3}}
=> 0
{{1,2},{3,4}}
=> 1
{{1,2},{3},{4}}
=> 0
{{1,3,4},{2}}
=> 0
{{1,3},{2,4}}
=> 0
{{1,3},{2},{4}}
=> 0
{{1,4},{2,3}}
=> 0
{{1},{2,3,4}}
=> 0
{{1},{2,3},{4}}
=> 0
{{1,4},{2},{3}}
=> 0
{{1},{2,4},{3}}
=> 0
{{1},{2},{3,4}}
=> 0
{{1},{2},{3},{4}}
=> 0
{{1,2,3,4,5}}
=> 0
{{1,2,3,4},{5}}
=> 0
{{1,2,3,5},{4}}
=> 0
{{1,2,3},{4,5}}
=> 3
{{1,2,3},{4},{5}}
=> 0
{{1,2,4,5},{3}}
=> 0
{{1,2,4},{3,5}}
=> 1
{{1,2,4},{3},{5}}
=> 0
{{1,2,5},{3,4}}
=> 1
{{1,2},{3,4,5}}
=> 3
{{1,2},{3,4},{5}}
=> 1
{{1,2,5},{3},{4}}
=> 0
{{1,2},{3,5},{4}}
=> 1
{{1,2},{3},{4,5}}
=> 1
{{1,2},{3},{4},{5}}
=> 0
{{1,3,4,5},{2}}
=> 0
{{1,3,4},{2,5}}
=> 0
{{1,3,4},{2},{5}}
=> 0
{{1,3,5},{2,4}}
=> 0
{{1,3},{2,4,5}}
=> 1
{{1,3},{2,4},{5}}
=> 0
{{1,3,5},{2},{4}}
=> 0
{{1,3},{2,5},{4}}
=> 0
{{1,3},{2},{4,5}}
=> 1
{{1,3},{2},{4},{5}}
=> 0
{{1,4,5},{2,3}}
=> 1
{{1,4},{2,3,5}}
=> 0
{{1,4},{2,3},{5}}
=> 0
Description
The number of occurrences of the pattern {{1,2},{3,4}} in a set partition.
Matching statistic: St000455
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 15% ●values known / values provided: 54%●distinct values known / distinct values provided: 15%
Mp00223: Permutations —runsort⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 15% ●values known / values provided: 54%●distinct values known / distinct values provided: 15%
Values
{{1,2}}
=> [2,1] => [1,2] => ([],2)
=> ? ∊ {0,0}
{{1},{2}}
=> [1,2] => [1,2] => ([],2)
=> ? ∊ {0,0}
{{1,2,3}}
=> [2,3,1] => [1,2,3] => ([],3)
=> ? ∊ {0,0,0}
{{1,2},{3}}
=> [2,1,3] => [1,3,2] => ([(1,2)],3)
=> 0
{{1,3},{2}}
=> [3,2,1] => [1,2,3] => ([],3)
=> ? ∊ {0,0,0}
{{1},{2,3}}
=> [1,3,2] => [1,3,2] => ([(1,2)],3)
=> 0
{{1},{2},{3}}
=> [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {0,0,0}
{{1,2,3,4}}
=> [2,3,4,1] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,1}
{{1,2,3},{4}}
=> [2,3,1,4] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 0
{{1,2,4},{3}}
=> [2,4,3,1] => [1,2,4,3] => ([(2,3)],4)
=> 0
{{1,2},{3,4}}
=> [2,1,4,3] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 0
{{1,2},{3},{4}}
=> [2,1,3,4] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> 0
{{1,3,4},{2}}
=> [3,2,4,1] => [1,2,4,3] => ([(2,3)],4)
=> 0
{{1,3},{2,4}}
=> [3,4,1,2] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,1}
{{1,3},{2},{4}}
=> [3,2,1,4] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 0
{{1,4},{2,3}}
=> [4,3,2,1] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,1}
{{1},{2,3,4}}
=> [1,3,4,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> 0
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
=> 0
{{1,4},{2},{3}}
=> [4,2,3,1] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,1}
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 0
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,4,3] => ([(2,3)],4)
=> 0
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,1}
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [1,2,3,4,5] => ([],5)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,3,3}
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> 0
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [1,2,3,5,4] => ([(3,4)],5)
=> 0
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> 0
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 0
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [1,2,4,3,5] => ([(3,4)],5)
=> 0
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [1,3,2,4,5] => ([(3,4)],5)
=> 0
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,3,3}
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> 0
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 0
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> 0
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> 0
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> 0
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,3,3}
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> 0
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> 0
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [1,2,3,5,4] => ([(3,4)],5)
=> 0
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,3,3}
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [1,2,3,4,5] => ([],5)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,3,3}
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> 0
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> 0
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> 0
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> 0
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> 0
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 0
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> 0
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [1,2,3,5,4] => ([(3,4)],5)
=> 0
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> 0
{{1,5},{2,3,4}}
=> [5,3,4,2,1] => [1,2,3,4,5] => ([],5)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,3,3}
{{1},{2,3,4,5}}
=> [1,3,4,5,2] => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> 0
{{1},{2,3,4},{5}}
=> [1,3,4,2,5] => [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> 0
{{1,5},{2,3},{4}}
=> [5,3,2,4,1] => [1,2,4,3,5] => ([(3,4)],5)
=> 0
{{1},{2,3,5},{4}}
=> [1,3,5,4,2] => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,3,3}
{{1},{2,3},{4,5}}
=> [1,3,2,5,4] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> 0
{{1},{2,3},{4},{5}}
=> [1,3,2,4,5] => [1,3,2,4,5] => ([(3,4)],5)
=> 0
{{1,4,5},{2},{3}}
=> [4,2,3,5,1] => [1,2,3,5,4] => ([(3,4)],5)
=> 0
{{1,4},{2,5},{3}}
=> [4,5,3,1,2] => [1,2,3,4,5] => ([],5)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,3,3}
{{1,4},{2},{3,5}}
=> [4,2,5,1,3] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> 0
{{1,4},{2},{3},{5}}
=> [4,2,3,1,5] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> 0
{{1,5},{2,4},{3}}
=> [5,4,3,2,1] => [1,2,3,4,5] => ([],5)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,3,3}
{{1},{2,4,5},{3}}
=> [1,4,3,5,2] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> 0
{{1},{2,4},{3,5}}
=> [1,4,5,2,3] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 0
{{1},{2,4},{3},{5}}
=> [1,4,3,2,5] => [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,3,3}
{{1,5},{2},{3,4}}
=> [5,2,4,3,1] => [1,2,4,3,5] => ([(3,4)],5)
=> 0
{{1},{2,5},{3,4}}
=> [1,5,4,3,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> 0
{{1},{2},{3,4,5}}
=> [1,2,4,5,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> 0
{{1},{2},{3,4},{5}}
=> [1,2,4,3,5] => [1,2,4,3,5] => ([(3,4)],5)
=> 0
{{1,5},{2},{3},{4}}
=> [5,2,3,4,1] => [1,2,3,4,5] => ([],5)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,3,3}
{{1},{2,5},{3},{4}}
=> [1,5,3,4,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> 0
{{1},{2},{3},{4},{5}}
=> [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,3,3}
{{1,2,3,4,5,6}}
=> [2,3,4,5,6,1] => [1,2,3,4,5,6] => ([],6)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,6,6,9}
{{1,2,3,5},{4},{6}}
=> [2,3,5,4,1,6] => [1,6,2,3,5,4] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,6,6,9}
{{1,2,3},{4},{5,6}}
=> [2,3,1,4,6,5] => [1,4,6,2,3,5] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,6,6,9}
{{1,2,4,5},{3},{6}}
=> [2,4,3,5,1,6] => [1,6,2,4,3,5] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,6,6,9}
{{1,2,4},{3,5,6}}
=> [2,4,5,1,6,3] => [1,6,2,4,5,3] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,6,6,9}
{{1,2,4},{3,5},{6}}
=> [2,4,5,1,3,6] => [1,3,6,2,4,5] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,6,6,9}
{{1,2,4},{3,6},{5}}
=> [2,4,6,1,5,3] => [1,5,2,4,6,3] => ([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,6,6,9}
{{1,2,4},{3},{5,6}}
=> [2,4,3,1,6,5] => [1,6,2,4,3,5] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,6,6,9}
{{1,2,4},{3},{5},{6}}
=> [2,4,3,1,5,6] => [1,5,6,2,4,3] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,6,6,9}
{{1,2,5,6},{3,4}}
=> [2,5,4,3,6,1] => [1,2,5,3,6,4] => ([(2,5),(3,4),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,6,6,9}
{{1,2,5},{3,4},{6}}
=> [2,5,4,3,1,6] => [1,6,2,5,3,4] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,6,6,9}
{{1,2,6},{3,4},{5}}
=> [2,6,4,3,5,1] => [1,2,6,3,5,4] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,6,6,9}
{{1,2},{3,4,6},{5}}
=> [2,1,4,6,5,3] => [1,4,6,2,3,5] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,6,6,9}
{{1,2,5},{3},{4,6}}
=> [2,5,3,6,1,4] => [1,4,2,5,3,6] => ([(2,5),(3,4),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,6,6,9}
{{1,2,5},{3},{4},{6}}
=> [2,5,3,4,1,6] => [1,6,2,5,3,4] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,6,6,9}
{{1,2},{3,5},{4},{6}}
=> [2,1,5,4,3,6] => [1,5,2,3,6,4] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,6,6,9}
{{1,2,6},{3},{4,5}}
=> [2,6,3,5,4,1] => [1,2,6,3,5,4] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,6,6,9}
{{1,2},{3},{4,5,6}}
=> [2,1,3,5,6,4] => [1,3,5,6,2,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,6,6,9}
{{1,2},{3},{4,5},{6}}
=> [2,1,3,5,4,6] => [1,3,5,2,4,6] => ([(2,5),(3,4),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,6,6,9}
{{1,2},{3},{4,6},{5}}
=> [2,1,3,6,5,4] => [1,3,6,2,4,5] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,6,6,9}
{{1,2},{3},{4},{5,6}}
=> [2,1,3,4,6,5] => [1,3,4,6,2,5] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,6,6,9}
{{1,3,4,5},{2},{6}}
=> [3,2,4,5,1,6] => [1,6,2,4,5,3] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,6,6,9}
{{1,3,4},{2,5,6}}
=> [3,5,4,1,6,2] => [1,6,2,3,5,4] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,6,6,9}
{{1,3,4},{2,5},{6}}
=> [3,5,4,1,2,6] => [1,2,6,3,5,4] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,6,6,9}
{{1,3,4,6},{2},{5}}
=> [3,2,4,6,5,1] => [1,2,4,6,3,5] => ([(2,5),(3,4),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,6,6,9}
{{1,3,4},{2,6},{5}}
=> [3,6,4,1,5,2] => [1,5,2,3,6,4] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,6,6,9}
{{1,3,4},{2},{5,6}}
=> [3,2,4,1,6,5] => [1,6,2,4,3,5] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,6,6,9}
{{1,3,4},{2},{5},{6}}
=> [3,2,4,1,5,6] => [1,5,6,2,4,3] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,6,6,9}
Description
The second largest eigenvalue of a graph if it is integral.
This statistic is undefined if the second largest eigenvalue of the graph is not integral.
Chapter 4 of [1] provides lots of context.
Matching statistic: St001877
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00206: Posets —antichains of maximal size⟶ Lattices
St001877: Lattices ⟶ ℤResult quality: 23% ●values known / values provided: 32%●distinct values known / distinct values provided: 23%
Mp00065: Permutations —permutation poset⟶ Posets
Mp00206: Posets —antichains of maximal size⟶ Lattices
St001877: Lattices ⟶ ℤResult quality: 23% ●values known / values provided: 32%●distinct values known / distinct values provided: 23%
Values
{{1,2}}
=> [2,1] => ([],2)
=> ([],1)
=> ? ∊ {0,0}
{{1},{2}}
=> [1,2] => ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {0,0}
{{1,2,3}}
=> [2,3,1] => ([(1,2)],3)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0}
{{1,2},{3}}
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([],1)
=> ? ∊ {0,0,0,0}
{{1,3},{2}}
=> [3,2,1] => ([],3)
=> ([],1)
=> ? ∊ {0,0,0,0}
{{1},{2,3}}
=> [1,3,2] => ([(0,1),(0,2)],3)
=> ([],1)
=> ? ∊ {0,0,0,0}
{{1},{2},{3}}
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 0
{{1,2,3,4}}
=> [2,3,4,1] => ([(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 0
{{1,2,3},{4}}
=> [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0}
{{1,2,4},{3}}
=> [2,4,3,1] => ([(1,2),(1,3)],4)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0}
{{1,2},{3,4}}
=> [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0}
{{1,2},{3},{4}}
=> [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0}
{{1,3,4},{2}}
=> [3,2,4,1] => ([(1,3),(2,3)],4)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0}
{{1,3},{2,4}}
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
{{1,3},{2},{4}}
=> [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0}
{{1,4},{2,3}}
=> [4,3,2,1] => ([],4)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0}
{{1},{2,3,4}}
=> [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0}
{{1},{2,3},{4}}
=> [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0}
{{1,4},{2},{3}}
=> [4,2,3,1] => ([(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0}
{{1},{2,4},{3}}
=> [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0}
{{1},{2},{3,4}}
=> [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0}
{{1},{2},{3},{4}}
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 0
{{1,2,3,4,5}}
=> [2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 0
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 0
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => ([(1,4),(4,2),(4,3)],5)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,3,3}
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> 0
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,3,3}
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => ([(1,2),(1,3),(2,4),(3,4)],5)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,3,3}
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => ([(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,3,3}
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => ([(1,2),(1,3),(1,4)],5)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,3,3}
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,2),(2,1)],3)
=> 0
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,3,3}
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => ([(1,3),(1,4),(4,2)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,3,3}
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,3,3}
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => ([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,3,3}
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,3,3}
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => ([(1,4),(2,4),(4,3)],5)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,3,3}
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => ([(0,4),(1,2),(1,3)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,3,3}
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,3,3}
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => ([(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 0
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => ([(0,3),(1,2),(1,4),(3,4)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,3,3}
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => ([(0,3),(0,4),(1,2),(1,4)],5)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,3,3}
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,3,3}
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => ([(0,4),(1,4),(2,4),(4,3)],5)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,3,3}
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => ([(1,4),(2,4),(3,4)],5)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,3,3}
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => ([(0,4),(1,4),(2,3)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,3,3}
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,3,3}
{{1,5},{2,3,4}}
=> [5,3,4,2,1] => ([(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,3,3}
{{1},{2,3,4,5}}
=> [1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> 0
{{1},{2,3,4},{5}}
=> [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,3,3}
{{1,5},{2,3},{4}}
=> [5,3,2,4,1] => ([(2,4),(3,4)],5)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,3,3}
{{1},{2,3,5},{4}}
=> [1,3,5,4,2] => ([(0,3),(0,4),(4,1),(4,2)],5)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,3,3}
{{1},{2,3},{4,5}}
=> [1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,3,3}
{{1},{2,3},{4},{5}}
=> [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,3,3}
{{1,4,5},{2},{3}}
=> [4,2,3,5,1] => ([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,3,3}
{{1,4},{2,5},{3}}
=> [4,5,3,1,2] => ([(1,4),(2,3)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
{{1,4},{2},{3,5}}
=> [4,2,5,1,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,3,3}
{{1,4},{2},{3},{5}}
=> [4,2,3,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,3,3}
{{1,5},{2,4},{3}}
=> [5,4,3,2,1] => ([],5)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,3,3}
{{1},{2,4,5},{3}}
=> [1,4,3,5,2] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,3,3}
{{1},{2,4},{3,5}}
=> [1,4,5,2,3] => ([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
{{1},{2,4},{3},{5}}
=> [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,3,3}
{{1,5},{2},{3},{4}}
=> [5,2,3,4,1] => ([(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 0
{{1},{2},{3},{4},{5}}
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
{{1,2,3,4,5,6}}
=> [2,3,4,5,6,1] => ([(1,5),(3,4),(4,2),(5,3)],6)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
{{1,2,3,4,5},{6}}
=> [2,3,4,5,1,6] => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ([(0,3),(2,1),(3,2)],4)
=> 0
{{1,2,3,4},{5,6}}
=> [2,3,4,1,6,5] => ([(0,4),(0,5),(1,2),(2,3),(3,4),(3,5)],6)
=> ([(0,3),(2,1),(3,2)],4)
=> 0
{{1,2,3,4},{5},{6}}
=> [2,3,4,1,5,6] => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(0,2),(2,1)],3)
=> 0
{{1,2,3,5},{4,6}}
=> [2,3,5,6,1,4] => ([(0,5),(1,4),(3,2),(4,3),(4,5)],6)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 1
{{1,2,3},{4,5,6}}
=> [2,3,1,5,6,4] => ([(0,4),(0,5),(1,3),(3,4),(3,5),(5,2)],6)
=> ([(0,3),(2,1),(3,2)],4)
=> 0
{{1,2,3},{4,5},{6}}
=> [2,3,1,5,4,6] => ([(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(0,2),(2,1)],3)
=> 0
{{1,2,3},{4},{5,6}}
=> [2,3,1,4,6,5] => ([(0,5),(1,2),(2,5),(5,3),(5,4)],6)
=> ([(0,2),(2,1)],3)
=> 0
{{1,2,4,6},{3,5}}
=> [2,4,5,6,3,1] => ([(1,3),(1,5),(4,2),(5,4)],6)
=> ([(0,2),(2,1)],3)
=> 0
{{1,2,4},{3,5,6}}
=> [2,4,5,1,6,3] => ([(0,4),(0,5),(1,2),(1,4),(2,3),(3,5)],6)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 1
{{1,2,4},{3,5},{6}}
=> [2,4,5,1,3,6] => ([(0,4),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1
{{1,2},{3,4,5,6}}
=> [2,1,4,5,6,3] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(5,2)],6)
=> ([(0,3),(2,1),(3,2)],4)
=> 0
{{1,2},{3,4,5},{6}}
=> [2,1,4,5,3,6] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(4,2),(5,3)],6)
=> ([(0,2),(2,1)],3)
=> 0
{{1,2},{3,4},{5,6}}
=> [2,1,4,3,6,5] => ([(0,4),(0,5),(1,4),(1,5),(4,2),(4,3),(5,2),(5,3)],6)
=> ([(0,2),(2,1)],3)
=> 0
{{1,2,5},{3,6},{4}}
=> [2,5,6,4,1,3] => ([(0,5),(1,3),(1,4),(1,5),(4,2)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
{{1,2},{3,5},{4,6}}
=> [2,1,5,6,3,4] => ([(0,4),(0,5),(1,4),(1,5),(4,3),(5,2)],6)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1
{{1,2},{3},{4,5,6}}
=> [2,1,3,5,6,4] => ([(0,5),(1,5),(4,2),(5,3),(5,4)],6)
=> ([(0,2),(2,1)],3)
=> 0
{{1,2,6},{3},{4},{5}}
=> [2,6,3,4,5,1] => ([(1,3),(1,5),(4,2),(5,4)],6)
=> ([(0,2),(2,1)],3)
=> 0
{{1,3,4,5},{2,6}}
=> [3,6,4,5,1,2] => ([(0,4),(1,3),(1,5),(5,2)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
{{1,3,4},{2,6},{5}}
=> [3,6,4,1,5,2] => ([(0,4),(0,5),(1,2),(1,3),(3,5)],6)
=> ([(0,2),(2,1)],3)
=> 0
{{1,3,5,6},{2,4}}
=> [3,4,5,2,6,1] => ([(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,2),(2,1)],3)
=> 0
{{1,3,5},{2,4},{6}}
=> [3,4,5,2,1,6] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,2),(2,1)],3)
=> 0
{{1,3},{2,4,5,6}}
=> [3,4,1,5,6,2] => ([(0,4),(1,3),(1,5),(4,5),(5,2)],6)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 1
{{1,3},{2,4,5},{6}}
=> [3,4,1,5,2,6] => ([(0,3),(1,2),(1,4),(2,5),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1
{{1,3,6},{2,4},{5}}
=> [3,4,6,2,5,1] => ([(1,5),(2,3),(3,4),(3,5)],6)
=> ([(0,3),(2,1),(3,2)],4)
=> 0
{{1,3},{2,4},{5,6}}
=> [3,4,1,2,6,5] => ([(0,3),(1,2),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1
{{1,3},{2,4},{5},{6}}
=> [3,4,1,2,5,6] => ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
{{1,3},{2,5},{4,6}}
=> [3,5,1,6,2,4] => ([(0,3),(0,5),(1,2),(1,4),(2,5),(3,4)],6)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> 2
{{1,3,6},{2},{4},{5}}
=> [3,2,6,4,5,1] => ([(1,4),(1,5),(2,4),(2,5),(5,3)],6)
=> ([(0,2),(2,1)],3)
=> 0
{{1,4,5},{2,3,6}}
=> [4,3,6,5,1,2] => ([(0,4),(0,5),(1,4),(1,5),(2,3)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
{{1,4},{2,3,6},{5}}
=> [4,3,6,1,5,2] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5)],6)
=> ([(0,2),(2,1)],3)
=> 0
{{1,5},{2,3,4,6}}
=> [5,3,4,6,1,2] => ([(0,5),(1,3),(2,4),(4,5)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
{{1,6},{2,3,4,5}}
=> [6,3,4,5,2,1] => ([(3,4),(4,5)],6)
=> ([(0,2),(2,1)],3)
=> 0
Description
Number of indecomposable injective modules with projective dimension 2.
Matching statistic: St001604
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001604: Integer partitions ⟶ ℤResult quality: 15% ●values known / values provided: 25%●distinct values known / distinct values provided: 15%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001604: Integer partitions ⟶ ℤResult quality: 15% ●values known / values provided: 25%●distinct values known / distinct values provided: 15%
Values
{{1,2}}
=> [2]
=> []
=> ?
=> ? ∊ {0,0}
{{1},{2}}
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,0}
{{1,2,3}}
=> [3]
=> []
=> ?
=> ? ∊ {0,0,0,0,0}
{{1,2},{3}}
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0}
{{1,3},{2}}
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0}
{{1},{2,3}}
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0}
{{1},{2},{3}}
=> [1,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0}
{{1,2,3,4}}
=> [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
{{1,2,3},{4}}
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
{{1,2,4},{3}}
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
{{1,2},{3,4}}
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
{{1,2},{3},{4}}
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
{{1,3,4},{2}}
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
{{1,3},{2,4}}
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
{{1,3},{2},{4}}
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
{{1,4},{2,3}}
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
{{1},{2,3,4}}
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
{{1},{2,3},{4}}
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
{{1,4},{2},{3}}
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
{{1},{2,4},{3}}
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
{{1},{2},{3,4}}
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
{{1},{2},{3},{4}}
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
{{1,2,3,4,5}}
=> [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3}
{{1,2,3,4},{5}}
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3}
{{1,2,3,5},{4}}
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3}
{{1,2,3},{4,5}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3}
{{1,2,3},{4},{5}}
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3}
{{1,2,4,5},{3}}
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3}
{{1,2,4},{3,5}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3}
{{1,2,4},{3},{5}}
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3}
{{1,2,5},{3,4}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3}
{{1,2},{3,4,5}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3}
{{1,2},{3,4},{5}}
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3}
{{1,2,5},{3},{4}}
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3}
{{1,2},{3,5},{4}}
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3}
{{1,2},{3},{4,5}}
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3}
{{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3}
{{1,3,4,5},{2}}
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3}
{{1,3,4},{2,5}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3}
{{1,3,4},{2},{5}}
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3}
{{1,3,5},{2,4}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3}
{{1,3},{2,4,5}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3}
{{1,3},{2,4},{5}}
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3}
{{1,3,5},{2},{4}}
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3}
{{1,3},{2,5},{4}}
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3}
{{1,3},{2},{4,5}}
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3}
{{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3}
{{1,4,5},{2,3}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3}
{{1,4},{2,3,5}}
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3}
{{1,4},{2,3},{5}}
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3}
{{1},{2},{3},{4},{5}}
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
{{1,2},{3},{4},{5},{6}}
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
{{1,3},{2},{4},{5},{6}}
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
{{1},{2,3},{4},{5},{6}}
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
{{1,4},{2},{3},{5},{6}}
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
{{1},{2,4},{3},{5},{6}}
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
{{1},{2},{3,4},{5},{6}}
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
{{1,5},{2},{3},{4},{6}}
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
{{1},{2,5},{3},{4},{6}}
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
{{1},{2},{3,5},{4},{6}}
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
{{1},{2},{3},{4,5},{6}}
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
{{1,6},{2},{3},{4},{5}}
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
{{1},{2,6},{3},{4},{5}}
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
{{1},{2},{3,6},{4},{5}}
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
{{1},{2},{3},{4,6},{5}}
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
{{1},{2},{3},{4},{5,6}}
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
{{1},{2},{3},{4},{5},{6}}
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
{{1,2,3},{4},{5},{6},{7}}
=> [3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
{{1,2,4},{3},{5},{6},{7}}
=> [3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
{{1,2},{3,4},{5,6},{7}}
=> [2,2,2,1]
=> [2,2,1]
=> [2,1]
=> 0
{{1,2},{3,4},{5,7},{6}}
=> [2,2,2,1]
=> [2,2,1]
=> [2,1]
=> 0
{{1,2},{3,4},{5},{6,7}}
=> [2,2,2,1]
=> [2,2,1]
=> [2,1]
=> 0
{{1,2},{3,4},{5},{6},{7}}
=> [2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
{{1,2,5},{3},{4},{6},{7}}
=> [3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
{{1,2},{3,5},{4,6},{7}}
=> [2,2,2,1]
=> [2,2,1]
=> [2,1]
=> 0
{{1,2},{3,5},{4,7},{6}}
=> [2,2,2,1]
=> [2,2,1]
=> [2,1]
=> 0
{{1,2},{3,5},{4},{6,7}}
=> [2,2,2,1]
=> [2,2,1]
=> [2,1]
=> 0
{{1,2},{3,5},{4},{6},{7}}
=> [2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
{{1,2},{3,6},{4,5},{7}}
=> [2,2,2,1]
=> [2,2,1]
=> [2,1]
=> 0
{{1,2},{3,7},{4,5},{6}}
=> [2,2,2,1]
=> [2,2,1]
=> [2,1]
=> 0
{{1,2},{3},{4,5},{6,7}}
=> [2,2,2,1]
=> [2,2,1]
=> [2,1]
=> 0
{{1,2},{3},{4,5},{6},{7}}
=> [2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
{{1,2,6},{3},{4},{5},{7}}
=> [3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
{{1,2},{3,6},{4,7},{5}}
=> [2,2,2,1]
=> [2,2,1]
=> [2,1]
=> 0
{{1,2},{3,6},{4},{5,7}}
=> [2,2,2,1]
=> [2,2,1]
=> [2,1]
=> 0
{{1,2},{3,6},{4},{5},{7}}
=> [2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
{{1,2},{3,7},{4,6},{5}}
=> [2,2,2,1]
=> [2,2,1]
=> [2,1]
=> 0
{{1,2},{3},{4,6},{5,7}}
=> [2,2,2,1]
=> [2,2,1]
=> [2,1]
=> 0
{{1,2},{3},{4,6},{5},{7}}
=> [2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
{{1,2},{3,7},{4},{5,6}}
=> [2,2,2,1]
=> [2,2,1]
=> [2,1]
=> 0
{{1,2},{3},{4,7},{5,6}}
=> [2,2,2,1]
=> [2,2,1]
=> [2,1]
=> 0
{{1,2},{3},{4},{5,6},{7}}
=> [2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
{{1,2,7},{3},{4},{5},{6}}
=> [3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
{{1,2},{3,7},{4},{5},{6}}
=> [2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
{{1,2},{3},{4,7},{5},{6}}
=> [2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
{{1,2},{3},{4},{5,7},{6}}
=> [2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
{{1,2},{3},{4},{5},{6,7}}
=> [2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
{{1,2},{3},{4},{5},{6},{7}}
=> [2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
{{1,3,4},{2},{5},{6},{7}}
=> [3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
{{1,3},{2,4},{5,6},{7}}
=> [2,2,2,1]
=> [2,2,1]
=> [2,1]
=> 0
Description
The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons.
Equivalently, this is the multiplicity of the irreducible representation corresponding to a partition in the cycle index of the dihedral group.
This statistic is only defined for partitions of size at least 3, to avoid ambiguity.
Matching statistic: St000741
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000741: Graphs ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 15%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000741: Graphs ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 15%
Values
{{1,2}}
=> [2,1] => [1,2] => ([],2)
=> 1 = 0 + 1
{{1},{2}}
=> [1,2] => [1,2] => ([],2)
=> 1 = 0 + 1
{{1,2,3}}
=> [2,3,1] => [1,2,3] => ([],3)
=> 1 = 0 + 1
{{1,2},{3}}
=> [2,1,3] => [1,2,3] => ([],3)
=> 1 = 0 + 1
{{1,3},{2}}
=> [3,2,1] => [1,3,2] => ([(1,2)],3)
=> ? = 0 + 1
{{1},{2,3}}
=> [1,3,2] => [1,2,3] => ([],3)
=> 1 = 0 + 1
{{1},{2},{3}}
=> [1,2,3] => [1,2,3] => ([],3)
=> 1 = 0 + 1
{{1,2,3,4}}
=> [2,3,4,1] => [1,2,3,4] => ([],4)
=> 1 = 0 + 1
{{1,2,3},{4}}
=> [2,3,1,4] => [1,2,3,4] => ([],4)
=> 1 = 0 + 1
{{1,2,4},{3}}
=> [2,4,3,1] => [1,2,4,3] => ([(2,3)],4)
=> ? ∊ {0,0,0,1} + 1
{{1,2},{3,4}}
=> [2,1,4,3] => [1,2,3,4] => ([],4)
=> 1 = 0 + 1
{{1,2},{3},{4}}
=> [2,1,3,4] => [1,2,3,4] => ([],4)
=> 1 = 0 + 1
{{1,3,4},{2}}
=> [3,2,4,1] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> 1 = 0 + 1
{{1,3},{2,4}}
=> [3,4,1,2] => [1,3,2,4] => ([(2,3)],4)
=> ? ∊ {0,0,0,1} + 1
{{1,3},{2},{4}}
=> [3,2,1,4] => [1,3,2,4] => ([(2,3)],4)
=> ? ∊ {0,0,0,1} + 1
{{1,4},{2,3}}
=> [4,3,2,1] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 1 = 0 + 1
{{1},{2,3,4}}
=> [1,3,4,2] => [1,2,3,4] => ([],4)
=> 1 = 0 + 1
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,2,3,4] => ([],4)
=> 1 = 0 + 1
{{1,4},{2},{3}}
=> [4,2,3,1] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 1 = 0 + 1
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,2,4,3] => ([(2,3)],4)
=> ? ∊ {0,0,0,1} + 1
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,3,4] => ([],4)
=> 1 = 0 + 1
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => ([],4)
=> 1 = 0 + 1
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [1,2,3,4,5] => ([],5)
=> 1 = 0 + 1
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [1,2,3,4,5] => ([],5)
=> 1 = 0 + 1
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [1,2,3,5,4] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,3,3} + 1
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [1,2,3,4,5] => ([],5)
=> 1 = 0 + 1
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [1,2,3,4,5] => ([],5)
=> 1 = 0 + 1
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> 1 = 0 + 1
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [1,2,4,3,5] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,3,3} + 1
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [1,2,4,3,5] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,3,3} + 1
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> 1 = 0 + 1
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [1,2,3,4,5] => ([],5)
=> 1 = 0 + 1
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [1,2,3,4,5] => ([],5)
=> 1 = 0 + 1
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> 1 = 0 + 1
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [1,2,3,5,4] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,3,3} + 1
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [1,2,3,4,5] => ([],5)
=> 1 = 0 + 1
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [1,2,3,4,5] => ([],5)
=> 1 = 0 + 1
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,3,3} + 1
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> 1 = 0 + 1
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> 1 = 0 + 1
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> 1 = 0 + 1
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [1,3,2,4,5] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,3,3} + 1
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [1,3,2,4,5] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,3,3} + 1
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> 1 = 0 + 1
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,3,3} + 1
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [1,3,2,4,5] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,3,3} + 1
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [1,3,2,4,5] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,3,3} + 1
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,3,3} + 1
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> 1 = 0 + 1
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> 1 = 0 + 1
{{1,5},{2,3,4}}
=> [5,3,4,2,1] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,3,3} + 1
{{1},{2,3,4,5}}
=> [1,3,4,5,2] => [1,2,3,4,5] => ([],5)
=> 1 = 0 + 1
{{1},{2,3,4},{5}}
=> [1,3,4,2,5] => [1,2,3,4,5] => ([],5)
=> 1 = 0 + 1
{{1,5},{2,3},{4}}
=> [5,3,2,4,1] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,3,3} + 1
{{1},{2,3,5},{4}}
=> [1,3,5,4,2] => [1,2,3,5,4] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,3,3} + 1
{{1},{2,3},{4,5}}
=> [1,3,2,5,4] => [1,2,3,4,5] => ([],5)
=> 1 = 0 + 1
{{1},{2,3},{4},{5}}
=> [1,3,2,4,5] => [1,2,3,4,5] => ([],5)
=> 1 = 0 + 1
{{1,4,5},{2},{3}}
=> [4,2,3,5,1] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,3,3} + 1
{{1,4},{2,5},{3}}
=> [4,5,3,1,2] => [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> 1 = 0 + 1
{{1,4},{2},{3,5}}
=> [4,2,5,1,3] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> 1 = 0 + 1
{{1,4},{2},{3},{5}}
=> [4,2,3,1,5] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> 1 = 0 + 1
{{1,5},{2,4},{3}}
=> [5,4,3,2,1] => [1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
{{1},{2,4,5},{3}}
=> [1,4,3,5,2] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> 1 = 0 + 1
{{1},{2,4},{3,5}}
=> [1,4,5,2,3] => [1,2,4,3,5] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,3,3} + 1
{{1},{2,4},{3},{5}}
=> [1,4,3,2,5] => [1,2,4,3,5] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,3,3} + 1
{{1,5},{2},{3,4}}
=> [5,2,4,3,1] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,3,3} + 1
{{1},{2,5},{3,4}}
=> [1,5,4,3,2] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> 1 = 0 + 1
{{1},{2},{3,4,5}}
=> [1,2,4,5,3] => [1,2,3,4,5] => ([],5)
=> 1 = 0 + 1
{{1},{2},{3,4},{5}}
=> [1,2,4,3,5] => [1,2,3,4,5] => ([],5)
=> 1 = 0 + 1
{{1,5},{2},{3},{4}}
=> [5,2,3,4,1] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,3,3} + 1
{{1},{2,5},{3},{4}}
=> [1,5,3,4,2] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> 1 = 0 + 1
{{1},{2},{3,5},{4}}
=> [1,2,5,4,3] => [1,2,3,5,4] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,3,3} + 1
{{1},{2},{3},{4,5}}
=> [1,2,3,5,4] => [1,2,3,4,5] => ([],5)
=> 1 = 0 + 1
{{1},{2},{3},{4},{5}}
=> [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> 1 = 0 + 1
{{1,2,3,4,5,6}}
=> [2,3,4,5,6,1] => [1,2,3,4,5,6] => ([],6)
=> 1 = 0 + 1
{{1,2,3,4,6},{5}}
=> [2,3,4,6,5,1] => [1,2,3,4,6,5] => ([(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,6,6,9} + 1
{{1,2,3,5},{4,6}}
=> [2,3,5,6,1,4] => [1,2,3,5,4,6] => ([(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,6,6,9} + 1
{{1,2,3,5},{4},{6}}
=> [2,3,5,4,1,6] => [1,2,3,5,4,6] => ([(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,6,6,9} + 1
{{1,2,3},{4,6},{5}}
=> [2,3,1,6,5,4] => [1,2,3,4,6,5] => ([(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,6,6,9} + 1
{{1,2,4,5,6},{3}}
=> [2,4,3,5,6,1] => [1,2,4,5,6,3] => ([(2,5),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,6,6,9} + 1
{{1,2,4},{3,5,6}}
=> [2,4,5,1,6,3] => [1,2,4,3,5,6] => ([(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,6,6,9} + 1
{{1,2,4},{3,5},{6}}
=> [2,4,5,1,3,6] => [1,2,4,3,5,6] => ([(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,6,6,9} + 1
{{1,2,4},{3,6},{5}}
=> [2,4,6,1,5,3] => [1,2,4,3,6,5] => ([(2,5),(3,4)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,6,6,9} + 1
{{1,2,4},{3},{5,6}}
=> [2,4,3,1,6,5] => [1,2,4,3,5,6] => ([(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,6,6,9} + 1
{{1,2,4},{3},{5},{6}}
=> [2,4,3,1,5,6] => [1,2,4,3,5,6] => ([(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,6,6,9} + 1
{{1,2,5,6},{3,4}}
=> [2,5,4,3,6,1] => [1,2,5,6,3,4] => ([(2,4),(2,5),(3,4),(3,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,6,6,9} + 1
{{1,2,6},{3,4,5}}
=> [2,6,4,5,3,1] => [1,2,6,3,4,5] => ([(2,5),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,6,6,9} + 1
{{1,2,6},{3,4},{5}}
=> [2,6,4,3,5,1] => [1,2,6,3,4,5] => ([(2,5),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,6,6,9} + 1
{{1,2},{3,4,6},{5}}
=> [2,1,4,6,5,3] => [1,2,3,4,6,5] => ([(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,6,6,9} + 1
{{1,2,5,6},{3},{4}}
=> [2,5,3,4,6,1] => [1,2,5,6,3,4] => ([(2,4),(2,5),(3,4),(3,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,6,6,9} + 1
{{1,2},{3,5},{4,6}}
=> [2,1,5,6,3,4] => [1,2,3,5,4,6] => ([(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,6,6,9} + 1
{{1,2},{3,5},{4},{6}}
=> [2,1,5,4,3,6] => [1,2,3,5,4,6] => ([(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,6,6,9} + 1
{{1,2,6},{3},{4,5}}
=> [2,6,3,5,4,1] => [1,2,6,3,4,5] => ([(2,5),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,6,6,9} + 1
{{1,2,6},{3},{4},{5}}
=> [2,6,3,4,5,1] => [1,2,6,3,4,5] => ([(2,5),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,6,6,9} + 1
{{1,2},{3},{4,6},{5}}
=> [2,1,3,6,5,4] => [1,2,3,4,6,5] => ([(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,6,6,9} + 1
{{1,3,4,5,6},{2}}
=> [3,2,4,5,6,1] => [1,3,4,5,6,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,6,6,9} + 1
{{1,3,4,5},{2,6}}
=> [3,6,4,5,1,2] => [1,3,4,5,2,6] => ([(2,5),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,6,6,9} + 1
{{1,3,4,5},{2},{6}}
=> [3,2,4,5,1,6] => [1,3,4,5,2,6] => ([(2,5),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,6,6,9} + 1
{{1,3,4,6},{2,5}}
=> [3,5,4,6,2,1] => [1,3,4,6,2,5] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,6,6,9} + 1
{{1,3,4,6},{2},{5}}
=> [3,2,4,6,5,1] => [1,3,4,6,2,5] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,6,6,9} + 1
Description
The Colin de Verdière graph invariant.
Matching statistic: St001719
(load all 18 compositions to match this statistic)
(load all 18 compositions to match this statistic)
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
St001719: Lattices ⟶ ℤResult quality: 8% ●values known / values provided: 13%●distinct values known / distinct values provided: 8%
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
St001719: Lattices ⟶ ℤResult quality: 8% ●values known / values provided: 13%●distinct values known / distinct values provided: 8%
Values
{{1,2}}
=> [2,1] => [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
{{1},{2}}
=> [1,2] => [1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
{{1,2,3}}
=> [2,3,1] => [3,1,2] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> 1 = 0 + 1
{{1,2},{3}}
=> [2,1,3] => [2,1,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> 1 = 0 + 1
{{1,3},{2}}
=> [3,2,1] => [2,3,1] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> 1 = 0 + 1
{{1},{2,3}}
=> [1,3,2] => [1,3,2] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> 1 = 0 + 1
{{1},{2},{3}}
=> [1,2,3] => [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> 1 = 0 + 1
{{1,2,3,4}}
=> [2,3,4,1] => [4,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> 1 = 0 + 1
{{1,2,3},{4}}
=> [2,3,1,4] => [3,1,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> 1 = 0 + 1
{{1,2,4},{3}}
=> [2,4,3,1] => [3,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> 1 = 0 + 1
{{1,2},{3,4}}
=> [2,1,4,3] => [2,1,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> 1 = 0 + 1
{{1,2},{3},{4}}
=> [2,1,3,4] => [2,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> 1 = 0 + 1
{{1,3,4},{2}}
=> [3,2,4,1] => [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 1 = 0 + 1
{{1,3},{2,4}}
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 1 = 0 + 1
{{1,3},{2},{4}}
=> [3,2,1,4] => [2,3,1,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> 1 = 0 + 1
{{1,4},{2,3}}
=> [4,3,2,1] => [3,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> 1 = 0 + 1
{{1},{2,3,4}}
=> [1,3,4,2] => [1,4,2,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> 1 = 0 + 1
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,7),(4,6),(5,6),(5,7),(6,8),(7,8)],9)
=> 1 = 0 + 1
{{1,4},{2},{3}}
=> [4,2,3,1] => [2,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> 1 = 0 + 1
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,3,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> 1 = 0 + 1
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> 1 = 0 + 1
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 1 + 1
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [5,1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3} + 1
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [4,1,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3} + 1
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [4,5,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3} + 1
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [3,1,2,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(8,9)],10)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3} + 1
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [3,1,2,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,7),(4,7),(5,6),(5,9),(6,10),(7,8),(8,9),(9,10)],11)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3} + 1
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [3,5,1,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> 1 = 0 + 1
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [4,1,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> 1 = 0 + 1
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [3,4,1,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,7),(4,6),(5,6),(6,9),(7,9),(9,8)],10)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3} + 1
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [4,3,5,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(8,9)],10)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3} + 1
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [2,1,5,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(8,9)],10)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3} + 1
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [2,1,4,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(7,9),(8,10),(9,10)],11)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3} + 1
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [3,4,5,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3} + 1
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [2,1,4,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(8,9)],10)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3} + 1
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [2,1,3,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(1,9),(2,7),(3,7),(4,6),(5,6),(6,9),(7,8),(8,10),(9,10)],11)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3} + 1
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [2,1,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,11),(3,10),(4,9),(4,12),(5,10),(5,12),(7,6),(8,6),(9,7),(10,8),(11,9),(12,7),(12,8)],13)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3} + 1
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [2,5,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> 1 = 0 + 1
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [4,1,3,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 1 = 0 + 1
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [2,4,1,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
=> 1 = 0 + 1
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [4,2,5,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 1 = 0 + 1
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [3,1,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 1 = 0 + 1
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [3,1,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
=> 1 = 0 + 1
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [2,4,5,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> 1 = 0 + 1
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [3,1,4,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> 1 = 0 + 1
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [2,3,1,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(8,9)],10)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3} + 1
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [2,3,1,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,7),(4,7),(5,6),(5,9),(6,10),(7,8),(8,9),(9,10)],11)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3} + 1
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [3,2,5,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> 1 = 0 + 1
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [4,1,5,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> 1 = 0 + 1
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => [3,2,4,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,9),(5,7),(6,9),(8,7),(9,8)],10)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3} + 1
{{1,5},{2,3,4}}
=> [5,3,4,2,1] => [4,2,3,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,9),(5,7),(6,9),(8,7),(9,8)],10)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3} + 1
{{1},{2,3,4,5}}
=> [1,3,4,5,2] => [1,5,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3} + 1
{{1},{2,3,4},{5}}
=> [1,3,4,2,5] => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3} + 1
{{1,5},{2,3},{4}}
=> [5,3,2,4,1] => [3,2,4,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,6),(3,7),(4,7),(5,6),(5,8),(6,10),(7,8),(8,10),(10,9)],11)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3} + 1
{{1},{2,3,5},{4}}
=> [1,3,5,4,2] => [1,4,5,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,7),(4,6),(5,6),(6,9),(7,9),(9,8)],10)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3} + 1
{{1},{2,3},{4,5}}
=> [1,3,2,5,4] => [1,3,2,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(7,9),(8,10),(9,10)],11)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3} + 1
{{1},{2,3},{4},{5}}
=> [1,3,2,4,5] => [1,3,2,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,12),(3,12),(4,9),(5,10),(5,11),(7,6),(8,6),(9,8),(10,7),(11,7),(11,8),(12,9),(12,11)],13)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3} + 1
{{1,4,5},{2},{3}}
=> [4,2,3,5,1] => [2,3,5,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> 1 = 0 + 1
{{1,4},{2,5},{3}}
=> [4,5,3,1,2] => [3,4,1,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> 1 = 0 + 1
{{1,4},{2},{3,5}}
=> [4,2,5,1,3] => [2,4,1,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 1 = 0 + 1
{{1,4},{2},{3},{5}}
=> [4,2,3,1,5] => [2,3,4,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3} + 1
{{1,5},{2,4},{3}}
=> [5,4,3,2,1] => [3,4,2,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,9),(5,7),(6,9),(8,7),(9,8)],10)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3} + 1
{{1},{2,4,5},{3}}
=> [1,4,3,5,2] => [1,3,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
=> 1 = 0 + 1
{{1},{2,4},{3,5}}
=> [1,4,5,2,3] => [1,4,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
=> 1 = 0 + 1
{{1},{2,4},{3},{5}}
=> [1,4,3,2,5] => [1,3,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3} + 1
{{1,5},{2},{3,4}}
=> [5,2,4,3,1] => [2,4,3,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,9),(4,8),(5,7),(6,8),(6,9),(8,10),(9,10),(10,7)],11)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3} + 1
{{1},{2,5},{3,4}}
=> [1,5,4,3,2] => [1,4,3,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,9),(5,7),(6,9),(8,7),(9,8)],10)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3} + 1
{{1},{2},{3,4,5}}
=> [1,2,4,5,3] => [1,2,5,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,7),(4,7),(5,6),(5,9),(6,10),(7,8),(8,9),(9,10)],11)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3} + 1
{{1},{2},{3,4},{5}}
=> [1,2,4,3,5] => [1,2,4,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,12),(3,12),(4,9),(5,10),(5,11),(7,6),(8,6),(9,8),(10,7),(11,7),(11,8),(12,9),(12,11)],13)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3} + 1
{{1,5},{2},{3},{4}}
=> [5,2,3,4,1] => [2,3,4,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3} + 1
{{1},{2,5},{3},{4}}
=> [1,5,3,4,2] => [1,3,4,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3} + 1
{{1},{2},{3,5},{4}}
=> [1,2,5,4,3] => [1,2,4,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,7),(4,7),(5,6),(5,9),(6,10),(7,8),(8,9),(9,10)],11)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3} + 1
{{1},{2},{3},{4,5}}
=> [1,2,3,5,4] => [1,2,3,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,11),(3,10),(4,9),(4,12),(5,10),(5,12),(7,6),(8,6),(9,7),(10,8),(11,9),(12,7),(12,8)],13)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3} + 1
{{1},{2},{3},{4},{5}}
=> [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3} + 1
{{1,2,3,4,5,6}}
=> [2,3,4,5,6,1] => [6,1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(2,13),(3,7),(4,13),(4,16),(5,14),(5,17),(6,16),(6,17),(8,12),(9,12),(10,8),(11,9),(12,7),(13,10),(14,11),(15,8),(15,9),(16,10),(16,15),(17,11),(17,15)],18)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,6,6,9} + 1
{{1,2,3,4,5},{6}}
=> [2,3,4,5,1,6] => [5,1,2,3,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,13),(3,12),(4,7),(5,12),(5,14),(6,13),(6,14),(8,11),(9,8),(10,8),(11,7),(12,9),(13,10),(14,9),(14,10)],15)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,6,6,9} + 1
{{1,2,3,4,6},{5}}
=> [2,3,4,6,5,1] => [5,6,1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(2,13),(3,12),(4,11),(5,11),(5,14),(6,12),(6,14),(8,10),(9,10),(10,7),(11,8),(12,9),(13,7),(14,8),(14,9)],15)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,6,6,9} + 1
{{1,2,3,4},{5,6}}
=> [2,3,4,1,6,5] => [4,1,2,3,6,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,8),(4,7),(5,7),(6,8),(6,9),(7,12),(8,11),(9,11),(10,12),(11,10)],13)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,6,6,9} + 1
{{1,2,3,4},{5},{6}}
=> [2,3,4,1,5,6] => [4,1,2,3,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,12),(2,11),(3,13),(4,10),(5,11),(5,12),(6,9),(6,13),(8,10),(9,7),(10,9),(11,8),(12,8),(13,7)],14)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,6,6,9} + 1
{{1,2,3,5,6},{4}}
=> [2,3,5,4,6,1] => [4,6,1,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,10),(3,10),(4,8),(5,7),(6,7),(6,8),(7,9),(8,9),(9,10)],11)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,6,6,9} + 1
{{1,2,3,5},{4,6}}
=> [2,3,5,6,1,4] => [5,1,2,3,6,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,10),(3,10),(4,8),(5,7),(6,7),(6,8),(7,9),(8,9),(9,10)],11)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,6,6,9} + 1
{{1,2,3,5},{4},{6}}
=> [2,3,5,4,1,6] => [4,5,1,2,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,8),(4,7),(5,7),(6,8),(6,9),(7,12),(8,11),(9,11),(11,12),(12,10)],13)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,6,6,9} + 1
{{1,2,3,6},{4,5}}
=> [2,3,6,5,4,1] => [5,4,6,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,8),(3,7),(4,9),(5,9),(6,7),(6,8),(7,11),(8,11),(9,10),(10,12),(11,12)],13)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,6,6,9} + 1
{{1,2,3},{4,5,6}}
=> [2,3,1,5,6,4] => [3,1,2,6,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,7),(4,7),(5,8),(6,8),(7,9),(8,10),(9,11),(10,11)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,6,6,9} + 1
{{1,2,3},{4,5},{6}}
=> [2,3,1,5,4,6] => [3,1,2,5,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,10),(3,7),(4,7),(5,8),(6,8),(7,10),(8,9),(8,11),(9,12),(10,11),(11,12)],13)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,6,6,9} + 1
{{1,2,3,6},{4},{5}}
=> [2,3,6,4,5,1] => [4,5,6,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,10),(3,13),(4,12),(5,12),(5,13),(6,10),(6,11),(8,7),(9,7),(10,8),(11,8),(12,9),(13,9)],14)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,6,6,9} + 1
{{1,2,3},{4,6},{5}}
=> [2,3,1,6,5,4] => [3,1,2,5,6,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,7),(4,7),(5,8),(6,8),(7,9),(8,10),(9,11),(10,11)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,6,6,9} + 1
{{1,2,3},{4},{5,6}}
=> [2,3,1,4,6,5] => [3,1,2,4,6,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,8),(4,8),(5,11),(6,9),(6,10),(7,10),(8,11),(9,12),(10,12),(11,9)],13)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,6,6,9} + 1
{{1,2,4,5},{3,6}}
=> [2,4,6,5,1,3] => [5,1,2,4,6,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 0 + 1
{{1,2,4,6},{3,5}}
=> [2,4,5,6,3,1] => [5,3,6,1,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 0 + 1
{{1,2,4},{3,5,6}}
=> [2,4,5,1,6,3] => [4,1,2,6,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 0 + 1
{{1,2,5},{3},{4,6}}
=> [2,5,3,6,1,4] => [3,5,1,2,6,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 0 + 1
{{1,3,4,5},{2,6}}
=> [3,6,4,5,1,2] => [5,1,3,4,6,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 0 + 1
{{1,3,4,6},{2,5}}
=> [3,5,4,6,2,1] => [5,2,6,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 0 + 1
{{1,3,4},{2,5,6}}
=> [3,5,4,1,6,2] => [4,1,3,6,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 1 = 0 + 1
{{1,3,4},{2,5},{6}}
=> [3,5,4,1,2,6] => [4,1,3,5,2,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 1 = 0 + 1
{{1,3,4},{2,6},{5}}
=> [3,6,4,1,5,2] => [4,1,3,5,6,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1 = 0 + 1
{{1,3,5,6},{2,4}}
=> [3,4,5,2,6,1] => [4,2,6,1,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 1 = 0 + 1
{{1,3,5},{2,4,6}}
=> [3,4,5,6,1,2] => [5,1,3,6,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 1 = 0 + 1
{{1,3,5},{2,4},{6}}
=> [3,4,5,2,1,6] => [4,2,5,1,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 1 = 0 + 1
Description
The number of shortest chains of small intervals from the bottom to the top in a lattice.
An interval $[a, b]$ in a lattice is small if $b$ is a join of elements covering $a$.
Matching statistic: St001720
(load all 18 compositions to match this statistic)
(load all 18 compositions to match this statistic)
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
St001720: Lattices ⟶ ℤResult quality: 8% ●values known / values provided: 13%●distinct values known / distinct values provided: 8%
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
St001720: Lattices ⟶ ℤResult quality: 8% ●values known / values provided: 13%●distinct values known / distinct values provided: 8%
Values
{{1,2}}
=> [2,1] => [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
{{1},{2}}
=> [1,2] => [1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
{{1,2,3}}
=> [2,3,1] => [3,1,2] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> 2 = 0 + 2
{{1,2},{3}}
=> [2,1,3] => [2,1,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> 2 = 0 + 2
{{1,3},{2}}
=> [3,2,1] => [2,3,1] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> 2 = 0 + 2
{{1},{2,3}}
=> [1,3,2] => [1,3,2] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> 2 = 0 + 2
{{1},{2},{3}}
=> [1,2,3] => [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> 2 = 0 + 2
{{1,2,3,4}}
=> [2,3,4,1] => [4,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> 2 = 0 + 2
{{1,2,3},{4}}
=> [2,3,1,4] => [3,1,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> 2 = 0 + 2
{{1,2,4},{3}}
=> [2,4,3,1] => [3,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> 2 = 0 + 2
{{1,2},{3,4}}
=> [2,1,4,3] => [2,1,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> 2 = 0 + 2
{{1,2},{3},{4}}
=> [2,1,3,4] => [2,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> 2 = 0 + 2
{{1,3,4},{2}}
=> [3,2,4,1] => [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 0 + 2
{{1,3},{2,4}}
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 0 + 2
{{1,3},{2},{4}}
=> [3,2,1,4] => [2,3,1,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> 2 = 0 + 2
{{1,4},{2,3}}
=> [4,3,2,1] => [3,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> 2 = 0 + 2
{{1},{2,3,4}}
=> [1,3,4,2] => [1,4,2,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> 2 = 0 + 2
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,7),(4,6),(5,6),(5,7),(6,8),(7,8)],9)
=> 2 = 0 + 2
{{1,4},{2},{3}}
=> [4,2,3,1] => [2,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> 2 = 0 + 2
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,3,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> 2 = 0 + 2
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> 2 = 0 + 2
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 1 + 2
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [5,1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3} + 2
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [4,1,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3} + 2
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [4,5,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3} + 2
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [3,1,2,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(8,9)],10)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3} + 2
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [3,1,2,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,7),(4,7),(5,6),(5,9),(6,10),(7,8),(8,9),(9,10)],11)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3} + 2
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [3,5,1,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> 2 = 0 + 2
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [4,1,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> 2 = 0 + 2
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [3,4,1,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,7),(4,6),(5,6),(6,9),(7,9),(9,8)],10)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3} + 2
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [4,3,5,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(8,9)],10)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3} + 2
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [2,1,5,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(8,9)],10)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3} + 2
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [2,1,4,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(7,9),(8,10),(9,10)],11)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3} + 2
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [3,4,5,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3} + 2
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [2,1,4,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(8,9)],10)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3} + 2
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [2,1,3,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(1,9),(2,7),(3,7),(4,6),(5,6),(6,9),(7,8),(8,10),(9,10)],11)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3} + 2
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [2,1,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,11),(3,10),(4,9),(4,12),(5,10),(5,12),(7,6),(8,6),(9,7),(10,8),(11,9),(12,7),(12,8)],13)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3} + 2
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [2,5,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> 2 = 0 + 2
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [4,1,3,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 2 = 0 + 2
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [2,4,1,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
=> 2 = 0 + 2
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [4,2,5,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 2 = 0 + 2
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [3,1,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 2 = 0 + 2
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [3,1,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
=> 2 = 0 + 2
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [2,4,5,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> 2 = 0 + 2
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [3,1,4,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> 2 = 0 + 2
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [2,3,1,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(8,9)],10)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3} + 2
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [2,3,1,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,7),(4,7),(5,6),(5,9),(6,10),(7,8),(8,9),(9,10)],11)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3} + 2
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [3,2,5,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> 2 = 0 + 2
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [4,1,5,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> 2 = 0 + 2
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => [3,2,4,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,9),(5,7),(6,9),(8,7),(9,8)],10)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3} + 2
{{1,5},{2,3,4}}
=> [5,3,4,2,1] => [4,2,3,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,9),(5,7),(6,9),(8,7),(9,8)],10)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3} + 2
{{1},{2,3,4,5}}
=> [1,3,4,5,2] => [1,5,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3} + 2
{{1},{2,3,4},{5}}
=> [1,3,4,2,5] => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3} + 2
{{1,5},{2,3},{4}}
=> [5,3,2,4,1] => [3,2,4,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,6),(3,7),(4,7),(5,6),(5,8),(6,10),(7,8),(8,10),(10,9)],11)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3} + 2
{{1},{2,3,5},{4}}
=> [1,3,5,4,2] => [1,4,5,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,7),(4,6),(5,6),(6,9),(7,9),(9,8)],10)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3} + 2
{{1},{2,3},{4,5}}
=> [1,3,2,5,4] => [1,3,2,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(7,9),(8,10),(9,10)],11)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3} + 2
{{1},{2,3},{4},{5}}
=> [1,3,2,4,5] => [1,3,2,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,12),(3,12),(4,9),(5,10),(5,11),(7,6),(8,6),(9,8),(10,7),(11,7),(11,8),(12,9),(12,11)],13)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3} + 2
{{1,4,5},{2},{3}}
=> [4,2,3,5,1] => [2,3,5,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> 2 = 0 + 2
{{1,4},{2,5},{3}}
=> [4,5,3,1,2] => [3,4,1,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> 2 = 0 + 2
{{1,4},{2},{3,5}}
=> [4,2,5,1,3] => [2,4,1,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 2 = 0 + 2
{{1,4},{2},{3},{5}}
=> [4,2,3,1,5] => [2,3,4,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3} + 2
{{1,5},{2,4},{3}}
=> [5,4,3,2,1] => [3,4,2,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,9),(5,7),(6,9),(8,7),(9,8)],10)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3} + 2
{{1},{2,4,5},{3}}
=> [1,4,3,5,2] => [1,3,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
=> 2 = 0 + 2
{{1},{2,4},{3,5}}
=> [1,4,5,2,3] => [1,4,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
=> 2 = 0 + 2
{{1},{2,4},{3},{5}}
=> [1,4,3,2,5] => [1,3,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3} + 2
{{1,5},{2},{3,4}}
=> [5,2,4,3,1] => [2,4,3,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,9),(4,8),(5,7),(6,8),(6,9),(8,10),(9,10),(10,7)],11)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3} + 2
{{1},{2,5},{3,4}}
=> [1,5,4,3,2] => [1,4,3,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,9),(5,7),(6,9),(8,7),(9,8)],10)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3} + 2
{{1},{2},{3,4,5}}
=> [1,2,4,5,3] => [1,2,5,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,7),(4,7),(5,6),(5,9),(6,10),(7,8),(8,9),(9,10)],11)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3} + 2
{{1},{2},{3,4},{5}}
=> [1,2,4,3,5] => [1,2,4,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,12),(3,12),(4,9),(5,10),(5,11),(7,6),(8,6),(9,8),(10,7),(11,7),(11,8),(12,9),(12,11)],13)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3} + 2
{{1,5},{2},{3},{4}}
=> [5,2,3,4,1] => [2,3,4,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3} + 2
{{1},{2,5},{3},{4}}
=> [1,5,3,4,2] => [1,3,4,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3} + 2
{{1},{2},{3,5},{4}}
=> [1,2,5,4,3] => [1,2,4,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,7),(4,7),(5,6),(5,9),(6,10),(7,8),(8,9),(9,10)],11)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3} + 2
{{1},{2},{3},{4,5}}
=> [1,2,3,5,4] => [1,2,3,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,11),(3,10),(4,9),(4,12),(5,10),(5,12),(7,6),(8,6),(9,7),(10,8),(11,9),(12,7),(12,8)],13)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3} + 2
{{1},{2},{3},{4},{5}}
=> [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3} + 2
{{1,2,3,4,5,6}}
=> [2,3,4,5,6,1] => [6,1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(2,13),(3,7),(4,13),(4,16),(5,14),(5,17),(6,16),(6,17),(8,12),(9,12),(10,8),(11,9),(12,7),(13,10),(14,11),(15,8),(15,9),(16,10),(16,15),(17,11),(17,15)],18)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,6,6,9} + 2
{{1,2,3,4,5},{6}}
=> [2,3,4,5,1,6] => [5,1,2,3,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,13),(3,12),(4,7),(5,12),(5,14),(6,13),(6,14),(8,11),(9,8),(10,8),(11,7),(12,9),(13,10),(14,9),(14,10)],15)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,6,6,9} + 2
{{1,2,3,4,6},{5}}
=> [2,3,4,6,5,1] => [5,6,1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(2,13),(3,12),(4,11),(5,11),(5,14),(6,12),(6,14),(8,10),(9,10),(10,7),(11,8),(12,9),(13,7),(14,8),(14,9)],15)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,6,6,9} + 2
{{1,2,3,4},{5,6}}
=> [2,3,4,1,6,5] => [4,1,2,3,6,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,8),(4,7),(5,7),(6,8),(6,9),(7,12),(8,11),(9,11),(10,12),(11,10)],13)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,6,6,9} + 2
{{1,2,3,4},{5},{6}}
=> [2,3,4,1,5,6] => [4,1,2,3,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,12),(2,11),(3,13),(4,10),(5,11),(5,12),(6,9),(6,13),(8,10),(9,7),(10,9),(11,8),(12,8),(13,7)],14)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,6,6,9} + 2
{{1,2,3,5,6},{4}}
=> [2,3,5,4,6,1] => [4,6,1,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,10),(3,10),(4,8),(5,7),(6,7),(6,8),(7,9),(8,9),(9,10)],11)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,6,6,9} + 2
{{1,2,3,5},{4,6}}
=> [2,3,5,6,1,4] => [5,1,2,3,6,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,10),(3,10),(4,8),(5,7),(6,7),(6,8),(7,9),(8,9),(9,10)],11)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,6,6,9} + 2
{{1,2,3,5},{4},{6}}
=> [2,3,5,4,1,6] => [4,5,1,2,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,8),(4,7),(5,7),(6,8),(6,9),(7,12),(8,11),(9,11),(11,12),(12,10)],13)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,6,6,9} + 2
{{1,2,3,6},{4,5}}
=> [2,3,6,5,4,1] => [5,4,6,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,8),(3,7),(4,9),(5,9),(6,7),(6,8),(7,11),(8,11),(9,10),(10,12),(11,12)],13)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,6,6,9} + 2
{{1,2,3},{4,5,6}}
=> [2,3,1,5,6,4] => [3,1,2,6,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,7),(4,7),(5,8),(6,8),(7,9),(8,10),(9,11),(10,11)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,6,6,9} + 2
{{1,2,3},{4,5},{6}}
=> [2,3,1,5,4,6] => [3,1,2,5,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,10),(3,7),(4,7),(5,8),(6,8),(7,10),(8,9),(8,11),(9,12),(10,11),(11,12)],13)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,6,6,9} + 2
{{1,2,3,6},{4},{5}}
=> [2,3,6,4,5,1] => [4,5,6,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,10),(3,13),(4,12),(5,12),(5,13),(6,10),(6,11),(8,7),(9,7),(10,8),(11,8),(12,9),(13,9)],14)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,6,6,9} + 2
{{1,2,3},{4,6},{5}}
=> [2,3,1,6,5,4] => [3,1,2,5,6,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,7),(4,7),(5,8),(6,8),(7,9),(8,10),(9,11),(10,11)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,6,6,9} + 2
{{1,2,3},{4},{5,6}}
=> [2,3,1,4,6,5] => [3,1,2,4,6,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,8),(4,8),(5,11),(6,9),(6,10),(7,10),(8,11),(9,12),(10,12),(11,9)],13)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,6,6,9} + 2
{{1,2,4,5},{3,6}}
=> [2,4,6,5,1,3] => [5,1,2,4,6,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 2 = 0 + 2
{{1,2,4,6},{3,5}}
=> [2,4,5,6,3,1] => [5,3,6,1,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 2 = 0 + 2
{{1,2,4},{3,5,6}}
=> [2,4,5,1,6,3] => [4,1,2,6,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 2 = 0 + 2
{{1,2,5},{3},{4,6}}
=> [2,5,3,6,1,4] => [3,5,1,2,6,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 2 = 0 + 2
{{1,3,4,5},{2,6}}
=> [3,6,4,5,1,2] => [5,1,3,4,6,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 2 = 0 + 2
{{1,3,4,6},{2,5}}
=> [3,5,4,6,2,1] => [5,2,6,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 2 = 0 + 2
{{1,3,4},{2,5,6}}
=> [3,5,4,1,6,2] => [4,1,3,6,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 2 = 0 + 2
{{1,3,4},{2,5},{6}}
=> [3,5,4,1,2,6] => [4,1,3,5,2,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 2 = 0 + 2
{{1,3,4},{2,6},{5}}
=> [3,6,4,1,5,2] => [4,1,3,5,6,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 2 = 0 + 2
{{1,3,5,6},{2,4}}
=> [3,4,5,2,6,1] => [4,2,6,1,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 2 = 0 + 2
{{1,3,5},{2,4,6}}
=> [3,4,5,6,1,2] => [5,1,3,6,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 2 = 0 + 2
{{1,3,5},{2,4},{6}}
=> [3,4,5,2,1,6] => [4,2,5,1,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 2 = 0 + 2
Description
The minimal length of a chain of small intervals in a lattice.
An interval $[a, b]$ is small if $b$ is a join of elements covering $a$.
Matching statistic: St000454
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00011: Binary trees —to graph⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 15%
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00011: Binary trees —to graph⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 15%
Values
{{1,2}}
=> [2,1] => [[.,.],.]
=> ([(0,1)],2)
=> 1 = 0 + 1
{{1},{2}}
=> [1,2] => [.,[.,.]]
=> ([(0,1)],2)
=> 1 = 0 + 1
{{1,2,3}}
=> [2,3,1] => [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ? ∊ {0,0,0,0,0} + 1
{{1,2},{3}}
=> [2,1,3] => [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ? ∊ {0,0,0,0,0} + 1
{{1,3},{2}}
=> [3,2,1] => [[[.,.],.],.]
=> ([(0,2),(1,2)],3)
=> ? ∊ {0,0,0,0,0} + 1
{{1},{2,3}}
=> [1,3,2] => [.,[[.,.],.]]
=> ([(0,2),(1,2)],3)
=> ? ∊ {0,0,0,0,0} + 1
{{1},{2},{3}}
=> [1,2,3] => [.,[.,[.,.]]]
=> ([(0,2),(1,2)],3)
=> ? ∊ {0,0,0,0,0} + 1
{{1,2,3,4}}
=> [2,3,4,1] => [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1} + 1
{{1,2,3},{4}}
=> [2,3,1,4] => [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1} + 1
{{1,2,4},{3}}
=> [2,4,3,1] => [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1} + 1
{{1,2},{3,4}}
=> [2,1,4,3] => [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1} + 1
{{1,2},{3},{4}}
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1} + 1
{{1,3,4},{2}}
=> [3,2,4,1] => [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1} + 1
{{1,3},{2,4}}
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1} + 1
{{1,3},{2},{4}}
=> [3,2,1,4] => [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1} + 1
{{1,4},{2,3}}
=> [4,3,2,1] => [[[[.,.],.],.],.]
=> ([(0,3),(1,2),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1} + 1
{{1},{2,3,4}}
=> [1,3,4,2] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1} + 1
{{1},{2,3},{4}}
=> [1,3,2,4] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1} + 1
{{1,4},{2},{3}}
=> [4,2,3,1] => [[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1} + 1
{{1},{2,4},{3}}
=> [1,4,3,2] => [.,[[[.,.],.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1} + 1
{{1},{2},{3,4}}
=> [1,2,4,3] => [.,[.,[[.,.],.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1} + 1
{{1},{2},{3},{4}}
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1} + 1
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3} + 1
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3} + 1
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3} + 1
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3} + 1
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3} + 1
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3} + 1
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3} + 1
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3} + 1
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3} + 1
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3} + 1
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3} + 1
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3} + 1
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3} + 1
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3} + 1
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3} + 1
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [[[.,.],.],[.,[.,.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3} + 1
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [[.,[.,.]],[[.,.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3} + 1
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [[[.,.],.],[.,[.,.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3} + 1
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [[[.,.],.],[.,[.,.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3} + 1
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [[.,[.,.]],[.,[.,.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3} + 1
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [[.,[.,.]],[.,[.,.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3} + 1
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [[[.,.],.],[[.,.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3} + 1
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [[.,[.,.]],[[.,.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3} + 1
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [[[.,.],.],[[.,.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3} + 1
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [[[.,.],.],[.,[.,.]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3} + 1
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3} + 1
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3} + 1
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => [[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3} + 1
{{1,5},{2,3,4}}
=> [5,3,4,2,1] => [[[[.,.],.],[.,.]],.]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3} + 1
{{1},{2,3,4,5}}
=> [1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3} + 1
{{1},{2,3,5,6},{4}}
=> [1,3,5,4,6,2] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2 = 1 + 1
{{1},{2,3,5},{4,6}}
=> [1,3,5,6,2,4] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2 = 1 + 1
{{1},{2,3,5},{4},{6}}
=> [1,3,5,4,2,6] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2 = 1 + 1
{{1},{2,3},{4,5,6}}
=> [1,3,2,5,6,4] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2 = 1 + 1
{{1},{2,3},{4,5},{6}}
=> [1,3,2,5,4,6] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2 = 1 + 1
{{1,6},{2,4},{3,5}}
=> [6,4,5,2,3,1] => [[[[.,.],[.,.]],[.,.]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2 = 1 + 1
{{1,6},{2},{3,4,5}}
=> [6,2,4,5,3,1] => [[[.,.],[[.,.],[.,.]]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2 = 1 + 1
{{1,6},{2},{3,4},{5}}
=> [6,2,4,3,5,1] => [[[.,.],[[.,.],[.,.]]],.]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2 = 1 + 1
{{1},{2,5,6},{3},{4}}
=> [1,5,3,4,6,2] => [.,[[[.,.],[.,.]],[.,.]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2 = 1 + 1
{{1},{2,5},{3},{4,6}}
=> [1,5,3,6,2,4] => [.,[[[.,.],[.,.]],[.,.]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2 = 1 + 1
{{1},{2,5},{3},{4},{6}}
=> [1,5,3,4,2,6] => [.,[[[.,.],[.,.]],[.,.]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2 = 1 + 1
{{1,2,5,6,7},{3,4}}
=> [2,5,4,3,6,7,1] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2 = 1 + 1
{{1,2,5,6},{3,4,7}}
=> [2,5,4,7,6,1,3] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2 = 1 + 1
{{1,2,5,6},{3,4},{7}}
=> [2,5,4,3,6,1,7] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2 = 1 + 1
{{1,2,5,7},{3,4,6}}
=> [2,5,4,6,7,3,1] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2 = 1 + 1
{{1,2,5},{3,4,6,7}}
=> [2,5,4,6,1,7,3] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2 = 1 + 1
{{1,2,5},{3,4,6},{7}}
=> [2,5,4,6,1,3,7] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2 = 1 + 1
{{1,2,5,7},{3,4},{6}}
=> [2,5,4,3,7,6,1] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2 = 1 + 1
{{1,2,5},{3,4,7},{6}}
=> [2,5,4,7,1,6,3] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2 = 1 + 1
{{1,2,5},{3,4},{6,7}}
=> [2,5,4,3,1,7,6] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2 = 1 + 1
{{1,2,5},{3,4},{6},{7}}
=> [2,5,4,3,1,6,7] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2 = 1 + 1
{{1,2,5,6,7},{3},{4}}
=> [2,5,3,4,6,7,1] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2 = 1 + 1
{{1,2,5,6},{3,7},{4}}
=> [2,5,7,4,6,1,3] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2 = 1 + 1
{{1,2,5,6},{3},{4,7}}
=> [2,5,3,7,6,1,4] => [[.,.],[[.,[.,.]],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2 = 1 + 1
{{1,2,5,6},{3},{4},{7}}
=> [2,5,3,4,6,1,7] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2 = 1 + 1
{{1,2,5,7},{3,6},{4}}
=> [2,5,6,4,7,3,1] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2 = 1 + 1
{{1,2,5},{3,6,7},{4}}
=> [2,5,6,4,1,7,3] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2 = 1 + 1
{{1,2,5},{3,6},{4,7}}
=> [2,5,6,7,1,3,4] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2 = 1 + 1
{{1,2,5},{3,6},{4},{7}}
=> [2,5,6,4,1,3,7] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2 = 1 + 1
{{1,2,5,7},{3},{4,6}}
=> [2,5,3,6,7,4,1] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2 = 1 + 1
{{1,2,5},{3,7},{4,6}}
=> [2,5,7,6,1,4,3] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2 = 1 + 1
{{1,2,5},{3},{4,6,7}}
=> [2,5,3,6,1,7,4] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2 = 1 + 1
{{1,2,5},{3},{4,6},{7}}
=> [2,5,3,6,1,4,7] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2 = 1 + 1
{{1,2,5,7},{3},{4},{6}}
=> [2,5,3,4,7,6,1] => [[.,.],[[.,[.,.]],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2 = 1 + 1
{{1,2,5},{3,7},{4},{6}}
=> [2,5,7,4,1,6,3] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2 = 1 + 1
{{1,2,5},{3},{4,7},{6}}
=> [2,5,3,7,1,6,4] => [[.,.],[[.,[.,.]],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2 = 1 + 1
{{1,2,5},{3},{4},{6,7}}
=> [2,5,3,4,1,7,6] => [[.,.],[[.,[.,.]],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2 = 1 + 1
{{1,2,5},{3},{4},{6},{7}}
=> [2,5,3,4,1,6,7] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2 = 1 + 1
{{1,2},{3,5,6,7},{4}}
=> [2,1,5,4,6,7,3] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2 = 1 + 1
{{1,2},{3,5,6},{4,7}}
=> [2,1,5,7,6,3,4] => [[.,.],[[.,[.,.]],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2 = 1 + 1
{{1,2},{3,5,6},{4},{7}}
=> [2,1,5,4,6,3,7] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2 = 1 + 1
{{1,2},{3,5,7},{4,6}}
=> [2,1,5,6,7,4,3] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2 = 1 + 1
{{1,2},{3,5},{4,6,7}}
=> [2,1,5,6,3,7,4] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2 = 1 + 1
{{1,2},{3,5},{4,6},{7}}
=> [2,1,5,6,3,4,7] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2 = 1 + 1
{{1,2},{3,5,7},{4},{6}}
=> [2,1,5,4,7,6,3] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2 = 1 + 1
{{1,2},{3,5},{4,7},{6}}
=> [2,1,5,7,3,6,4] => [[.,.],[[.,[.,.]],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2 = 1 + 1
{{1,2},{3,5},{4},{6,7}}
=> [2,1,5,4,3,7,6] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2 = 1 + 1
{{1,2},{3,5},{4},{6},{7}}
=> [2,1,5,4,3,6,7] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2 = 1 + 1
Description
The largest eigenvalue of a graph if it is integral.
If a graph is $d$-regular, then its largest eigenvalue equals $d$. One can show that the largest eigenvalue always lies between the average degree and the maximal degree.
This statistic is undefined if the largest eigenvalue of the graph is not integral.
Matching statistic: St001001
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001001: Dyck paths ⟶ ℤResult quality: 8% ●values known / values provided: 10%●distinct values known / distinct values provided: 8%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001001: Dyck paths ⟶ ℤResult quality: 8% ●values known / values provided: 10%●distinct values known / distinct values provided: 8%
Values
{{1,2}}
=> [2]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 0
{{1},{2}}
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 0
{{1,2,3}}
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 0
{{1,2},{3}}
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 0
{{1,3},{2}}
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 0
{{1},{2,3}}
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 0
{{1},{2},{3}}
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 0
{{1,2,3,4}}
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> ? ∊ {0,1}
{{1,2,3},{4}}
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 0
{{1,2,4},{3}}
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 0
{{1,2},{3,4}}
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 0
{{1,2},{3},{4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 0
{{1,3,4},{2}}
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 0
{{1,3},{2,4}}
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 0
{{1,3},{2},{4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 0
{{1,4},{2,3}}
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 0
{{1},{2,3,4}}
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 0
{{1},{2,3},{4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 0
{{1,4},{2},{3}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 0
{{1},{2,4},{3}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 0
{{1},{2},{3,4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 0
{{1},{2},{3},{4}}
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,1}
{{1,2,3,4,5}}
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3}
{{1,2,3,4},{5}}
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3}
{{1,2,3,5},{4}}
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3}
{{1,2,3},{4,5}}
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 0
{{1,2,3},{4},{5}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0
{{1,2,4,5},{3}}
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3}
{{1,2,4},{3,5}}
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 0
{{1,2,4},{3},{5}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0
{{1,2,5},{3,4}}
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 0
{{1,2},{3,4,5}}
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 0
{{1,2},{3,4},{5}}
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 0
{{1,2,5},{3},{4}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0
{{1,2},{3,5},{4}}
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 0
{{1,2},{3},{4,5}}
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 0
{{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3}
{{1,3,4,5},{2}}
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3}
{{1,3,4},{2,5}}
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 0
{{1,3,4},{2},{5}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0
{{1,3,5},{2,4}}
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 0
{{1,3},{2,4,5}}
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 0
{{1,3},{2,4},{5}}
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 0
{{1,3,5},{2},{4}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0
{{1,3},{2,5},{4}}
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 0
{{1,3},{2},{4,5}}
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 0
{{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3}
{{1,4,5},{2,3}}
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 0
{{1,4},{2,3,5}}
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 0
{{1,4},{2,3},{5}}
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 0
{{1,5},{2,3,4}}
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 0
{{1},{2,3,4,5}}
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3}
{{1},{2,3,4},{5}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0
{{1,5},{2,3},{4}}
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 0
{{1},{2,3,5},{4}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0
{{1},{2,3},{4,5}}
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 0
{{1},{2,3},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3}
{{1,4,5},{2},{3}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0
{{1,4},{2,5},{3}}
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 0
{{1,4},{2},{3,5}}
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 0
{{1,4},{2},{3},{5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3}
{{1,5},{2,4},{3}}
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 0
{{1},{2,4},{3},{5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3}
{{1},{2},{3,4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3}
{{1,5},{2},{3},{4}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3}
{{1},{2,5},{3},{4}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3}
{{1},{2},{3,5},{4}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3}
{{1},{2},{3},{4,5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3}
{{1},{2},{3},{4},{5}}
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3}
{{1,2,3,4,5,6}}
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,6,6,9}
{{1,2,3,4,5},{6}}
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,6,6,9}
{{1,2,3,4,6},{5}}
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,6,6,9}
{{1,2,3,4},{5,6}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,6,6,9}
{{1,2,3,4},{5},{6}}
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,6,6,9}
{{1,2,3,5,6},{4}}
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,6,6,9}
{{1,2,3,5},{4,6}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,6,6,9}
{{1,2,3,5},{4},{6}}
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,6,6,9}
{{1,2,3,6},{4,5}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,6,6,9}
{{1,2,3},{4,5,6}}
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,6,6,9}
{{1,2,3,6},{4},{5}}
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,6,6,9}
{{1,2,3},{4},{5},{6}}
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,6,6,9}
{{1,2,4,5,6},{3}}
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,6,6,9}
{{1,2,4,5},{3,6}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,6,6,9}
{{1,2,4,5},{3},{6}}
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,6,6,9}
{{1,2,4,6},{3,5}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,6,6,9}
{{1,2,4},{3,5,6}}
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,6,6,9}
{{1,2,4,6},{3},{5}}
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,6,6,9}
{{1,2,4},{3},{5},{6}}
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,6,6,9}
{{1,2,5,6},{3,4}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,6,6,9}
{{1,2,5},{3,4,6}}
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,6,6,9}
{{1,2,6},{3,4,5}}
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,6,6,9}
{{1,2},{3,4,5,6}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,6,6,9}
{{1,2},{3,4},{5,6}}
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,6,6,9}
{{1,2},{3,4},{5},{6}}
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,6,6,9}
{{1,2,5,6},{3},{4}}
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,6,6,9}
{{1,2,5},{3},{4},{6}}
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,6,6,9}
{{1,2},{3,5},{4,6}}
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,6,6,9}
{{1,2},{3,5},{4},{6}}
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,6,6,9}
{{1,2},{3,6},{4,5}}
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,6,6,9}
{{1,2},{3},{4,5},{6}}
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,6,6,9}
Description
The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001371
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001371: Binary words ⟶ ℤResult quality: 8% ●values known / values provided: 10%●distinct values known / distinct values provided: 8%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001371: Binary words ⟶ ℤResult quality: 8% ●values known / values provided: 10%●distinct values known / distinct values provided: 8%
Values
{{1,2}}
=> [2]
=> [1,1,0,0,1,0]
=> 110010 => 0
{{1},{2}}
=> [1,1]
=> [1,0,1,1,0,0]
=> 101100 => 0
{{1,2,3}}
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => 0
{{1,2},{3}}
=> [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 0
{{1,3},{2}}
=> [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 0
{{1},{2,3}}
=> [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 0
{{1},{2},{3}}
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 0
{{1,2,3,4}}
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1111000010 => ? ∊ {0,1}
{{1,2,3},{4}}
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => 0
{{1,2,4},{3}}
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => 0
{{1,2},{3,4}}
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 11001100 => 0
{{1,2},{3},{4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 0
{{1,3,4},{2}}
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => 0
{{1,3},{2,4}}
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 11001100 => 0
{{1,3},{2},{4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 0
{{1,4},{2,3}}
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 11001100 => 0
{{1},{2,3,4}}
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => 0
{{1},{2,3},{4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 0
{{1,4},{2},{3}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 0
{{1},{2,4},{3}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 0
{{1},{2},{3,4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 0
{{1},{2},{3},{4}}
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1011110000 => ? ∊ {0,1}
{{1,2,3,4,5}}
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 111110000010 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3}
{{1,2,3,4},{5}}
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3}
{{1,2,3,5},{4}}
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3}
{{1,2,3},{4,5}}
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => 0
{{1,2,3},{4},{5}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
{{1,2,4,5},{3}}
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3}
{{1,2,4},{3,5}}
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => 0
{{1,2,4},{3},{5}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
{{1,2,5},{3,4}}
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => 0
{{1,2},{3,4,5}}
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => 0
{{1,2},{3,4},{5}}
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 0
{{1,2,5},{3},{4}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
{{1,2},{3,5},{4}}
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 0
{{1,2},{3},{4,5}}
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 0
{{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3}
{{1,3,4,5},{2}}
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3}
{{1,3,4},{2,5}}
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => 0
{{1,3,4},{2},{5}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
{{1,3,5},{2,4}}
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => 0
{{1,3},{2,4,5}}
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => 0
{{1,3},{2,4},{5}}
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 0
{{1,3,5},{2},{4}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
{{1,3},{2,5},{4}}
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 0
{{1,3},{2},{4,5}}
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 0
{{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3}
{{1,4,5},{2,3}}
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => 0
{{1,4},{2,3,5}}
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => 0
{{1,4},{2,3},{5}}
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 0
{{1,5},{2,3,4}}
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => 0
{{1},{2,3,4,5}}
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3}
{{1},{2,3,4},{5}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
{{1,5},{2,3},{4}}
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 0
{{1},{2,3,5},{4}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
{{1},{2,3},{4,5}}
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 0
{{1},{2,3},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3}
{{1,4,5},{2},{3}}
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
{{1,4},{2,5},{3}}
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 0
{{1,4},{2},{3,5}}
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 0
{{1,4},{2},{3},{5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3}
{{1,5},{2,4},{3}}
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 0
{{1},{2,4},{3},{5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3}
{{1},{2},{3,4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3}
{{1,5},{2},{3},{4}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3}
{{1},{2,5},{3},{4}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3}
{{1},{2},{3,5},{4}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3}
{{1},{2},{3},{4,5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3}
{{1},{2},{3},{4},{5}}
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 101111100000 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,3,3}
{{1,2,3,4,5,6}}
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 11111100000010 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,6,6,9}
{{1,2,3,4,5},{6}}
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> 111101000010 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,6,6,9}
{{1,2,3,4,6},{5}}
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> 111101000010 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,6,6,9}
{{1,2,3,4},{5,6}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1110010010 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,6,6,9}
{{1,2,3,4},{5},{6}}
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1101100010 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,6,6,9}
{{1,2,3,5,6},{4}}
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> 111101000010 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,6,6,9}
{{1,2,3,5},{4,6}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1110010010 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,6,6,9}
{{1,2,3,5},{4},{6}}
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1101100010 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,6,6,9}
{{1,2,3,6},{4,5}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1110010010 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,6,6,9}
{{1,2,3},{4,5,6}}
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1110001100 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,6,6,9}
{{1,2,3,6},{4},{5}}
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1101100010 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,6,6,9}
{{1,2,3},{4},{5},{6}}
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,6,6,9}
{{1,2,4,5,6},{3}}
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> 111101000010 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,6,6,9}
{{1,2,4,5},{3,6}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1110010010 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,6,6,9}
{{1,2,4,5},{3},{6}}
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1101100010 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,6,6,9}
{{1,2,4,6},{3,5}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1110010010 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,6,6,9}
{{1,2,4},{3,5,6}}
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1110001100 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,6,6,9}
{{1,2,4,6},{3},{5}}
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1101100010 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,6,6,9}
{{1,2,4},{3},{5},{6}}
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,6,6,9}
{{1,2,5,6},{3,4}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1110010010 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,6,6,9}
{{1,2,5},{3,4,6}}
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1110001100 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,6,6,9}
{{1,2,6},{3,4,5}}
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1110001100 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,6,6,9}
{{1,2},{3,4,5,6}}
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1110010010 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,6,6,9}
{{1,2},{3,4},{5,6}}
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1100111000 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,6,6,9}
{{1,2},{3,4},{5},{6}}
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> 1011011000 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,6,6,9}
{{1,2,5,6},{3},{4}}
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1101100010 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,6,6,9}
{{1,2,5},{3},{4},{6}}
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,6,6,9}
{{1,2},{3,5},{4,6}}
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1100111000 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,6,6,9}
{{1,2},{3,5},{4},{6}}
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> 1011011000 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,6,6,9}
{{1,2},{3,6},{4,5}}
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1100111000 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,6,6,9}
{{1,2},{3},{4,5},{6}}
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> 1011011000 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,6,6,9}
Description
The length of the longest Yamanouchi prefix of a binary word.
This is the largest index $i$ such that in each of the prefixes $w_1$, $w_1w_2$, $w_1w_2\dots w_i$ the number of zeros is greater than or equal to the number of ones.
The following 22 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001730The number of times the path corresponding to a binary word crosses the base line. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001845The number of join irreducibles minus the rank of a lattice. St001862The number of crossings of a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001875The number of simple modules with projective dimension at most 1. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St000068The number of minimal elements in a poset. St001866The nesting alignments of a signed permutation. St001490The number of connected components of a skew partition. St001301The first Betti number of the order complex associated with the poset. St000908The length of the shortest maximal antichain in a poset. St000914The sum of the values of the Möbius function of a poset. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St001867The number of alignments of type EN of a signed permutation. St001396Number of triples of incomparable elements in a finite poset. St001532The leading coefficient of the Poincare polynomial of the poset cone.
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