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Your data matches 21 different statistics following compositions of up to 3 maps.
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Matching statistic: St000075
St000075: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> 1
[[1,2]]
=> 1
[[1],[2]]
=> 1
[[1,2,3]]
=> 1
[[1,3],[2]]
=> 2
[[1,2],[3]]
=> 2
[[1],[2],[3]]
=> 1
[[1,2,3,4]]
=> 1
[[1,3,4],[2]]
=> 3
[[1,2,4],[3]]
=> 3
[[1,2,3],[4]]
=> 3
[[1,3],[2,4]]
=> 2
[[1,2],[3,4]]
=> 2
[[1,4],[2],[3]]
=> 3
[[1,3],[2],[4]]
=> 3
[[1,2],[3],[4]]
=> 3
[[1],[2],[3],[4]]
=> 1
[[1,2,3,4,5]]
=> 1
[[1,3,4,5],[2]]
=> 4
[[1,2,4,5],[3]]
=> 4
[[1,2,3,5],[4]]
=> 4
[[1,2,3,4],[5]]
=> 4
[[1,3,5],[2,4]]
=> 2
[[1,2,5],[3,4]]
=> 3
[[1,3,4],[2,5]]
=> 3
[[1,2,4],[3,5]]
=> 2
[[1,2,3],[4,5]]
=> 3
[[1,4,5],[2],[3]]
=> 4
[[1,3,5],[2],[4]]
=> 2
[[1,2,5],[3],[4]]
=> 4
[[1,3,4],[2],[5]]
=> 4
[[1,2,4],[3],[5]]
=> 2
[[1,2,3],[4],[5]]
=> 4
[[1,4],[2,5],[3]]
=> 3
[[1,3],[2,5],[4]]
=> 2
[[1,2],[3,5],[4]]
=> 3
[[1,3],[2,4],[5]]
=> 3
[[1,2],[3,4],[5]]
=> 2
[[1,5],[2],[3],[4]]
=> 4
[[1,4],[2],[3],[5]]
=> 4
[[1,3],[2],[4],[5]]
=> 4
[[1,2],[3],[4],[5]]
=> 4
[[1],[2],[3],[4],[5]]
=> 1
[[1,2,3,4,5,6]]
=> 1
[[1,3,4,5,6],[2]]
=> 5
[[1,2,4,5,6],[3]]
=> 5
[[1,2,3,5,6],[4]]
=> 5
[[1,2,3,4,6],[5]]
=> 5
[[1,2,3,4,5],[6]]
=> 5
[[1,3,5,6],[2,4]]
=> 5
Description
The orbit size of a standard tableau under promotion.
Matching statistic: St000454
Mp00134: Standard tableaux —descent word⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 50% ●values known / values provided: 50%●distinct values known / distinct values provided: 83%
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 50% ●values known / values provided: 50%●distinct values known / distinct values provided: 83%
Values
[[1]]
=> => [] => ?
=> ? = 1 - 1
[[1,2]]
=> 0 => [1] => ([],1)
=> 0 = 1 - 1
[[1],[2]]
=> 1 => [1] => ([],1)
=> 0 = 1 - 1
[[1,2,3]]
=> 00 => [2] => ([],2)
=> 0 = 1 - 1
[[1,3],[2]]
=> 10 => [1,1] => ([(0,1)],2)
=> 1 = 2 - 1
[[1,2],[3]]
=> 01 => [1,1] => ([(0,1)],2)
=> 1 = 2 - 1
[[1],[2],[3]]
=> 11 => [2] => ([],2)
=> 0 = 1 - 1
[[1,2,3,4]]
=> 000 => [3] => ([],3)
=> 0 = 1 - 1
[[1,3,4],[2]]
=> 100 => [1,2] => ([(1,2)],3)
=> 1 = 2 - 1
[[1,2,4],[3]]
=> 010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 3 - 1
[[1,2,3],[4]]
=> 001 => [2,1] => ([(0,2),(1,2)],3)
=> ? ∊ {3,3} - 1
[[1,3],[2,4]]
=> 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 3 - 1
[[1,2],[3,4]]
=> 010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 3 - 1
[[1,4],[2],[3]]
=> 110 => [2,1] => ([(0,2),(1,2)],3)
=> ? ∊ {3,3} - 1
[[1,3],[2],[4]]
=> 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 3 - 1
[[1,2],[3],[4]]
=> 011 => [1,2] => ([(1,2)],3)
=> 1 = 2 - 1
[[1],[2],[3],[4]]
=> 111 => [3] => ([],3)
=> 0 = 1 - 1
[[1,2,3,4,5]]
=> 0000 => [4] => ([],4)
=> 0 = 1 - 1
[[1,3,4,5],[2]]
=> 1000 => [1,3] => ([(2,3)],4)
=> 1 = 2 - 1
[[1,2,4,5],[3]]
=> 0100 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[[1,2,3,5],[4]]
=> 0010 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,3,3,4,4,4,4,4,4} - 1
[[1,2,3,4],[5]]
=> 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,3,3,4,4,4,4,4,4} - 1
[[1,3,5],[2,4]]
=> 1010 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[[1,2,5],[3,4]]
=> 0100 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[[1,3,4],[2,5]]
=> 1001 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,3,3,4,4,4,4,4,4} - 1
[[1,2,4],[3,5]]
=> 0101 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[[1,2,3],[4,5]]
=> 0010 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,3,3,4,4,4,4,4,4} - 1
[[1,4,5],[2],[3]]
=> 1100 => [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,3,3,4,4,4,4,4,4} - 1
[[1,3,5],[2],[4]]
=> 1010 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[[1,2,5],[3],[4]]
=> 0110 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,3,3,4,4,4,4,4,4} - 1
[[1,3,4],[2],[5]]
=> 1001 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,3,3,4,4,4,4,4,4} - 1
[[1,2,4],[3],[5]]
=> 0101 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[[1,2,3],[4],[5]]
=> 0011 => [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,3,3,4,4,4,4,4,4} - 1
[[1,4],[2,5],[3]]
=> 1101 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,3,3,4,4,4,4,4,4} - 1
[[1,3],[2,5],[4]]
=> 1010 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[[1,2],[3,5],[4]]
=> 0110 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,3,3,4,4,4,4,4,4} - 1
[[1,3],[2,4],[5]]
=> 1011 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[[1,2],[3,4],[5]]
=> 0101 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[[1,5],[2],[3],[4]]
=> 1110 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,3,3,4,4,4,4,4,4} - 1
[[1,4],[2],[3],[5]]
=> 1101 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,2,2,3,3,4,4,4,4,4,4} - 1
[[1,3],[2],[4],[5]]
=> 1011 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[[1,2],[3],[4],[5]]
=> 0111 => [1,3] => ([(2,3)],4)
=> 1 = 2 - 1
[[1],[2],[3],[4],[5]]
=> 1111 => [4] => ([],4)
=> 0 = 1 - 1
[[1,2,3,4,5,6]]
=> 00000 => [5] => ([],5)
=> 0 = 1 - 1
[[1,3,4,5,6],[2]]
=> 10000 => [1,4] => ([(3,4)],5)
=> 1 = 2 - 1
[[1,2,4,5,6],[3]]
=> 01000 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[[1,2,3,5,6],[4]]
=> 00100 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,2,3,4,6],[5]]
=> 00010 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[[1,2,3,4,5],[6]]
=> 00001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 3 - 1
[[1,3,5,6],[2,4]]
=> 10100 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[[1,2,5,6],[3,4]]
=> 01000 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[[1,3,4,6],[2,5]]
=> 10010 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,2,4,6],[3,5]]
=> 01010 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 5 - 1
[[1,2,3,6],[4,5]]
=> 00100 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,3,4,5],[2,6]]
=> 10001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,2,4,5],[3,6]]
=> 01001 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,2,3,5],[4,6]]
=> 00101 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,2,3,4],[5,6]]
=> 00010 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[[1,4,5,6],[2],[3]]
=> 11000 => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {2,2,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,3,5,6],[2],[4]]
=> 10100 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[[1,2,5,6],[3],[4]]
=> 01100 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,3,4,6],[2],[5]]
=> 10010 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,2,4,6],[3],[5]]
=> 01010 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 5 - 1
[[1,2,3,6],[4],[5]]
=> 00110 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,3,4,5],[2],[6]]
=> 10001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,2,4,5],[3],[6]]
=> 01001 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,2,3,5],[4],[6]]
=> 00101 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,2,3,4],[5],[6]]
=> 00011 => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {2,2,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,3,5],[2,4,6]]
=> 10101 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 5 - 1
[[1,2,5],[3,4,6]]
=> 01001 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,3,4],[2,5,6]]
=> 10010 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,2,4],[3,5,6]]
=> 01010 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 5 - 1
[[1,2,3],[4,5,6]]
=> 00100 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,4,6],[2,5],[3]]
=> 11010 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,3,6],[2,5],[4]]
=> 10100 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[[1,2,6],[3,5],[4]]
=> 01100 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,3,6],[2,4],[5]]
=> 10110 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,2,6],[3,4],[5]]
=> 01010 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 5 - 1
[[1,4,5],[2,6],[3]]
=> 11001 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,3,5],[2,6],[4]]
=> 10101 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 5 - 1
[[1,2,5],[3,6],[4]]
=> 01101 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,3,4],[2,6],[5]]
=> 10010 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,2,4],[3,6],[5]]
=> 01010 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 5 - 1
[[1,2,3],[4,6],[5]]
=> 00110 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,3,5],[2,4],[6]]
=> 10101 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 5 - 1
[[1,2,5],[3,4],[6]]
=> 01001 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,3,4],[2,5],[6]]
=> 10011 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,2,4],[3,5],[6]]
=> 01011 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[[1,2,3],[4,5],[6]]
=> 00101 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,5,6],[2],[3],[4]]
=> 11100 => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {2,2,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,4,6],[2],[3],[5]]
=> 11010 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,3,6],[2],[4],[5]]
=> 10110 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,2,6],[3],[4],[5]]
=> 01110 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,4,5],[2],[3],[6]]
=> 11001 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,3,5],[2],[4],[6]]
=> 10101 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 5 - 1
[[1,2,5],[3],[4],[6]]
=> 01101 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,3,4],[2],[5],[6]]
=> 10011 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,2,4],[3],[5],[6]]
=> 01011 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[[1,2,3],[4],[5],[6]]
=> 00111 => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {2,2,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,3],[2,5],[4,6]]
=> 10101 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 5 - 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: St001232
Mp00134: Standard tableaux —descent word⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 39% ●values known / values provided: 39%●distinct values known / distinct values provided: 83%
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 39% ●values known / values provided: 39%●distinct values known / distinct values provided: 83%
Values
[[1]]
=> => [] => ?
=> ? = 1 - 1
[[1,2]]
=> 0 => [1] => [1,0]
=> 0 = 1 - 1
[[1],[2]]
=> 1 => [1] => [1,0]
=> 0 = 1 - 1
[[1,2,3]]
=> 00 => [2] => [1,1,0,0]
=> 0 = 1 - 1
[[1,3],[2]]
=> 10 => [1,1] => [1,0,1,0]
=> 1 = 2 - 1
[[1,2],[3]]
=> 01 => [1,1] => [1,0,1,0]
=> 1 = 2 - 1
[[1],[2],[3]]
=> 11 => [2] => [1,1,0,0]
=> 0 = 1 - 1
[[1,2,3,4]]
=> 000 => [3] => [1,1,1,0,0,0]
=> 0 = 1 - 1
[[1,3,4],[2]]
=> 100 => [1,2] => [1,0,1,1,0,0]
=> 2 = 3 - 1
[[1,2,4],[3]]
=> 010 => [1,1,1] => [1,0,1,0,1,0]
=> ? ∊ {3,3,3,3} - 1
[[1,2,3],[4]]
=> 001 => [2,1] => [1,1,0,0,1,0]
=> 1 = 2 - 1
[[1,3],[2,4]]
=> 101 => [1,1,1] => [1,0,1,0,1,0]
=> ? ∊ {3,3,3,3} - 1
[[1,2],[3,4]]
=> 010 => [1,1,1] => [1,0,1,0,1,0]
=> ? ∊ {3,3,3,3} - 1
[[1,4],[2],[3]]
=> 110 => [2,1] => [1,1,0,0,1,0]
=> 1 = 2 - 1
[[1,3],[2],[4]]
=> 101 => [1,1,1] => [1,0,1,0,1,0]
=> ? ∊ {3,3,3,3} - 1
[[1,2],[3],[4]]
=> 011 => [1,2] => [1,0,1,1,0,0]
=> 2 = 3 - 1
[[1],[2],[3],[4]]
=> 111 => [3] => [1,1,1,0,0,0]
=> 0 = 1 - 1
[[1,2,3,4,5]]
=> 0000 => [4] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[[1,3,4,5],[2]]
=> 1000 => [1,3] => [1,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[[1,2,4,5],[3]]
=> 0100 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,4,4} - 1
[[1,2,3,5],[4]]
=> 0010 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,4,4} - 1
[[1,2,3,4],[5]]
=> 0001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[[1,3,5],[2,4]]
=> 1010 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,4,4} - 1
[[1,2,5],[3,4]]
=> 0100 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,4,4} - 1
[[1,3,4],[2,5]]
=> 1001 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 3 = 4 - 1
[[1,2,4],[3,5]]
=> 0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,4,4} - 1
[[1,2,3],[4,5]]
=> 0010 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,4,4} - 1
[[1,4,5],[2],[3]]
=> 1100 => [2,2] => [1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[[1,3,5],[2],[4]]
=> 1010 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,4,4} - 1
[[1,2,5],[3],[4]]
=> 0110 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 3 = 4 - 1
[[1,3,4],[2],[5]]
=> 1001 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 3 = 4 - 1
[[1,2,4],[3],[5]]
=> 0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,4,4} - 1
[[1,2,3],[4],[5]]
=> 0011 => [2,2] => [1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[[1,4],[2,5],[3]]
=> 1101 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,4,4} - 1
[[1,3],[2,5],[4]]
=> 1010 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,4,4} - 1
[[1,2],[3,5],[4]]
=> 0110 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 3 = 4 - 1
[[1,3],[2,4],[5]]
=> 1011 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,4,4} - 1
[[1,2],[3,4],[5]]
=> 0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,4,4} - 1
[[1,5],[2],[3],[4]]
=> 1110 => [3,1] => [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[[1,4],[2],[3],[5]]
=> 1101 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,4,4} - 1
[[1,3],[2],[4],[5]]
=> 1011 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> ? ∊ {2,2,2,2,3,3,3,3,4,4,4,4,4,4} - 1
[[1,2],[3],[4],[5]]
=> 0111 => [1,3] => [1,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[[1],[2],[3],[4],[5]]
=> 1111 => [4] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[[1,2,3,4,5,6]]
=> 00000 => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[[1,3,4,5,6],[2]]
=> 10000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 4 = 5 - 1
[[1,2,4,5,6],[3]]
=> 01000 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> ? ∊ {2,2,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,2,3,5,6],[4]]
=> 00100 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> ? ∊ {2,2,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,2,3,4,6],[5]]
=> 00010 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> ? ∊ {2,2,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,2,3,4,5],[6]]
=> 00001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1 = 2 - 1
[[1,3,5,6],[2,4]]
=> 10100 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {2,2,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,2,5,6],[3,4]]
=> 01000 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> ? ∊ {2,2,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,3,4,6],[2,5]]
=> 10010 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> ? ∊ {2,2,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,2,4,6],[3,5]]
=> 01010 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {2,2,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,2,3,6],[4,5]]
=> 00100 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> ? ∊ {2,2,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,3,4,5],[2,6]]
=> 10001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 4 = 5 - 1
[[1,2,4,5],[3,6]]
=> 01001 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> ? ∊ {2,2,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,2,3,5],[4,6]]
=> 00101 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> ? ∊ {2,2,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,2,3,4],[5,6]]
=> 00010 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> ? ∊ {2,2,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,4,5,6],[2],[3]]
=> 11000 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[[1,3,5,6],[2],[4]]
=> 10100 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {2,2,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,2,5,6],[3],[4]]
=> 01100 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 4 = 5 - 1
[[1,3,4,6],[2],[5]]
=> 10010 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> ? ∊ {2,2,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,2,4,6],[3],[5]]
=> 01010 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {2,2,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,2,3,6],[4],[5]]
=> 00110 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3 = 4 - 1
[[1,3,4,5],[2],[6]]
=> 10001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 4 = 5 - 1
[[1,2,4,5],[3],[6]]
=> 01001 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> ? ∊ {2,2,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,2,3,5],[4],[6]]
=> 00101 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> ? ∊ {2,2,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,2,3,4],[5],[6]]
=> 00011 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2 = 3 - 1
[[1,3,5],[2,4,6]]
=> 10101 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {2,2,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,2,5],[3,4,6]]
=> 01001 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> ? ∊ {2,2,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,3,4],[2,5,6]]
=> 10010 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> ? ∊ {2,2,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,2,4],[3,5,6]]
=> 01010 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {2,2,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,2,3],[4,5,6]]
=> 00100 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> ? ∊ {2,2,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,4,6],[2,5],[3]]
=> 11010 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> ? ∊ {2,2,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,3,6],[2,5],[4]]
=> 10100 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {2,2,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,2,6],[3,5],[4]]
=> 01100 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 4 = 5 - 1
[[1,3,6],[2,4],[5]]
=> 10110 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> ? ∊ {2,2,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,2,6],[3,4],[5]]
=> 01010 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {2,2,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,4,5],[2,6],[3]]
=> 11001 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3 = 4 - 1
[[1,3,5],[2,6],[4]]
=> 10101 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {2,2,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,2,5],[3,6],[4]]
=> 01101 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> ? ∊ {2,2,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,3,4],[2,6],[5]]
=> 10010 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> ? ∊ {2,2,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,2,4],[3,6],[5]]
=> 01010 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {2,2,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,2,3],[4,6],[5]]
=> 00110 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3 = 4 - 1
[[1,3,5],[2,4],[6]]
=> 10101 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {2,2,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,2,5],[3,4],[6]]
=> 01001 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> ? ∊ {2,2,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,3,4],[2,5],[6]]
=> 10011 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 4 = 5 - 1
[[1,5,6],[2],[3],[4]]
=> 11100 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2 = 3 - 1
[[1,2,6],[3],[4],[5]]
=> 01110 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 4 = 5 - 1
[[1,4,5],[2],[3],[6]]
=> 11001 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3 = 4 - 1
[[1,3,4],[2],[5],[6]]
=> 10011 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 4 = 5 - 1
[[1,2,3],[4],[5],[6]]
=> 00111 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[[1,2],[3,6],[4],[5]]
=> 01110 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 4 = 5 - 1
[[1,6],[2],[3],[4],[5]]
=> 11110 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1 = 2 - 1
[[1,2],[3],[4],[5],[6]]
=> 01111 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 4 = 5 - 1
[[1],[2],[3],[4],[5],[6]]
=> 11111 => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Matching statistic: St001633
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St001633: Posets ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 50%
Mp00209: Permutations —pattern poset⟶ Posets
St001633: Posets ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 50%
Values
[[1]]
=> [1] => ([],1)
=> 0 = 1 - 1
[[1,2]]
=> [1,2] => ([(0,1)],2)
=> 0 = 1 - 1
[[1],[2]]
=> [2,1] => ([(0,1)],2)
=> 0 = 1 - 1
[[1,2,3]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[[1,3],[2]]
=> [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[[1,2],[3]]
=> [3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[[1],[2],[3]]
=> [3,2,1] => ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[[1,2,3,4]]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[[1,3,4],[2]]
=> [2,1,3,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[[1,2,4],[3]]
=> [3,1,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? ∊ {2,2,3} - 1
[[1,2,3],[4]]
=> [4,1,2,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[[1,3],[2,4]]
=> [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? ∊ {2,2,3} - 1
[[1,2],[3,4]]
=> [3,4,1,2] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 2 = 3 - 1
[[1,4],[2],[3]]
=> [3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[[1,3],[2],[4]]
=> [4,2,1,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? ∊ {2,2,3} - 1
[[1,2],[3],[4]]
=> [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[[1],[2],[3],[4]]
=> [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[[1,2,3,4,5]]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[[1,3,4,5],[2]]
=> [2,1,3,4,5] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4} - 1
[[1,2,4,5],[3]]
=> [3,1,2,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4} - 1
[[1,2,3,5],[4]]
=> [4,1,2,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4} - 1
[[1,2,3,4],[5]]
=> [5,1,2,3,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4} - 1
[[1,3,5],[2,4]]
=> [2,4,1,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4} - 1
[[1,2,5],[3,4]]
=> [3,4,1,2,5] => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(5,9),(5,10),(6,9),(6,10),(7,10),(8,9),(8,10),(9,4),(10,4)],11)
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4} - 1
[[1,3,4],[2,5]]
=> [2,5,1,3,4] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,9),(2,10),(3,6),(3,9),(4,7),(4,9),(4,10),(5,1),(5,6),(5,7),(5,10),(6,11),(6,12),(7,11),(7,12),(9,12),(10,11),(10,12),(11,8),(12,8)],13)
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4} - 1
[[1,2,4],[3,5]]
=> [3,5,1,2,4] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,7),(2,10),(3,6),(3,10),(4,6),(4,8),(4,10),(5,1),(5,7),(5,8),(5,10),(6,12),(7,11),(7,12),(8,11),(8,12),(10,11),(10,12),(11,9),(12,9)],13)
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4} - 1
[[1,2,3],[4,5]]
=> [4,5,1,2,3] => ([(0,3),(0,4),(1,8),(2,7),(2,8),(3,1),(3,5),(4,2),(4,5),(5,7),(5,8),(7,6),(8,6)],9)
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4} - 1
[[1,4,5],[2],[3]]
=> [3,2,1,4,5] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4} - 1
[[1,3,5],[2],[4]]
=> [4,2,1,3,5] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4} - 1
[[1,2,5],[3],[4]]
=> [4,3,1,2,5] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4} - 1
[[1,3,4],[2],[5]]
=> [5,2,1,3,4] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4} - 1
[[1,2,4],[3],[5]]
=> [5,3,1,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4} - 1
[[1,2,3],[4],[5]]
=> [5,4,1,2,3] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4} - 1
[[1,4],[2,5],[3]]
=> [3,2,5,1,4] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,7),(2,10),(3,6),(3,10),(4,6),(4,8),(4,10),(5,1),(5,7),(5,8),(5,10),(6,12),(7,11),(7,12),(8,11),(8,12),(10,11),(10,12),(11,9),(12,9)],13)
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4} - 1
[[1,3],[2,5],[4]]
=> [4,2,5,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,6),(2,9),(2,11),(3,6),(3,9),(3,10),(4,7),(4,9),(4,10),(4,11),(5,7),(5,9),(5,10),(5,11),(6,13),(7,12),(7,13),(9,12),(9,13),(10,12),(10,13),(11,12),(11,13),(12,8),(13,8)],14)
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4} - 1
[[1,2],[3,5],[4]]
=> [4,3,5,1,2] => ([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4} - 1
[[1,3],[2,4],[5]]
=> [5,2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4} - 1
[[1,2],[3,4],[5]]
=> [5,3,4,1,2] => ([(0,1),(0,2),(0,3),(1,7),(1,8),(2,5),(2,8),(3,5),(3,7),(3,8),(5,9),(6,4),(7,6),(7,9),(8,6),(8,9),(9,4)],10)
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4} - 1
[[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4} - 1
[[1,4],[2],[3],[5]]
=> [5,3,2,1,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4} - 1
[[1,3],[2],[4],[5]]
=> [5,4,2,1,3] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4} - 1
[[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4} - 1
[[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[[1,3,4,5,6],[2]]
=> [2,1,3,4,5,6] => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,2,4,5,6],[3]]
=> [3,1,2,4,5,6] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,2,3,5,6],[4]]
=> [4,1,2,3,5,6] => ([(0,3),(0,4),(0,5),(1,9),(1,13),(2,8),(2,13),(3,11),(4,2),(4,6),(4,11),(5,1),(5,6),(5,11),(6,8),(6,9),(6,13),(8,10),(8,12),(9,10),(9,12),(10,7),(11,13),(12,7),(13,12)],14)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,2,3,4,6],[5]]
=> [5,1,2,3,4,6] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,3,5,6],[2,4]]
=> [2,4,1,3,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,6),(1,14),(1,18),(2,13),(2,14),(2,18),(3,12),(3,14),(3,18),(4,11),(4,14),(4,18),(5,8),(5,9),(5,10),(5,16),(6,5),(6,11),(6,12),(6,13),(6,18),(8,15),(8,19),(9,15),(9,19),(10,15),(10,19),(11,8),(11,16),(11,17),(12,9),(12,16),(12,17),(13,10),(13,16),(13,17),(14,17),(15,7),(16,15),(16,19),(17,19),(18,16),(18,17),(19,7)],20)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,2,5,6],[3,4]]
=> [3,4,1,2,5,6] => ([(0,2),(0,3),(0,4),(1,8),(1,9),(2,1),(2,10),(2,11),(3,6),(3,7),(3,11),(4,6),(4,7),(4,10),(6,14),(7,12),(7,14),(8,13),(8,15),(9,13),(9,15),(10,8),(10,12),(10,14),(11,9),(11,12),(11,14),(12,13),(12,15),(13,5),(14,15),(15,5)],16)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,3,4,6],[2,5]]
=> [2,5,1,3,4,6] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,12),(1,20),(1,21),(2,13),(2,14),(3,8),(3,13),(3,15),(4,9),(4,10),(4,13),(4,15),(5,7),(5,8),(5,9),(5,14),(6,1),(6,7),(6,10),(6,14),(6,15),(7,12),(7,16),(7,18),(7,20),(8,18),(8,21),(9,16),(9,18),(9,21),(10,16),(10,20),(10,21),(12,17),(12,19),(13,21),(14,20),(14,21),(15,18),(15,20),(15,21),(16,17),(16,19),(17,11),(18,17),(18,19),(19,11),(20,17),(20,19),(21,19)],22)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,2,4,6],[3,5]]
=> [3,5,1,2,4,6] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(1,18),(1,21),(2,11),(2,12),(2,19),(3,9),(3,13),(3,19),(4,10),(4,13),(4,19),(5,10),(5,12),(5,14),(5,19),(6,1),(6,9),(6,11),(6,14),(6,19),(8,17),(8,20),(9,18),(9,21),(10,16),(10,21),(11,15),(11,18),(11,21),(12,15),(12,16),(13,21),(14,8),(14,15),(14,16),(14,18),(15,17),(15,20),(16,17),(16,20),(17,7),(18,17),(18,20),(19,16),(19,18),(19,21),(20,7),(21,20)],22)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,2,3,6],[4,5]]
=> [4,5,1,2,3,6] => ([(0,3),(0,4),(0,5),(1,14),(2,6),(2,8),(2,14),(3,9),(3,10),(4,2),(4,10),(4,11),(5,1),(5,9),(5,11),(6,13),(6,15),(8,13),(8,15),(9,12),(9,14),(10,8),(10,12),(11,6),(11,12),(11,14),(12,13),(12,15),(13,7),(14,15),(15,7)],16)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,3,4,5],[2,6]]
=> [2,6,1,3,4,5] => ([(0,2),(0,3),(0,4),(0,6),(1,15),(1,17),(2,12),(2,13),(3,7),(3,12),(4,8),(4,12),(4,13),(5,1),(5,10),(5,11),(5,14),(6,5),(6,7),(6,8),(6,13),(7,10),(7,16),(8,11),(8,14),(8,16),(10,15),(10,17),(11,15),(11,17),(12,16),(13,14),(13,16),(14,15),(14,17),(15,9),(16,17),(17,9)],18)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,2,4,5],[3,6]]
=> [3,6,1,2,4,5] => ([(0,3),(0,4),(0,5),(0,6),(1,11),(1,18),(2,12),(2,17),(2,18),(3,7),(3,14),(4,1),(4,10),(4,13),(4,14),(5,2),(5,9),(5,13),(5,14),(6,7),(6,9),(6,10),(7,17),(9,15),(9,17),(10,15),(10,17),(10,18),(11,16),(11,19),(12,16),(12,19),(13,11),(13,12),(13,15),(13,18),(14,17),(14,18),(15,16),(15,19),(16,8),(17,19),(18,16),(18,19),(19,8)],20)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,2,3,5],[4,6]]
=> [4,6,1,2,3,5] => ([(0,2),(0,3),(0,4),(0,6),(1,15),(1,17),(2,7),(2,14),(3,9),(3,14),(4,9),(4,10),(4,14),(5,1),(5,11),(5,12),(5,16),(6,5),(6,7),(6,10),(6,14),(7,11),(7,16),(9,13),(10,12),(10,13),(10,16),(11,15),(11,17),(12,15),(12,17),(13,17),(14,13),(14,16),(15,8),(16,15),(16,17),(17,8)],18)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,2,3,4],[5,6]]
=> [5,6,1,2,3,4] => ([(0,4),(0,5),(1,7),(2,9),(2,11),(3,2),(3,10),(4,3),(4,6),(5,1),(5,6),(6,7),(6,10),(7,11),(9,8),(10,9),(10,11),(11,8)],12)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,4,5,6],[2],[3]]
=> [3,2,1,4,5,6] => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,3,5,6],[2],[4]]
=> [4,2,1,3,5,6] => ([(0,1),(0,2),(0,4),(0,5),(1,9),(1,16),(2,10),(2,16),(3,6),(3,7),(3,15),(4,9),(4,11),(4,16),(5,3),(5,10),(5,11),(5,16),(6,13),(7,13),(7,14),(9,12),(10,6),(10,15),(11,7),(11,12),(11,15),(12,14),(13,8),(14,8),(15,13),(15,14),(16,12),(16,15)],17)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,2,5,6],[3],[4]]
=> [4,3,1,2,5,6] => ([(0,3),(0,4),(0,5),(1,13),(2,6),(2,8),(3,9),(3,10),(4,2),(4,10),(4,11),(5,1),(5,9),(5,11),(6,14),(6,15),(8,14),(9,12),(9,13),(10,8),(10,12),(11,6),(11,12),(11,13),(12,14),(12,15),(13,15),(14,7),(15,7)],16)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,3,4,6],[2],[5]]
=> [5,2,1,3,4,6] => ([(0,2),(0,3),(0,4),(0,5),(1,10),(1,17),(2,7),(2,13),(3,9),(3,11),(3,13),(4,1),(4,8),(4,11),(4,13),(5,7),(5,8),(5,9),(7,16),(8,12),(8,16),(8,17),(9,12),(9,16),(10,14),(11,10),(11,12),(11,17),(12,14),(12,15),(13,16),(13,17),(14,6),(15,6),(16,15),(17,14),(17,15)],18)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,2,4,6],[3],[5]]
=> [5,3,1,2,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(1,19),(2,9),(2,12),(2,19),(3,8),(3,12),(3,19),(4,6),(4,8),(4,10),(4,19),(5,6),(5,9),(5,11),(5,19),(6,13),(6,17),(6,18),(8,16),(8,17),(9,16),(9,18),(10,13),(10,17),(11,13),(11,18),(12,16),(13,15),(14,7),(15,7),(16,14),(17,14),(17,15),(18,14),(18,15),(19,16),(19,17),(19,18)],20)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,2,3,6],[4],[5]]
=> [5,4,1,2,3,6] => ([(0,3),(0,4),(0,5),(1,13),(2,6),(2,8),(3,9),(3,10),(4,2),(4,10),(4,11),(5,1),(5,9),(5,11),(6,14),(6,15),(8,14),(9,12),(9,13),(10,8),(10,12),(11,6),(11,12),(11,13),(12,14),(12,15),(13,15),(14,7),(15,7)],16)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,3,4,5],[2],[6]]
=> [6,2,1,3,4,5] => ([(0,3),(0,4),(0,5),(1,14),(2,1),(2,6),(2,7),(3,9),(3,11),(4,9),(4,10),(5,2),(5,10),(5,11),(6,13),(6,14),(7,13),(7,14),(9,12),(10,6),(10,12),(11,7),(11,12),(12,13),(13,8),(14,8)],15)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,2,4,5],[3],[6]]
=> [6,3,1,2,4,5] => ([(0,2),(0,3),(0,4),(0,5),(1,10),(1,17),(2,7),(2,13),(3,9),(3,11),(3,13),(4,1),(4,8),(4,11),(4,13),(5,7),(5,8),(5,9),(7,16),(8,12),(8,16),(8,17),(9,12),(9,16),(10,14),(11,10),(11,12),(11,17),(12,14),(12,15),(13,16),(13,17),(14,6),(15,6),(16,15),(17,14),(17,15)],18)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,2,3,5],[4],[6]]
=> [6,4,1,2,3,5] => ([(0,1),(0,2),(0,4),(0,5),(1,9),(1,16),(2,10),(2,16),(3,6),(3,7),(3,15),(4,9),(4,11),(4,16),(5,3),(5,10),(5,11),(5,16),(6,13),(7,13),(7,14),(9,12),(10,6),(10,15),(11,7),(11,12),(11,15),(12,14),(13,8),(14,8),(15,13),(15,14),(16,12),(16,15)],17)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
Description
The number of simple modules with projective dimension two in the incidence algebra of the poset.
Matching statistic: St000848
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
Mp00125: Posets —dual poset⟶ Posets
St000848: Posets ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 50%
Mp00209: Permutations —pattern poset⟶ Posets
Mp00125: Posets —dual poset⟶ Posets
St000848: Posets ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 50%
Values
[[1]]
=> [1] => ([],1)
=> ([],1)
=> ? = 1 - 1
[[1,2]]
=> [1,2] => ([(0,1)],2)
=> ([(0,1)],2)
=> 0 = 1 - 1
[[1],[2]]
=> [2,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> 0 = 1 - 1
[[1,2,3]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[[1,3],[2]]
=> [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[[1,2],[3]]
=> [3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[[1],[2],[3]]
=> [3,2,1] => ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[[1,2,3,4]]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[[1,3,4],[2]]
=> [2,1,3,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[[1,2,4],[3]]
=> [3,1,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([(0,2),(0,3),(1,6),(2,4),(2,5),(3,1),(3,4),(3,5),(4,6),(5,6)],7)
=> ? ∊ {2,2,3} - 1
[[1,2,3],[4]]
=> [4,1,2,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[[1,3],[2,4]]
=> [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ([(0,1),(0,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,7),(4,7),(5,7),(6,7)],8)
=> ? ∊ {2,2,3} - 1
[[1,2],[3,4]]
=> [3,4,1,2] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 2 = 3 - 1
[[1,4],[2],[3]]
=> [3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[[1,3],[2],[4]]
=> [4,2,1,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([(0,2),(0,3),(1,6),(2,4),(2,5),(3,1),(3,4),(3,5),(4,6),(5,6)],7)
=> ? ∊ {2,2,3} - 1
[[1,2],[3],[4]]
=> [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[[1],[2],[3],[4]]
=> [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[[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 - 1
[[1,3,4,5],[2]]
=> [2,1,3,4,5] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4} - 1
[[1,2,4,5],[3]]
=> [3,1,2,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4} - 1
[[1,2,3,5],[4]]
=> [4,1,2,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4} - 1
[[1,2,3,4],[5]]
=> [5,1,2,3,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4} - 1
[[1,3,5],[2,4]]
=> [2,4,1,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ([(0,2),(0,3),(1,4),(1,9),(1,10),(1,11),(2,5),(2,6),(2,7),(2,8),(3,1),(3,5),(3,6),(3,7),(3,8),(4,13),(5,11),(5,12),(6,10),(6,12),(7,9),(7,12),(8,4),(8,9),(8,10),(8,11),(8,12),(9,13),(10,13),(11,13),(12,13)],14)
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4} - 1
[[1,2,5],[3,4]]
=> [3,4,1,2,5] => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(5,9),(5,10),(6,9),(6,10),(7,10),(8,9),(8,10),(9,4),(10,4)],11)
=> ([(0,2),(0,3),(1,8),(1,9),(2,5),(2,6),(2,7),(3,1),(3,5),(3,6),(3,7),(4,10),(5,8),(5,9),(6,4),(6,9),(7,4),(7,8),(8,10),(9,10)],11)
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4} - 1
[[1,3,4],[2,5]]
=> [2,5,1,3,4] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,9),(2,10),(3,6),(3,9),(4,7),(4,9),(4,10),(5,1),(5,6),(5,7),(5,10),(6,11),(6,12),(7,11),(7,12),(9,12),(10,11),(10,12),(11,8),(12,8)],13)
=> ([(0,2),(0,3),(1,4),(1,5),(1,10),(2,6),(2,7),(2,8),(2,9),(3,1),(3,6),(3,7),(3,8),(3,9),(4,12),(5,12),(6,11),(7,10),(7,11),(8,5),(8,11),(9,4),(9,10),(9,11),(10,12),(11,12)],13)
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4} - 1
[[1,2,4],[3,5]]
=> [3,5,1,2,4] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,7),(2,10),(3,6),(3,10),(4,6),(4,8),(4,10),(5,1),(5,7),(5,8),(5,10),(6,12),(7,11),(7,12),(8,11),(8,12),(10,11),(10,12),(11,9),(12,9)],13)
=> ([(0,2),(0,3),(1,5),(1,10),(2,6),(2,7),(2,8),(2,9),(3,1),(3,6),(3,7),(3,8),(3,9),(4,12),(5,12),(6,11),(7,10),(7,11),(8,4),(8,11),(9,4),(9,5),(9,10),(9,11),(10,12),(11,12)],13)
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4} - 1
[[1,2,3],[4,5]]
=> [4,5,1,2,3] => ([(0,3),(0,4),(1,8),(2,7),(2,8),(3,1),(3,5),(4,2),(4,5),(5,7),(5,8),(7,6),(8,6)],9)
=> ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4} - 1
[[1,4,5],[2],[3]]
=> [3,2,1,4,5] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4} - 1
[[1,3,5],[2],[4]]
=> [4,2,1,3,5] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> ([(0,3),(0,4),(1,8),(1,10),(2,7),(2,9),(3,2),(3,5),(3,6),(4,1),(4,5),(4,6),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,11),(8,11),(9,11),(10,11)],12)
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4} - 1
[[1,2,5],[3],[4]]
=> [4,3,1,2,5] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ([(0,3),(0,4),(1,10),(2,7),(2,8),(3,1),(3,5),(3,6),(4,2),(4,5),(4,6),(5,8),(5,10),(6,7),(6,10),(7,9),(8,9),(10,9)],11)
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4} - 1
[[1,3,4],[2],[5]]
=> [5,2,1,3,4] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ([(0,3),(0,4),(1,10),(2,7),(2,8),(3,1),(3,5),(3,6),(4,2),(4,5),(4,6),(5,8),(5,10),(6,7),(6,10),(7,9),(8,9),(10,9)],11)
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4} - 1
[[1,2,4],[3],[5]]
=> [5,3,1,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> ([(0,3),(0,4),(1,8),(1,10),(2,7),(2,9),(3,2),(3,5),(3,6),(4,1),(4,5),(4,6),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,11),(8,11),(9,11),(10,11)],12)
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4} - 1
[[1,2,3],[4],[5]]
=> [5,4,1,2,3] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4} - 1
[[1,4],[2,5],[3]]
=> [3,2,5,1,4] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,7),(2,10),(3,6),(3,10),(4,6),(4,8),(4,10),(5,1),(5,7),(5,8),(5,10),(6,12),(7,11),(7,12),(8,11),(8,12),(10,11),(10,12),(11,9),(12,9)],13)
=> ([(0,2),(0,3),(1,5),(1,10),(2,6),(2,7),(2,8),(2,9),(3,1),(3,6),(3,7),(3,8),(3,9),(4,12),(5,12),(6,11),(7,10),(7,11),(8,4),(8,11),(9,4),(9,5),(9,10),(9,11),(10,12),(11,12)],13)
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4} - 1
[[1,3],[2,5],[4]]
=> [4,2,5,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,6),(2,9),(2,11),(3,6),(3,9),(3,10),(4,7),(4,9),(4,10),(4,11),(5,7),(5,9),(5,10),(5,11),(6,13),(7,12),(7,13),(9,12),(9,13),(10,12),(10,13),(11,12),(11,13),(12,8),(13,8)],14)
=> ([(0,2),(0,3),(1,9),(1,10),(2,5),(2,6),(2,7),(2,8),(3,1),(3,5),(3,6),(3,7),(3,8),(4,13),(5,11),(5,12),(6,9),(6,10),(6,11),(6,12),(7,4),(7,10),(7,11),(7,12),(8,4),(8,9),(8,11),(8,12),(9,13),(10,13),(11,13),(12,13)],14)
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4} - 1
[[1,2],[3,5],[4]]
=> [4,3,5,1,2] => ([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ([(0,2),(0,3),(1,5),(1,9),(2,6),(2,7),(2,8),(3,1),(3,6),(3,7),(3,8),(4,10),(5,10),(6,9),(7,4),(7,9),(8,4),(8,5),(9,10)],11)
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4} - 1
[[1,3],[2,4],[5]]
=> [5,2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ([(0,2),(0,3),(1,4),(1,9),(1,10),(1,11),(2,5),(2,6),(2,7),(2,8),(3,1),(3,5),(3,6),(3,7),(3,8),(4,13),(5,11),(5,12),(6,10),(6,12),(7,9),(7,12),(8,4),(8,9),(8,10),(8,11),(8,12),(9,13),(10,13),(11,13),(12,13)],14)
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4} - 1
[[1,2],[3,4],[5]]
=> [5,3,4,1,2] => ([(0,1),(0,2),(0,3),(1,7),(1,8),(2,5),(2,8),(3,5),(3,7),(3,8),(5,9),(6,4),(7,6),(7,9),(8,6),(8,9),(9,4)],10)
=> ([(0,2),(0,3),(1,5),(1,8),(2,6),(2,7),(3,1),(3,6),(3,7),(4,9),(5,9),(6,4),(6,8),(7,4),(7,5),(7,8),(8,9)],10)
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4} - 1
[[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4} - 1
[[1,4],[2],[3],[5]]
=> [5,3,2,1,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4} - 1
[[1,3],[2],[4],[5]]
=> [5,4,2,1,3] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4} - 1
[[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4} - 1
[[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[[1,3,4,5,6],[2]]
=> [2,1,3,4,5,6] => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,2,4,5,6],[3]]
=> [3,1,2,4,5,6] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ([(0,2),(0,5),(1,11),(2,6),(2,7),(3,4),(3,9),(3,12),(4,1),(4,8),(4,10),(5,3),(5,6),(5,7),(6,12),(7,9),(7,12),(8,11),(9,8),(9,10),(10,11),(12,10)],13)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,2,3,5,6],[4]]
=> [4,1,2,3,5,6] => ([(0,3),(0,4),(0,5),(1,9),(1,13),(2,8),(2,13),(3,11),(4,2),(4,6),(4,11),(5,1),(5,6),(5,11),(6,8),(6,9),(6,13),(8,10),(8,12),(9,10),(9,12),(10,7),(11,13),(12,7),(13,12)],14)
=> ([(0,2),(0,5),(1,12),(2,6),(2,7),(3,4),(3,8),(3,9),(3,13),(4,1),(4,10),(4,11),(5,3),(5,6),(5,7),(6,9),(6,13),(7,8),(7,13),(8,10),(9,11),(10,12),(11,12),(13,10),(13,11)],14)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,2,3,4,6],[5]]
=> [5,1,2,3,4,6] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ([(0,2),(0,5),(1,11),(2,6),(2,7),(3,4),(3,9),(3,12),(4,1),(4,8),(4,10),(5,3),(5,6),(5,7),(6,12),(7,9),(7,12),(8,11),(9,8),(9,10),(10,11),(12,10)],13)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,3,5,6],[2,4]]
=> [2,4,1,3,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,6),(1,14),(1,18),(2,13),(2,14),(2,18),(3,12),(3,14),(3,18),(4,11),(4,14),(4,18),(5,8),(5,9),(5,10),(5,16),(6,5),(6,11),(6,12),(6,13),(6,18),(8,15),(8,19),(9,15),(9,19),(10,15),(10,19),(11,8),(11,16),(11,17),(12,9),(12,16),(12,17),(13,10),(13,16),(13,17),(14,17),(15,7),(16,15),(16,19),(17,19),(18,16),(18,17),(19,7)],20)
=> ([(0,3),(0,4),(1,2),(1,10),(1,14),(1,15),(1,16),(2,5),(2,11),(2,12),(2,13),(3,6),(3,7),(3,8),(3,9),(4,1),(4,6),(4,7),(4,8),(4,9),(5,19),(6,16),(6,17),(7,15),(7,17),(8,14),(8,17),(9,10),(9,14),(9,15),(9,16),(9,17),(10,5),(10,11),(10,12),(10,13),(10,18),(11,19),(12,19),(13,19),(14,11),(14,18),(15,12),(15,18),(16,13),(16,18),(17,18),(18,19)],20)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,2,5,6],[3,4]]
=> [3,4,1,2,5,6] => ([(0,2),(0,3),(0,4),(1,8),(1,9),(2,1),(2,10),(2,11),(3,6),(3,7),(3,11),(4,6),(4,7),(4,10),(6,14),(7,12),(7,14),(8,13),(8,15),(9,13),(9,15),(10,8),(10,12),(10,14),(11,9),(11,12),(11,14),(12,13),(12,15),(13,5),(14,15),(15,5)],16)
=> ([(0,1),(0,2),(1,6),(1,7),(1,9),(2,4),(2,6),(2,7),(2,9),(3,12),(3,13),(4,3),(4,8),(4,14),(4,15),(5,11),(6,5),(6,15),(7,5),(7,14),(8,12),(8,13),(9,8),(9,14),(9,15),(11,10),(12,10),(13,10),(14,11),(14,12),(15,11),(15,13)],16)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,3,4,6],[2,5]]
=> [2,5,1,3,4,6] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,12),(1,20),(1,21),(2,13),(2,14),(3,8),(3,13),(3,15),(4,9),(4,10),(4,13),(4,15),(5,7),(5,8),(5,9),(5,14),(6,1),(6,7),(6,10),(6,14),(6,15),(7,12),(7,16),(7,18),(7,20),(8,18),(8,21),(9,16),(9,18),(9,21),(10,16),(10,20),(10,21),(12,17),(12,19),(13,21),(14,20),(14,21),(15,18),(15,20),(15,21),(16,17),(16,19),(17,11),(18,17),(18,19),(19,11),(20,17),(20,19),(21,19)],22)
=> ([(0,3),(0,4),(1,6),(1,16),(1,19),(2,1),(2,5),(2,9),(2,12),(2,13),(2,14),(2,15),(3,7),(3,8),(3,10),(3,11),(4,2),(4,7),(4,8),(4,10),(4,11),(5,16),(5,18),(6,20),(7,13),(7,17),(8,12),(8,14),(8,17),(9,6),(9,18),(9,21),(10,5),(10,14),(10,15),(10,17),(11,9),(11,12),(11,13),(11,15),(11,17),(12,19),(12,21),(13,21),(14,18),(14,19),(15,16),(15,19),(15,21),(16,20),(17,18),(17,21),(18,20),(19,20),(21,20)],22)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,2,4,6],[3,5]]
=> [3,5,1,2,4,6] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(1,18),(1,21),(2,11),(2,12),(2,19),(3,9),(3,13),(3,19),(4,10),(4,13),(4,19),(5,10),(5,12),(5,14),(5,19),(6,1),(6,9),(6,11),(6,14),(6,19),(8,17),(8,20),(9,18),(9,21),(10,16),(10,21),(11,15),(11,18),(11,21),(12,15),(12,16),(13,21),(14,8),(14,15),(14,16),(14,18),(15,17),(15,20),(16,17),(16,20),(17,7),(18,17),(18,20),(19,16),(19,18),(19,21),(20,7),(21,20)],22)
=> ([(0,3),(0,4),(1,15),(1,16),(2,1),(2,6),(2,7),(2,12),(2,13),(2,14),(3,8),(3,9),(3,10),(3,11),(4,2),(4,8),(4,9),(4,10),(4,11),(5,17),(5,19),(6,15),(6,21),(7,16),(7,19),(8,12),(8,18),(9,5),(9,14),(9,18),(10,6),(10,12),(10,13),(10,14),(10,18),(11,5),(11,7),(11,13),(11,18),(12,21),(13,15),(13,16),(13,17),(13,19),(13,21),(14,17),(14,21),(15,20),(16,20),(17,20),(18,19),(18,21),(19,20),(21,20)],22)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,2,3,6],[4,5]]
=> [4,5,1,2,3,6] => ([(0,3),(0,4),(0,5),(1,14),(2,6),(2,8),(2,14),(3,9),(3,10),(4,2),(4,10),(4,11),(5,1),(5,9),(5,11),(6,13),(6,15),(8,13),(8,15),(9,12),(9,14),(10,8),(10,12),(11,6),(11,12),(11,14),(12,13),(12,15),(13,7),(14,15),(15,7)],16)
=> ([(0,3),(0,4),(1,13),(2,1),(2,8),(2,11),(2,12),(3,6),(3,9),(3,10),(4,2),(4,6),(4,9),(4,10),(5,15),(6,11),(6,12),(7,5),(7,14),(8,5),(8,13),(9,7),(9,11),(10,7),(10,8),(10,12),(11,14),(12,13),(12,14),(13,15),(14,15)],16)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,3,4,5],[2,6]]
=> [2,6,1,3,4,5] => ([(0,2),(0,3),(0,4),(0,6),(1,15),(1,17),(2,12),(2,13),(3,7),(3,12),(4,8),(4,12),(4,13),(5,1),(5,10),(5,11),(5,14),(6,5),(6,7),(6,8),(6,13),(7,10),(7,16),(8,11),(8,14),(8,16),(10,15),(10,17),(11,15),(11,17),(12,16),(13,14),(13,16),(14,15),(14,17),(15,9),(16,17),(17,9)],18)
=> ([(0,3),(0,4),(1,2),(1,11),(1,12),(1,14),(2,5),(2,6),(2,13),(3,7),(3,8),(3,9),(3,10),(4,1),(4,7),(4,8),(4,9),(4,10),(5,17),(6,17),(7,15),(8,14),(8,15),(9,12),(9,14),(9,15),(10,11),(10,15),(11,6),(11,16),(12,5),(12,13),(12,16),(13,17),(14,13),(14,16),(15,16),(16,17)],18)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,2,4,5],[3,6]]
=> [3,6,1,2,4,5] => ([(0,3),(0,4),(0,5),(0,6),(1,11),(1,18),(2,12),(2,17),(2,18),(3,7),(3,14),(4,1),(4,10),(4,13),(4,14),(5,2),(5,9),(5,13),(5,14),(6,7),(6,9),(6,10),(7,17),(9,15),(9,17),(10,15),(10,17),(10,18),(11,16),(11,19),(12,16),(12,19),(13,11),(13,12),(13,15),(13,18),(14,17),(14,18),(15,16),(15,19),(16,8),(17,19),(18,16),(18,19),(19,8)],20)
=> ([(0,3),(0,4),(1,7),(1,15),(2,1),(2,6),(2,9),(2,13),(2,14),(3,8),(3,10),(3,11),(3,12),(4,2),(4,8),(4,10),(4,11),(4,12),(5,18),(6,15),(6,19),(7,16),(8,14),(8,17),(9,7),(9,18),(9,19),(10,5),(10,17),(11,6),(11,13),(11,17),(12,5),(12,9),(12,13),(12,14),(12,17),(13,15),(13,18),(14,19),(15,16),(17,18),(17,19),(18,16),(19,16)],20)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,2,3,5],[4,6]]
=> [4,6,1,2,3,5] => ([(0,2),(0,3),(0,4),(0,6),(1,15),(1,17),(2,7),(2,14),(3,9),(3,14),(4,9),(4,10),(4,14),(5,1),(5,11),(5,12),(5,16),(6,5),(6,7),(6,10),(6,14),(7,11),(7,16),(9,13),(10,12),(10,13),(10,16),(11,15),(11,17),(12,15),(12,17),(13,17),(14,13),(14,16),(15,8),(16,15),(16,17),(17,8)],18)
=> ([(0,3),(0,4),(1,2),(1,12),(1,14),(2,5),(2,13),(3,7),(3,8),(3,10),(3,11),(4,1),(4,7),(4,8),(4,10),(4,11),(5,17),(6,17),(7,15),(8,14),(8,15),(9,6),(9,16),(10,9),(10,15),(11,9),(11,12),(11,14),(11,15),(12,5),(12,6),(12,13),(12,16),(13,17),(14,13),(14,16),(15,16),(16,17)],18)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,2,3,4],[5,6]]
=> [5,6,1,2,3,4] => ([(0,4),(0,5),(1,7),(2,9),(2,11),(3,2),(3,10),(4,3),(4,6),(5,1),(5,6),(6,7),(6,10),(7,11),(9,8),(10,9),(10,11),(11,8)],12)
=> ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,4,5,6],[2],[3]]
=> [3,2,1,4,5,6] => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,3,5,6],[2],[4]]
=> [4,2,1,3,5,6] => ([(0,1),(0,2),(0,4),(0,5),(1,9),(1,16),(2,10),(2,16),(3,6),(3,7),(3,15),(4,9),(4,11),(4,16),(5,3),(5,10),(5,11),(5,16),(6,13),(7,13),(7,14),(9,12),(10,6),(10,15),(11,7),(11,12),(11,15),(12,14),(13,8),(14,8),(15,13),(15,14),(16,12),(16,15)],17)
=> ([(0,4),(0,5),(1,3),(1,11),(1,14),(2,9),(2,13),(3,6),(3,12),(4,1),(4,8),(4,10),(5,2),(5,8),(5,10),(6,15),(7,15),(8,13),(8,14),(9,7),(9,16),(10,9),(10,11),(10,13),(10,14),(11,6),(11,7),(11,12),(11,16),(12,15),(13,16),(14,12),(14,16),(16,15)],17)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,2,5,6],[3],[4]]
=> [4,3,1,2,5,6] => ([(0,3),(0,4),(0,5),(1,13),(2,6),(2,8),(3,9),(3,10),(4,2),(4,10),(4,11),(5,1),(5,9),(5,11),(6,14),(6,15),(8,14),(9,12),(9,13),(10,8),(10,12),(11,6),(11,12),(11,13),(12,14),(12,15),(13,15),(14,7),(15,7)],16)
=> ([(0,4),(0,5),(1,14),(2,1),(2,9),(2,12),(3,7),(3,8),(4,3),(4,10),(4,11),(5,2),(5,10),(5,11),(6,15),(7,13),(8,6),(8,13),(9,6),(9,14),(10,7),(10,12),(11,8),(11,9),(11,12),(12,13),(12,14),(13,15),(14,15)],16)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,3,4,6],[2],[5]]
=> [5,2,1,3,4,6] => ([(0,2),(0,3),(0,4),(0,5),(1,10),(1,17),(2,7),(2,13),(3,9),(3,11),(3,13),(4,1),(4,8),(4,11),(4,13),(5,7),(5,8),(5,9),(7,16),(8,12),(8,16),(8,17),(9,12),(9,16),(10,14),(11,10),(11,12),(11,17),(12,14),(12,15),(13,16),(13,17),(14,6),(15,6),(16,15),(17,14),(17,15)],18)
=> ([(0,4),(0,5),(1,8),(1,15),(2,7),(2,12),(3,1),(3,6),(3,9),(3,13),(4,2),(4,10),(4,11),(5,3),(5,10),(5,11),(6,14),(6,15),(7,17),(8,16),(9,8),(9,14),(9,17),(10,6),(10,12),(10,13),(11,7),(11,9),(11,12),(11,13),(12,14),(12,17),(13,15),(13,17),(14,16),(15,16),(17,16)],18)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,2,4,6],[3],[5]]
=> [5,3,1,2,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(1,19),(2,9),(2,12),(2,19),(3,8),(3,12),(3,19),(4,6),(4,8),(4,10),(4,19),(5,6),(5,9),(5,11),(5,19),(6,13),(6,17),(6,18),(8,16),(8,17),(9,16),(9,18),(10,13),(10,17),(11,13),(11,18),(12,16),(13,15),(14,7),(15,7),(16,14),(17,14),(17,15),(18,14),(18,15),(19,16),(19,17),(19,18)],20)
=> ([(0,4),(0,5),(1,13),(1,14),(2,8),(2,9),(2,15),(3,1),(3,6),(3,7),(3,16),(4,2),(4,10),(4,11),(5,3),(5,10),(5,11),(6,14),(6,18),(7,13),(7,17),(8,12),(8,17),(9,12),(9,18),(10,7),(10,8),(10,15),(10,16),(11,6),(11,9),(11,15),(11,16),(12,19),(13,19),(14,19),(15,17),(15,18),(16,12),(16,13),(16,14),(16,17),(16,18),(17,19),(18,19)],20)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,2,3,6],[4],[5]]
=> [5,4,1,2,3,6] => ([(0,3),(0,4),(0,5),(1,13),(2,6),(2,8),(3,9),(3,10),(4,2),(4,10),(4,11),(5,1),(5,9),(5,11),(6,14),(6,15),(8,14),(9,12),(9,13),(10,8),(10,12),(11,6),(11,12),(11,13),(12,14),(12,15),(13,15),(14,7),(15,7)],16)
=> ([(0,4),(0,5),(1,14),(2,1),(2,9),(2,12),(3,7),(3,8),(4,3),(4,10),(4,11),(5,2),(5,10),(5,11),(6,15),(7,13),(8,6),(8,13),(9,6),(9,14),(10,7),(10,12),(11,8),(11,9),(11,12),(12,13),(12,14),(13,15),(14,15)],16)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,3,4,5],[2],[6]]
=> [6,2,1,3,4,5] => ([(0,3),(0,4),(0,5),(1,14),(2,1),(2,6),(2,7),(3,9),(3,11),(4,9),(4,10),(5,2),(5,10),(5,11),(6,13),(6,14),(7,13),(7,14),(9,12),(10,6),(10,12),(11,7),(11,12),(12,13),(13,8),(14,8)],15)
=> ([(0,4),(0,5),(1,12),(2,3),(2,10),(2,11),(3,6),(3,7),(4,1),(4,8),(4,9),(5,2),(5,8),(5,9),(6,13),(7,13),(8,10),(8,12),(9,11),(9,12),(10,7),(10,14),(11,6),(11,14),(12,14),(14,13)],15)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,2,4,5],[3],[6]]
=> [6,3,1,2,4,5] => ([(0,2),(0,3),(0,4),(0,5),(1,10),(1,17),(2,7),(2,13),(3,9),(3,11),(3,13),(4,1),(4,8),(4,11),(4,13),(5,7),(5,8),(5,9),(7,16),(8,12),(8,16),(8,17),(9,12),(9,16),(10,14),(11,10),(11,12),(11,17),(12,14),(12,15),(13,16),(13,17),(14,6),(15,6),(16,15),(17,14),(17,15)],18)
=> ([(0,4),(0,5),(1,8),(1,15),(2,7),(2,12),(3,1),(3,6),(3,9),(3,13),(4,2),(4,10),(4,11),(5,3),(5,10),(5,11),(6,14),(6,15),(7,17),(8,16),(9,8),(9,14),(9,17),(10,6),(10,12),(10,13),(11,7),(11,9),(11,12),(11,13),(12,14),(12,17),(13,15),(13,17),(14,16),(15,16),(17,16)],18)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
Description
The balance constant multiplied with the number of linear extensions of a poset.
A pair of elements $x,y$ of a poset is $\alpha$-balanced if the proportion $P(x,y)$ of linear extensions where $x$ comes before $y$ is between $\alpha$ and $1-\alpha$. The balance constant of a poset is $\max\min(P(x,y), P(y,x)).$
Kislitsyn [1] conjectured that every poset which is not a chain is $1/3$-balanced. Brightwell, Felsner and Trotter [2] show that it is at least $(1-\sqrt 5)/10$-balanced.
Olson and Sagan [3] exhibit various posets that are $1/2$-balanced.
Matching statistic: St000849
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
Mp00125: Posets —dual poset⟶ Posets
St000849: Posets ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 50%
Mp00209: Permutations —pattern poset⟶ Posets
Mp00125: Posets —dual poset⟶ Posets
St000849: Posets ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 50%
Values
[[1]]
=> [1] => ([],1)
=> ([],1)
=> ? = 1 - 1
[[1,2]]
=> [1,2] => ([(0,1)],2)
=> ([(0,1)],2)
=> 0 = 1 - 1
[[1],[2]]
=> [2,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> 0 = 1 - 1
[[1,2,3]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[[1,3],[2]]
=> [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[[1,2],[3]]
=> [3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[[1],[2],[3]]
=> [3,2,1] => ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[[1,2,3,4]]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[[1,3,4],[2]]
=> [2,1,3,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[[1,2,4],[3]]
=> [3,1,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([(0,2),(0,3),(1,6),(2,4),(2,5),(3,1),(3,4),(3,5),(4,6),(5,6)],7)
=> ? ∊ {2,2,3} - 1
[[1,2,3],[4]]
=> [4,1,2,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[[1,3],[2,4]]
=> [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ([(0,1),(0,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,7),(4,7),(5,7),(6,7)],8)
=> ? ∊ {2,2,3} - 1
[[1,2],[3,4]]
=> [3,4,1,2] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 2 = 3 - 1
[[1,4],[2],[3]]
=> [3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[[1,3],[2],[4]]
=> [4,2,1,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([(0,2),(0,3),(1,6),(2,4),(2,5),(3,1),(3,4),(3,5),(4,6),(5,6)],7)
=> ? ∊ {2,2,3} - 1
[[1,2],[3],[4]]
=> [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[[1],[2],[3],[4]]
=> [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[[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 - 1
[[1,3,4,5],[2]]
=> [2,1,3,4,5] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4} - 1
[[1,2,4,5],[3]]
=> [3,1,2,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4} - 1
[[1,2,3,5],[4]]
=> [4,1,2,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4} - 1
[[1,2,3,4],[5]]
=> [5,1,2,3,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4} - 1
[[1,3,5],[2,4]]
=> [2,4,1,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ([(0,2),(0,3),(1,4),(1,9),(1,10),(1,11),(2,5),(2,6),(2,7),(2,8),(3,1),(3,5),(3,6),(3,7),(3,8),(4,13),(5,11),(5,12),(6,10),(6,12),(7,9),(7,12),(8,4),(8,9),(8,10),(8,11),(8,12),(9,13),(10,13),(11,13),(12,13)],14)
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4} - 1
[[1,2,5],[3,4]]
=> [3,4,1,2,5] => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(5,9),(5,10),(6,9),(6,10),(7,10),(8,9),(8,10),(9,4),(10,4)],11)
=> ([(0,2),(0,3),(1,8),(1,9),(2,5),(2,6),(2,7),(3,1),(3,5),(3,6),(3,7),(4,10),(5,8),(5,9),(6,4),(6,9),(7,4),(7,8),(8,10),(9,10)],11)
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4} - 1
[[1,3,4],[2,5]]
=> [2,5,1,3,4] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,9),(2,10),(3,6),(3,9),(4,7),(4,9),(4,10),(5,1),(5,6),(5,7),(5,10),(6,11),(6,12),(7,11),(7,12),(9,12),(10,11),(10,12),(11,8),(12,8)],13)
=> ([(0,2),(0,3),(1,4),(1,5),(1,10),(2,6),(2,7),(2,8),(2,9),(3,1),(3,6),(3,7),(3,8),(3,9),(4,12),(5,12),(6,11),(7,10),(7,11),(8,5),(8,11),(9,4),(9,10),(9,11),(10,12),(11,12)],13)
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4} - 1
[[1,2,4],[3,5]]
=> [3,5,1,2,4] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,7),(2,10),(3,6),(3,10),(4,6),(4,8),(4,10),(5,1),(5,7),(5,8),(5,10),(6,12),(7,11),(7,12),(8,11),(8,12),(10,11),(10,12),(11,9),(12,9)],13)
=> ([(0,2),(0,3),(1,5),(1,10),(2,6),(2,7),(2,8),(2,9),(3,1),(3,6),(3,7),(3,8),(3,9),(4,12),(5,12),(6,11),(7,10),(7,11),(8,4),(8,11),(9,4),(9,5),(9,10),(9,11),(10,12),(11,12)],13)
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4} - 1
[[1,2,3],[4,5]]
=> [4,5,1,2,3] => ([(0,3),(0,4),(1,8),(2,7),(2,8),(3,1),(3,5),(4,2),(4,5),(5,7),(5,8),(7,6),(8,6)],9)
=> ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4} - 1
[[1,4,5],[2],[3]]
=> [3,2,1,4,5] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4} - 1
[[1,3,5],[2],[4]]
=> [4,2,1,3,5] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> ([(0,3),(0,4),(1,8),(1,10),(2,7),(2,9),(3,2),(3,5),(3,6),(4,1),(4,5),(4,6),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,11),(8,11),(9,11),(10,11)],12)
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4} - 1
[[1,2,5],[3],[4]]
=> [4,3,1,2,5] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ([(0,3),(0,4),(1,10),(2,7),(2,8),(3,1),(3,5),(3,6),(4,2),(4,5),(4,6),(5,8),(5,10),(6,7),(6,10),(7,9),(8,9),(10,9)],11)
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4} - 1
[[1,3,4],[2],[5]]
=> [5,2,1,3,4] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ([(0,3),(0,4),(1,10),(2,7),(2,8),(3,1),(3,5),(3,6),(4,2),(4,5),(4,6),(5,8),(5,10),(6,7),(6,10),(7,9),(8,9),(10,9)],11)
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4} - 1
[[1,2,4],[3],[5]]
=> [5,3,1,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> ([(0,3),(0,4),(1,8),(1,10),(2,7),(2,9),(3,2),(3,5),(3,6),(4,1),(4,5),(4,6),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,11),(8,11),(9,11),(10,11)],12)
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4} - 1
[[1,2,3],[4],[5]]
=> [5,4,1,2,3] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4} - 1
[[1,4],[2,5],[3]]
=> [3,2,5,1,4] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,7),(2,10),(3,6),(3,10),(4,6),(4,8),(4,10),(5,1),(5,7),(5,8),(5,10),(6,12),(7,11),(7,12),(8,11),(8,12),(10,11),(10,12),(11,9),(12,9)],13)
=> ([(0,2),(0,3),(1,5),(1,10),(2,6),(2,7),(2,8),(2,9),(3,1),(3,6),(3,7),(3,8),(3,9),(4,12),(5,12),(6,11),(7,10),(7,11),(8,4),(8,11),(9,4),(9,5),(9,10),(9,11),(10,12),(11,12)],13)
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4} - 1
[[1,3],[2,5],[4]]
=> [4,2,5,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,6),(2,9),(2,11),(3,6),(3,9),(3,10),(4,7),(4,9),(4,10),(4,11),(5,7),(5,9),(5,10),(5,11),(6,13),(7,12),(7,13),(9,12),(9,13),(10,12),(10,13),(11,12),(11,13),(12,8),(13,8)],14)
=> ([(0,2),(0,3),(1,9),(1,10),(2,5),(2,6),(2,7),(2,8),(3,1),(3,5),(3,6),(3,7),(3,8),(4,13),(5,11),(5,12),(6,9),(6,10),(6,11),(6,12),(7,4),(7,10),(7,11),(7,12),(8,4),(8,9),(8,11),(8,12),(9,13),(10,13),(11,13),(12,13)],14)
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4} - 1
[[1,2],[3,5],[4]]
=> [4,3,5,1,2] => ([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ([(0,2),(0,3),(1,5),(1,9),(2,6),(2,7),(2,8),(3,1),(3,6),(3,7),(3,8),(4,10),(5,10),(6,9),(7,4),(7,9),(8,4),(8,5),(9,10)],11)
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4} - 1
[[1,3],[2,4],[5]]
=> [5,2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ([(0,2),(0,3),(1,4),(1,9),(1,10),(1,11),(2,5),(2,6),(2,7),(2,8),(3,1),(3,5),(3,6),(3,7),(3,8),(4,13),(5,11),(5,12),(6,10),(6,12),(7,9),(7,12),(8,4),(8,9),(8,10),(8,11),(8,12),(9,13),(10,13),(11,13),(12,13)],14)
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4} - 1
[[1,2],[3,4],[5]]
=> [5,3,4,1,2] => ([(0,1),(0,2),(0,3),(1,7),(1,8),(2,5),(2,8),(3,5),(3,7),(3,8),(5,9),(6,4),(7,6),(7,9),(8,6),(8,9),(9,4)],10)
=> ([(0,2),(0,3),(1,5),(1,8),(2,6),(2,7),(3,1),(3,6),(3,7),(4,9),(5,9),(6,4),(6,8),(7,4),(7,5),(7,8),(8,9)],10)
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4} - 1
[[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4} - 1
[[1,4],[2],[3],[5]]
=> [5,3,2,1,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4} - 1
[[1,3],[2],[4],[5]]
=> [5,4,2,1,3] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4} - 1
[[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? ∊ {2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4} - 1
[[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[[1,3,4,5,6],[2]]
=> [2,1,3,4,5,6] => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,2,4,5,6],[3]]
=> [3,1,2,4,5,6] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ([(0,2),(0,5),(1,11),(2,6),(2,7),(3,4),(3,9),(3,12),(4,1),(4,8),(4,10),(5,3),(5,6),(5,7),(6,12),(7,9),(7,12),(8,11),(9,8),(9,10),(10,11),(12,10)],13)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,2,3,5,6],[4]]
=> [4,1,2,3,5,6] => ([(0,3),(0,4),(0,5),(1,9),(1,13),(2,8),(2,13),(3,11),(4,2),(4,6),(4,11),(5,1),(5,6),(5,11),(6,8),(6,9),(6,13),(8,10),(8,12),(9,10),(9,12),(10,7),(11,13),(12,7),(13,12)],14)
=> ([(0,2),(0,5),(1,12),(2,6),(2,7),(3,4),(3,8),(3,9),(3,13),(4,1),(4,10),(4,11),(5,3),(5,6),(5,7),(6,9),(6,13),(7,8),(7,13),(8,10),(9,11),(10,12),(11,12),(13,10),(13,11)],14)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,2,3,4,6],[5]]
=> [5,1,2,3,4,6] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ([(0,2),(0,5),(1,11),(2,6),(2,7),(3,4),(3,9),(3,12),(4,1),(4,8),(4,10),(5,3),(5,6),(5,7),(6,12),(7,9),(7,12),(8,11),(9,8),(9,10),(10,11),(12,10)],13)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,3,5,6],[2,4]]
=> [2,4,1,3,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,6),(1,14),(1,18),(2,13),(2,14),(2,18),(3,12),(3,14),(3,18),(4,11),(4,14),(4,18),(5,8),(5,9),(5,10),(5,16),(6,5),(6,11),(6,12),(6,13),(6,18),(8,15),(8,19),(9,15),(9,19),(10,15),(10,19),(11,8),(11,16),(11,17),(12,9),(12,16),(12,17),(13,10),(13,16),(13,17),(14,17),(15,7),(16,15),(16,19),(17,19),(18,16),(18,17),(19,7)],20)
=> ([(0,3),(0,4),(1,2),(1,10),(1,14),(1,15),(1,16),(2,5),(2,11),(2,12),(2,13),(3,6),(3,7),(3,8),(3,9),(4,1),(4,6),(4,7),(4,8),(4,9),(5,19),(6,16),(6,17),(7,15),(7,17),(8,14),(8,17),(9,10),(9,14),(9,15),(9,16),(9,17),(10,5),(10,11),(10,12),(10,13),(10,18),(11,19),(12,19),(13,19),(14,11),(14,18),(15,12),(15,18),(16,13),(16,18),(17,18),(18,19)],20)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,2,5,6],[3,4]]
=> [3,4,1,2,5,6] => ([(0,2),(0,3),(0,4),(1,8),(1,9),(2,1),(2,10),(2,11),(3,6),(3,7),(3,11),(4,6),(4,7),(4,10),(6,14),(7,12),(7,14),(8,13),(8,15),(9,13),(9,15),(10,8),(10,12),(10,14),(11,9),(11,12),(11,14),(12,13),(12,15),(13,5),(14,15),(15,5)],16)
=> ([(0,1),(0,2),(1,6),(1,7),(1,9),(2,4),(2,6),(2,7),(2,9),(3,12),(3,13),(4,3),(4,8),(4,14),(4,15),(5,11),(6,5),(6,15),(7,5),(7,14),(8,12),(8,13),(9,8),(9,14),(9,15),(11,10),(12,10),(13,10),(14,11),(14,12),(15,11),(15,13)],16)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,3,4,6],[2,5]]
=> [2,5,1,3,4,6] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,12),(1,20),(1,21),(2,13),(2,14),(3,8),(3,13),(3,15),(4,9),(4,10),(4,13),(4,15),(5,7),(5,8),(5,9),(5,14),(6,1),(6,7),(6,10),(6,14),(6,15),(7,12),(7,16),(7,18),(7,20),(8,18),(8,21),(9,16),(9,18),(9,21),(10,16),(10,20),(10,21),(12,17),(12,19),(13,21),(14,20),(14,21),(15,18),(15,20),(15,21),(16,17),(16,19),(17,11),(18,17),(18,19),(19,11),(20,17),(20,19),(21,19)],22)
=> ([(0,3),(0,4),(1,6),(1,16),(1,19),(2,1),(2,5),(2,9),(2,12),(2,13),(2,14),(2,15),(3,7),(3,8),(3,10),(3,11),(4,2),(4,7),(4,8),(4,10),(4,11),(5,16),(5,18),(6,20),(7,13),(7,17),(8,12),(8,14),(8,17),(9,6),(9,18),(9,21),(10,5),(10,14),(10,15),(10,17),(11,9),(11,12),(11,13),(11,15),(11,17),(12,19),(12,21),(13,21),(14,18),(14,19),(15,16),(15,19),(15,21),(16,20),(17,18),(17,21),(18,20),(19,20),(21,20)],22)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,2,4,6],[3,5]]
=> [3,5,1,2,4,6] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(1,18),(1,21),(2,11),(2,12),(2,19),(3,9),(3,13),(3,19),(4,10),(4,13),(4,19),(5,10),(5,12),(5,14),(5,19),(6,1),(6,9),(6,11),(6,14),(6,19),(8,17),(8,20),(9,18),(9,21),(10,16),(10,21),(11,15),(11,18),(11,21),(12,15),(12,16),(13,21),(14,8),(14,15),(14,16),(14,18),(15,17),(15,20),(16,17),(16,20),(17,7),(18,17),(18,20),(19,16),(19,18),(19,21),(20,7),(21,20)],22)
=> ([(0,3),(0,4),(1,15),(1,16),(2,1),(2,6),(2,7),(2,12),(2,13),(2,14),(3,8),(3,9),(3,10),(3,11),(4,2),(4,8),(4,9),(4,10),(4,11),(5,17),(5,19),(6,15),(6,21),(7,16),(7,19),(8,12),(8,18),(9,5),(9,14),(9,18),(10,6),(10,12),(10,13),(10,14),(10,18),(11,5),(11,7),(11,13),(11,18),(12,21),(13,15),(13,16),(13,17),(13,19),(13,21),(14,17),(14,21),(15,20),(16,20),(17,20),(18,19),(18,21),(19,20),(21,20)],22)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,2,3,6],[4,5]]
=> [4,5,1,2,3,6] => ([(0,3),(0,4),(0,5),(1,14),(2,6),(2,8),(2,14),(3,9),(3,10),(4,2),(4,10),(4,11),(5,1),(5,9),(5,11),(6,13),(6,15),(8,13),(8,15),(9,12),(9,14),(10,8),(10,12),(11,6),(11,12),(11,14),(12,13),(12,15),(13,7),(14,15),(15,7)],16)
=> ([(0,3),(0,4),(1,13),(2,1),(2,8),(2,11),(2,12),(3,6),(3,9),(3,10),(4,2),(4,6),(4,9),(4,10),(5,15),(6,11),(6,12),(7,5),(7,14),(8,5),(8,13),(9,7),(9,11),(10,7),(10,8),(10,12),(11,14),(12,13),(12,14),(13,15),(14,15)],16)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,3,4,5],[2,6]]
=> [2,6,1,3,4,5] => ([(0,2),(0,3),(0,4),(0,6),(1,15),(1,17),(2,12),(2,13),(3,7),(3,12),(4,8),(4,12),(4,13),(5,1),(5,10),(5,11),(5,14),(6,5),(6,7),(6,8),(6,13),(7,10),(7,16),(8,11),(8,14),(8,16),(10,15),(10,17),(11,15),(11,17),(12,16),(13,14),(13,16),(14,15),(14,17),(15,9),(16,17),(17,9)],18)
=> ([(0,3),(0,4),(1,2),(1,11),(1,12),(1,14),(2,5),(2,6),(2,13),(3,7),(3,8),(3,9),(3,10),(4,1),(4,7),(4,8),(4,9),(4,10),(5,17),(6,17),(7,15),(8,14),(8,15),(9,12),(9,14),(9,15),(10,11),(10,15),(11,6),(11,16),(12,5),(12,13),(12,16),(13,17),(14,13),(14,16),(15,16),(16,17)],18)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,2,4,5],[3,6]]
=> [3,6,1,2,4,5] => ([(0,3),(0,4),(0,5),(0,6),(1,11),(1,18),(2,12),(2,17),(2,18),(3,7),(3,14),(4,1),(4,10),(4,13),(4,14),(5,2),(5,9),(5,13),(5,14),(6,7),(6,9),(6,10),(7,17),(9,15),(9,17),(10,15),(10,17),(10,18),(11,16),(11,19),(12,16),(12,19),(13,11),(13,12),(13,15),(13,18),(14,17),(14,18),(15,16),(15,19),(16,8),(17,19),(18,16),(18,19),(19,8)],20)
=> ([(0,3),(0,4),(1,7),(1,15),(2,1),(2,6),(2,9),(2,13),(2,14),(3,8),(3,10),(3,11),(3,12),(4,2),(4,8),(4,10),(4,11),(4,12),(5,18),(6,15),(6,19),(7,16),(8,14),(8,17),(9,7),(9,18),(9,19),(10,5),(10,17),(11,6),(11,13),(11,17),(12,5),(12,9),(12,13),(12,14),(12,17),(13,15),(13,18),(14,19),(15,16),(17,18),(17,19),(18,16),(19,16)],20)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,2,3,5],[4,6]]
=> [4,6,1,2,3,5] => ([(0,2),(0,3),(0,4),(0,6),(1,15),(1,17),(2,7),(2,14),(3,9),(3,14),(4,9),(4,10),(4,14),(5,1),(5,11),(5,12),(5,16),(6,5),(6,7),(6,10),(6,14),(7,11),(7,16),(9,13),(10,12),(10,13),(10,16),(11,15),(11,17),(12,15),(12,17),(13,17),(14,13),(14,16),(15,8),(16,15),(16,17),(17,8)],18)
=> ([(0,3),(0,4),(1,2),(1,12),(1,14),(2,5),(2,13),(3,7),(3,8),(3,10),(3,11),(4,1),(4,7),(4,8),(4,10),(4,11),(5,17),(6,17),(7,15),(8,14),(8,15),(9,6),(9,16),(10,9),(10,15),(11,9),(11,12),(11,14),(11,15),(12,5),(12,6),(12,13),(12,16),(13,17),(14,13),(14,16),(15,16),(16,17)],18)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,2,3,4],[5,6]]
=> [5,6,1,2,3,4] => ([(0,4),(0,5),(1,7),(2,9),(2,11),(3,2),(3,10),(4,3),(4,6),(5,1),(5,6),(6,7),(6,10),(7,11),(9,8),(10,9),(10,11),(11,8)],12)
=> ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,4,5,6],[2],[3]]
=> [3,2,1,4,5,6] => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,3,5,6],[2],[4]]
=> [4,2,1,3,5,6] => ([(0,1),(0,2),(0,4),(0,5),(1,9),(1,16),(2,10),(2,16),(3,6),(3,7),(3,15),(4,9),(4,11),(4,16),(5,3),(5,10),(5,11),(5,16),(6,13),(7,13),(7,14),(9,12),(10,6),(10,15),(11,7),(11,12),(11,15),(12,14),(13,8),(14,8),(15,13),(15,14),(16,12),(16,15)],17)
=> ([(0,4),(0,5),(1,3),(1,11),(1,14),(2,9),(2,13),(3,6),(3,12),(4,1),(4,8),(4,10),(5,2),(5,8),(5,10),(6,15),(7,15),(8,13),(8,14),(9,7),(9,16),(10,9),(10,11),(10,13),(10,14),(11,6),(11,7),(11,12),(11,16),(12,15),(13,16),(14,12),(14,16),(16,15)],17)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,2,5,6],[3],[4]]
=> [4,3,1,2,5,6] => ([(0,3),(0,4),(0,5),(1,13),(2,6),(2,8),(3,9),(3,10),(4,2),(4,10),(4,11),(5,1),(5,9),(5,11),(6,14),(6,15),(8,14),(9,12),(9,13),(10,8),(10,12),(11,6),(11,12),(11,13),(12,14),(12,15),(13,15),(14,7),(15,7)],16)
=> ([(0,4),(0,5),(1,14),(2,1),(2,9),(2,12),(3,7),(3,8),(4,3),(4,10),(4,11),(5,2),(5,10),(5,11),(6,15),(7,13),(8,6),(8,13),(9,6),(9,14),(10,7),(10,12),(11,8),(11,9),(11,12),(12,13),(12,14),(13,15),(14,15)],16)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,3,4,6],[2],[5]]
=> [5,2,1,3,4,6] => ([(0,2),(0,3),(0,4),(0,5),(1,10),(1,17),(2,7),(2,13),(3,9),(3,11),(3,13),(4,1),(4,8),(4,11),(4,13),(5,7),(5,8),(5,9),(7,16),(8,12),(8,16),(8,17),(9,12),(9,16),(10,14),(11,10),(11,12),(11,17),(12,14),(12,15),(13,16),(13,17),(14,6),(15,6),(16,15),(17,14),(17,15)],18)
=> ([(0,4),(0,5),(1,8),(1,15),(2,7),(2,12),(3,1),(3,6),(3,9),(3,13),(4,2),(4,10),(4,11),(5,3),(5,10),(5,11),(6,14),(6,15),(7,17),(8,16),(9,8),(9,14),(9,17),(10,6),(10,12),(10,13),(11,7),(11,9),(11,12),(11,13),(12,14),(12,17),(13,15),(13,17),(14,16),(15,16),(17,16)],18)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,2,4,6],[3],[5]]
=> [5,3,1,2,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(1,19),(2,9),(2,12),(2,19),(3,8),(3,12),(3,19),(4,6),(4,8),(4,10),(4,19),(5,6),(5,9),(5,11),(5,19),(6,13),(6,17),(6,18),(8,16),(8,17),(9,16),(9,18),(10,13),(10,17),(11,13),(11,18),(12,16),(13,15),(14,7),(15,7),(16,14),(17,14),(17,15),(18,14),(18,15),(19,16),(19,17),(19,18)],20)
=> ([(0,4),(0,5),(1,13),(1,14),(2,8),(2,9),(2,15),(3,1),(3,6),(3,7),(3,16),(4,2),(4,10),(4,11),(5,3),(5,10),(5,11),(6,14),(6,18),(7,13),(7,17),(8,12),(8,17),(9,12),(9,18),(10,7),(10,8),(10,15),(10,16),(11,6),(11,9),(11,15),(11,16),(12,19),(13,19),(14,19),(15,17),(15,18),(16,12),(16,13),(16,14),(16,17),(16,18),(17,19),(18,19)],20)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,2,3,6],[4],[5]]
=> [5,4,1,2,3,6] => ([(0,3),(0,4),(0,5),(1,13),(2,6),(2,8),(3,9),(3,10),(4,2),(4,10),(4,11),(5,1),(5,9),(5,11),(6,14),(6,15),(8,14),(9,12),(9,13),(10,8),(10,12),(11,6),(11,12),(11,13),(12,14),(12,15),(13,15),(14,7),(15,7)],16)
=> ([(0,4),(0,5),(1,14),(2,1),(2,9),(2,12),(3,7),(3,8),(4,3),(4,10),(4,11),(5,2),(5,10),(5,11),(6,15),(7,13),(8,6),(8,13),(9,6),(9,14),(10,7),(10,12),(11,8),(11,9),(11,12),(12,13),(12,14),(13,15),(14,15)],16)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,3,4,5],[2],[6]]
=> [6,2,1,3,4,5] => ([(0,3),(0,4),(0,5),(1,14),(2,1),(2,6),(2,7),(3,9),(3,11),(4,9),(4,10),(5,2),(5,10),(5,11),(6,13),(6,14),(7,13),(7,14),(9,12),(10,6),(10,12),(11,7),(11,12),(12,13),(13,8),(14,8)],15)
=> ([(0,4),(0,5),(1,12),(2,3),(2,10),(2,11),(3,6),(3,7),(4,1),(4,8),(4,9),(5,2),(5,8),(5,9),(6,13),(7,13),(8,10),(8,12),(9,11),(9,12),(10,7),(10,14),(11,6),(11,14),(12,14),(14,13)],15)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,2,4,5],[3],[6]]
=> [6,3,1,2,4,5] => ([(0,2),(0,3),(0,4),(0,5),(1,10),(1,17),(2,7),(2,13),(3,9),(3,11),(3,13),(4,1),(4,8),(4,11),(4,13),(5,7),(5,8),(5,9),(7,16),(8,12),(8,16),(8,17),(9,12),(9,16),(10,14),(11,10),(11,12),(11,17),(12,14),(12,15),(13,16),(13,17),(14,6),(15,6),(16,15),(17,14),(17,15)],18)
=> ([(0,4),(0,5),(1,8),(1,15),(2,7),(2,12),(3,1),(3,6),(3,9),(3,13),(4,2),(4,10),(4,11),(5,3),(5,10),(5,11),(6,14),(6,15),(7,17),(8,16),(9,8),(9,14),(9,17),(10,6),(10,12),(10,13),(11,7),(11,9),(11,12),(11,13),(12,14),(12,17),(13,15),(13,17),(14,16),(15,16),(17,16)],18)
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
Description
The number of 1/3-balanced pairs in a poset.
A pair of elements $x,y$ of a poset is $\alpha$-balanced if the proportion of linear extensions where $x$ comes before $y$ is between $\alpha$ and $1-\alpha$.
Kislitsyn [1] conjectured that every poset which is not a chain has a $1/3$-balanced pair. Brightwell, Felsner and Trotter [2] show that at least a $(1-\sqrt 5)/10$-balanced pair exists in posets which are not chains.
Olson and Sagan [3] show that posets corresponding to skew Ferrers diagrams have a $1/3$-balanced pair.
Matching statistic: St001811
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001811: Permutations ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 50%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001811: Permutations ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 50%
Values
[[1]]
=> [1] => [1,0]
=> [2,1] => 0 = 1 - 1
[[1,2]]
=> [2] => [1,1,0,0]
=> [2,3,1] => 0 = 1 - 1
[[1],[2]]
=> [1,1] => [1,0,1,0]
=> [3,1,2] => 0 = 1 - 1
[[1,2,3]]
=> [3] => [1,1,1,0,0,0]
=> [2,3,4,1] => 0 = 1 - 1
[[1,3],[2]]
=> [1,2] => [1,0,1,1,0,0]
=> [3,1,4,2] => 1 = 2 - 1
[[1,2],[3]]
=> [2,1] => [1,1,0,0,1,0]
=> [2,4,1,3] => 1 = 2 - 1
[[1],[2],[3]]
=> [1,1,1] => [1,0,1,0,1,0]
=> [4,1,2,3] => 0 = 1 - 1
[[1,2,3,4]]
=> [4] => [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 0 = 1 - 1
[[1,3,4],[2]]
=> [1,3] => [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 1 = 2 - 1
[[1,2,4],[3]]
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 2 = 3 - 1
[[1,2,3],[4]]
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 2 = 3 - 1
[[1,3],[2,4]]
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 2 = 3 - 1
[[1,2],[3,4]]
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 2 = 3 - 1
[[1,4],[2],[3]]
=> [1,1,2] => [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 2 = 3 - 1
[[1,3],[2],[4]]
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 2 = 3 - 1
[[1,2],[3],[4]]
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 1 = 2 - 1
[[1],[2],[3],[4]]
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 0 = 1 - 1
[[1,2,3,4,5]]
=> [5] => [1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4} - 1
[[1,3,4,5],[2]]
=> [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4} - 1
[[1,2,4,5],[3]]
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4} - 1
[[1,2,3,5],[4]]
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4} - 1
[[1,2,3,4],[5]]
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4} - 1
[[1,3,5],[2,4]]
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4} - 1
[[1,2,5],[3,4]]
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4} - 1
[[1,3,4],[2,5]]
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4} - 1
[[1,2,4],[3,5]]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4} - 1
[[1,2,3],[4,5]]
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4} - 1
[[1,4,5],[2],[3]]
=> [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4} - 1
[[1,3,5],[2],[4]]
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4} - 1
[[1,2,5],[3],[4]]
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4} - 1
[[1,3,4],[2],[5]]
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4} - 1
[[1,2,4],[3],[5]]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4} - 1
[[1,2,3],[4],[5]]
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4} - 1
[[1,4],[2,5],[3]]
=> [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4} - 1
[[1,3],[2,5],[4]]
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4} - 1
[[1,2],[3,5],[4]]
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4} - 1
[[1,3],[2,4],[5]]
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4} - 1
[[1,2],[3,4],[5]]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4} - 1
[[1,5],[2],[3],[4]]
=> [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4} - 1
[[1,4],[2],[3],[5]]
=> [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4} - 1
[[1,3],[2],[4],[5]]
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4} - 1
[[1,2],[3],[4],[5]]
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4} - 1
[[1],[2],[3],[4],[5]]
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4} - 1
[[1,2,3,4,5,6]]
=> [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,3,4,5,6],[2]]
=> [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,2,4,5,6],[3]]
=> [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,4,1,5,6,7,3] => ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,2,3,5,6],[4]]
=> [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,3,5,1,6,7,4] => ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,2,3,4,6],[5]]
=> [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,2,3,4,5],[6]]
=> [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,3,5,6],[2,4]]
=> [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,1,5,2,6,7,4] => ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,2,5,6],[3,4]]
=> [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,4,1,5,6,7,3] => ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,3,4,6],[2,5]]
=> [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [3,1,4,6,2,7,5] => ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,2,4,6],[3,5]]
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,4,1,6,3,7,5] => ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,2,3,6],[4,5]]
=> [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,3,5,1,6,7,4] => ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,3,4,5],[2,6]]
=> [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,2,4,5],[3,6]]
=> [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,4,1,5,7,3,6] => ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,2,3,5],[4,6]]
=> [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,3,5,1,7,4,6] => ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,2,3,4],[5,6]]
=> [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,4,5,6],[2],[3]]
=> [1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,1,2,5,6,7,3] => ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,3,5,6],[2],[4]]
=> [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,1,5,2,6,7,4] => ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,2,5,6],[3],[4]]
=> [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,5,1,3,6,7,4] => ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,3,4,6],[2],[5]]
=> [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [3,1,4,6,2,7,5] => ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,2,4,6],[3],[5]]
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,4,1,6,3,7,5] => ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,2,3,6],[4],[5]]
=> [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [2,3,6,1,4,7,5] => ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,3,4,5],[2],[6]]
=> [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,2,4,5],[3],[6]]
=> [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,4,1,5,7,3,6] => ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,2,3,5],[4],[6]]
=> [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,3,5,1,7,4,6] => ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
Description
The Castelnuovo-Mumford regularity of a permutation.
The ''Castelnuovo-Mumford regularity'' of a permutation $\sigma$ is the ''Castelnuovo-Mumford regularity'' of the ''matrix Schubert variety'' $X_\sigma$.
Equivalently, it is the difference between the degrees of the ''Grothendieck polynomial'' and the ''Schubert polynomial'' for $\sigma$. It can be computed by subtracting the ''Coxeter length'' [[St000018]] from the ''Rajchgot index'' [[St001759]].
Matching statistic: St001822
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001822: Signed permutations ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 50%
Mp00064: Permutations —reverse⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001822: Signed permutations ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 50%
Values
[[1]]
=> [1] => [1] => [1] => 0 = 1 - 1
[[1,2]]
=> [1,2] => [2,1] => [2,1] => 0 = 1 - 1
[[1],[2]]
=> [2,1] => [1,2] => [1,2] => 0 = 1 - 1
[[1,2,3]]
=> [1,2,3] => [3,2,1] => [3,2,1] => 1 = 2 - 1
[[1,3],[2]]
=> [2,1,3] => [3,1,2] => [3,1,2] => 0 = 1 - 1
[[1,2],[3]]
=> [3,1,2] => [2,1,3] => [2,1,3] => 1 = 2 - 1
[[1],[2],[3]]
=> [3,2,1] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[[1,2,3,4]]
=> [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 2 = 3 - 1
[[1,3,4],[2]]
=> [2,1,3,4] => [4,3,1,2] => [4,3,1,2] => 1 = 2 - 1
[[1,2,4],[3]]
=> [3,1,2,4] => [4,2,1,3] => [4,2,1,3] => 2 = 3 - 1
[[1,2,3],[4]]
=> [4,1,2,3] => [3,2,1,4] => [3,2,1,4] => 2 = 3 - 1
[[1,3],[2,4]]
=> [2,4,1,3] => [3,1,4,2] => [3,1,4,2] => 1 = 2 - 1
[[1,2],[3,4]]
=> [3,4,1,2] => [2,1,4,3] => [2,1,4,3] => 2 = 3 - 1
[[1,4],[2],[3]]
=> [3,2,1,4] => [4,1,2,3] => [4,1,2,3] => 0 = 1 - 1
[[1,3],[2],[4]]
=> [4,2,1,3] => [3,1,2,4] => [3,1,2,4] => 2 = 3 - 1
[[1,2],[3],[4]]
=> [4,3,1,2] => [2,1,3,4] => [2,1,3,4] => 2 = 3 - 1
[[1],[2],[3],[4]]
=> [4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[[1,2,3,4,5]]
=> [1,2,3,4,5] => [5,4,3,2,1] => [5,4,3,2,1] => ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4} - 1
[[1,3,4,5],[2]]
=> [2,1,3,4,5] => [5,4,3,1,2] => [5,4,3,1,2] => ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4} - 1
[[1,2,4,5],[3]]
=> [3,1,2,4,5] => [5,4,2,1,3] => [5,4,2,1,3] => ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4} - 1
[[1,2,3,5],[4]]
=> [4,1,2,3,5] => [5,3,2,1,4] => [5,3,2,1,4] => ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4} - 1
[[1,2,3,4],[5]]
=> [5,1,2,3,4] => [4,3,2,1,5] => [4,3,2,1,5] => ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4} - 1
[[1,3,5],[2,4]]
=> [2,4,1,3,5] => [5,3,1,4,2] => [5,3,1,4,2] => ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4} - 1
[[1,2,5],[3,4]]
=> [3,4,1,2,5] => [5,2,1,4,3] => [5,2,1,4,3] => ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4} - 1
[[1,3,4],[2,5]]
=> [2,5,1,3,4] => [4,3,1,5,2] => [4,3,1,5,2] => ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4} - 1
[[1,2,4],[3,5]]
=> [3,5,1,2,4] => [4,2,1,5,3] => [4,2,1,5,3] => ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4} - 1
[[1,2,3],[4,5]]
=> [4,5,1,2,3] => [3,2,1,5,4] => [3,2,1,5,4] => ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4} - 1
[[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [5,4,1,2,3] => [5,4,1,2,3] => ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4} - 1
[[1,3,5],[2],[4]]
=> [4,2,1,3,5] => [5,3,1,2,4] => [5,3,1,2,4] => ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4} - 1
[[1,2,5],[3],[4]]
=> [4,3,1,2,5] => [5,2,1,3,4] => [5,2,1,3,4] => ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4} - 1
[[1,3,4],[2],[5]]
=> [5,2,1,3,4] => [4,3,1,2,5] => [4,3,1,2,5] => ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4} - 1
[[1,2,4],[3],[5]]
=> [5,3,1,2,4] => [4,2,1,3,5] => [4,2,1,3,5] => ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4} - 1
[[1,2,3],[4],[5]]
=> [5,4,1,2,3] => [3,2,1,4,5] => [3,2,1,4,5] => ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4} - 1
[[1,4],[2,5],[3]]
=> [3,2,5,1,4] => [4,1,5,2,3] => [4,1,5,2,3] => ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4} - 1
[[1,3],[2,5],[4]]
=> [4,2,5,1,3] => [3,1,5,2,4] => [3,1,5,2,4] => ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4} - 1
[[1,2],[3,5],[4]]
=> [4,3,5,1,2] => [2,1,5,3,4] => [2,1,5,3,4] => ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4} - 1
[[1,3],[2,4],[5]]
=> [5,2,4,1,3] => [3,1,4,2,5] => [3,1,4,2,5] => ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4} - 1
[[1,2],[3,4],[5]]
=> [5,3,4,1,2] => [2,1,4,3,5] => [2,1,4,3,5] => ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4} - 1
[[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => [5,1,2,3,4] => [5,1,2,3,4] => ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4} - 1
[[1,4],[2],[3],[5]]
=> [5,3,2,1,4] => [4,1,2,3,5] => [4,1,2,3,5] => ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4} - 1
[[1,3],[2],[4],[5]]
=> [5,4,2,1,3] => [3,1,2,4,5] => [3,1,2,4,5] => ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4} - 1
[[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => [2,1,3,4,5] => [2,1,3,4,5] => ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4} - 1
[[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4} - 1
[[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,3,4,5,6],[2]]
=> [2,1,3,4,5,6] => [6,5,4,3,1,2] => [6,5,4,3,1,2] => ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,2,4,5,6],[3]]
=> [3,1,2,4,5,6] => [6,5,4,2,1,3] => [6,5,4,2,1,3] => ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,2,3,5,6],[4]]
=> [4,1,2,3,5,6] => [6,5,3,2,1,4] => [6,5,3,2,1,4] => ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,2,3,4,6],[5]]
=> [5,1,2,3,4,6] => [6,4,3,2,1,5] => [6,4,3,2,1,5] => ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => [5,4,3,2,1,6] => [5,4,3,2,1,6] => ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,3,5,6],[2,4]]
=> [2,4,1,3,5,6] => [6,5,3,1,4,2] => [6,5,3,1,4,2] => ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,2,5,6],[3,4]]
=> [3,4,1,2,5,6] => [6,5,2,1,4,3] => [6,5,2,1,4,3] => ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,3,4,6],[2,5]]
=> [2,5,1,3,4,6] => [6,4,3,1,5,2] => [6,4,3,1,5,2] => ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,2,4,6],[3,5]]
=> [3,5,1,2,4,6] => [6,4,2,1,5,3] => [6,4,2,1,5,3] => ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,2,3,6],[4,5]]
=> [4,5,1,2,3,6] => [6,3,2,1,5,4] => [6,3,2,1,5,4] => ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,3,4,5],[2,6]]
=> [2,6,1,3,4,5] => [5,4,3,1,6,2] => [5,4,3,1,6,2] => ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,2,4,5],[3,6]]
=> [3,6,1,2,4,5] => [5,4,2,1,6,3] => [5,4,2,1,6,3] => ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,2,3,5],[4,6]]
=> [4,6,1,2,3,5] => [5,3,2,1,6,4] => [5,3,2,1,6,4] => ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,2,3,4],[5,6]]
=> [5,6,1,2,3,4] => [4,3,2,1,6,5] => [4,3,2,1,6,5] => ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,4,5,6],[2],[3]]
=> [3,2,1,4,5,6] => [6,5,4,1,2,3] => [6,5,4,1,2,3] => ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,3,5,6],[2],[4]]
=> [4,2,1,3,5,6] => [6,5,3,1,2,4] => [6,5,3,1,2,4] => ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,2,5,6],[3],[4]]
=> [4,3,1,2,5,6] => [6,5,2,1,3,4] => [6,5,2,1,3,4] => ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,3,4,6],[2],[5]]
=> [5,2,1,3,4,6] => [6,4,3,1,2,5] => [6,4,3,1,2,5] => ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,2,4,6],[3],[5]]
=> [5,3,1,2,4,6] => [6,4,2,1,3,5] => [6,4,2,1,3,5] => ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,2,3,6],[4],[5]]
=> [5,4,1,2,3,6] => [6,3,2,1,4,5] => [6,3,2,1,4,5] => ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,3,4,5],[2],[6]]
=> [6,2,1,3,4,5] => [5,4,3,1,2,6] => [5,4,3,1,2,6] => ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,2,4,5],[3],[6]]
=> [6,3,1,2,4,5] => [5,4,2,1,3,6] => [5,4,2,1,3,6] => ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
[[1,2,3,5],[4],[6]]
=> [6,4,1,2,3,5] => [5,3,2,1,4,6] => [5,3,2,1,4,6] => ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12} - 1
Description
The number of alignments of a signed permutation.
An alignment of a signed permutation $n\in\mathfrak H_n$ is either a nesting alignment, [[St001866]], an alignment of type EN, [[St001867]], or an alignment of type NE, [[St001868]].
Let $\operatorname{al}$ be the number of alignments of $\pi$, let \operatorname{cr} be the number of crossings, [[St001862]], let \operatorname{wex} be the number of weak excedances, [[St001863]], and let \operatorname{neg} be the number of negative entries, [[St001429]]. Then, $\operatorname{al}+\operatorname{cr}=(n-\operatorname{wex})(\operatorname{wex}-1+\operatorname{neg})+\binom{\operatorname{neg}{2}$.
Matching statistic: St001686
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00082: Standard tableaux —to Gelfand-Tsetlin pattern⟶ Gelfand-Tsetlin patterns
St001686: Gelfand-Tsetlin patterns ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 50%
St001686: Gelfand-Tsetlin patterns ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 50%
Values
[[1]]
=> [[1]]
=> ? = 1
[[1,2]]
=> [[2,0],[1]]
=> 1
[[1],[2]]
=> [[1,1],[1]]
=> 1
[[1,2,3]]
=> [[3,0,0],[2,0],[1]]
=> 1
[[1,3],[2]]
=> [[2,1,0],[1,1],[1]]
=> 2
[[1,2],[3]]
=> [[2,1,0],[2,0],[1]]
=> 2
[[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> 1
[[1,2,3,4]]
=> [[4,0,0,0],[3,0,0],[2,0],[1]]
=> 1
[[1,3,4],[2]]
=> [[3,1,0,0],[2,1,0],[1,1],[1]]
=> 3
[[1,2,4],[3]]
=> [[3,1,0,0],[2,1,0],[2,0],[1]]
=> 3
[[1,2,3],[4]]
=> [[3,1,0,0],[3,0,0],[2,0],[1]]
=> 3
[[1,3],[2,4]]
=> [[2,2,0,0],[2,1,0],[1,1],[1]]
=> 2
[[1,2],[3,4]]
=> [[2,2,0,0],[2,1,0],[2,0],[1]]
=> 2
[[1,4],[2],[3]]
=> [[2,1,1,0],[1,1,1],[1,1],[1]]
=> 3
[[1,3],[2],[4]]
=> [[2,1,1,0],[2,1,0],[1,1],[1]]
=> 3
[[1,2],[3],[4]]
=> [[2,1,1,0],[2,1,0],[2,0],[1]]
=> 3
[[1],[2],[3],[4]]
=> [[1,1,1,1],[1,1,1],[1,1],[1]]
=> 1
[[1,2,3,4,5]]
=> [[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4}
[[1,3,4,5],[2]]
=> [[4,1,0,0,0],[3,1,0,0],[2,1,0],[1,1],[1]]
=> ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4}
[[1,2,4,5],[3]]
=> [[4,1,0,0,0],[3,1,0,0],[2,1,0],[2,0],[1]]
=> ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4}
[[1,2,3,5],[4]]
=> [[4,1,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4}
[[1,2,3,4],[5]]
=> [[4,1,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4}
[[1,3,5],[2,4]]
=> [[3,2,0,0,0],[2,2,0,0],[2,1,0],[1,1],[1]]
=> ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4}
[[1,2,5],[3,4]]
=> [[3,2,0,0,0],[2,2,0,0],[2,1,0],[2,0],[1]]
=> ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4}
[[1,3,4],[2,5]]
=> [[3,2,0,0,0],[3,1,0,0],[2,1,0],[1,1],[1]]
=> ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4}
[[1,2,4],[3,5]]
=> [[3,2,0,0,0],[3,1,0,0],[2,1,0],[2,0],[1]]
=> ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4}
[[1,2,3],[4,5]]
=> [[3,2,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4}
[[1,4,5],[2],[3]]
=> [[3,1,1,0,0],[2,1,1,0],[1,1,1],[1,1],[1]]
=> ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4}
[[1,3,5],[2],[4]]
=> [[3,1,1,0,0],[2,1,1,0],[2,1,0],[1,1],[1]]
=> ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4}
[[1,2,5],[3],[4]]
=> [[3,1,1,0,0],[2,1,1,0],[2,1,0],[2,0],[1]]
=> ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4}
[[1,3,4],[2],[5]]
=> [[3,1,1,0,0],[3,1,0,0],[2,1,0],[1,1],[1]]
=> ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4}
[[1,2,4],[3],[5]]
=> [[3,1,1,0,0],[3,1,0,0],[2,1,0],[2,0],[1]]
=> ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4}
[[1,2,3],[4],[5]]
=> [[3,1,1,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4}
[[1,4],[2,5],[3]]
=> [[2,2,1,0,0],[2,1,1,0],[1,1,1],[1,1],[1]]
=> ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4}
[[1,3],[2,5],[4]]
=> [[2,2,1,0,0],[2,1,1,0],[2,1,0],[1,1],[1]]
=> ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4}
[[1,2],[3,5],[4]]
=> [[2,2,1,0,0],[2,1,1,0],[2,1,0],[2,0],[1]]
=> ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4}
[[1,3],[2,4],[5]]
=> [[2,2,1,0,0],[2,2,0,0],[2,1,0],[1,1],[1]]
=> ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4}
[[1,2],[3,4],[5]]
=> [[2,2,1,0,0],[2,2,0,0],[2,1,0],[2,0],[1]]
=> ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4}
[[1,5],[2],[3],[4]]
=> [[2,1,1,1,0],[1,1,1,1],[1,1,1],[1,1],[1]]
=> ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4}
[[1,4],[2],[3],[5]]
=> [[2,1,1,1,0],[2,1,1,0],[1,1,1],[1,1],[1]]
=> ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4}
[[1,3],[2],[4],[5]]
=> [[2,1,1,1,0],[2,1,1,0],[2,1,0],[1,1],[1]]
=> ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4}
[[1,2],[3],[4],[5]]
=> [[2,1,1,1,0],[2,1,1,0],[2,1,0],[2,0],[1]]
=> ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4}
[[1],[2],[3],[4],[5]]
=> [[1,1,1,1,1],[1,1,1,1],[1,1,1],[1,1],[1]]
=> ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4}
[[1,2,3,4,5,6]]
=> [[6,0,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12}
[[1,3,4,5,6],[2]]
=> [[5,1,0,0,0,0],[4,1,0,0,0],[3,1,0,0],[2,1,0],[1,1],[1]]
=> ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12}
[[1,2,4,5,6],[3]]
=> [[5,1,0,0,0,0],[4,1,0,0,0],[3,1,0,0],[2,1,0],[2,0],[1]]
=> ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12}
[[1,2,3,5,6],[4]]
=> [[5,1,0,0,0,0],[4,1,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12}
[[1,2,3,4,6],[5]]
=> [[5,1,0,0,0,0],[4,1,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12}
[[1,2,3,4,5],[6]]
=> [[5,1,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12}
[[1,3,5,6],[2,4]]
=> [[4,2,0,0,0,0],[3,2,0,0,0],[2,2,0,0],[2,1,0],[1,1],[1]]
=> ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12}
[[1,2,5,6],[3,4]]
=> [[4,2,0,0,0,0],[3,2,0,0,0],[2,2,0,0],[2,1,0],[2,0],[1]]
=> ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12}
[[1,3,4,6],[2,5]]
=> [[4,2,0,0,0,0],[3,2,0,0,0],[3,1,0,0],[2,1,0],[1,1],[1]]
=> ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12}
[[1,2,4,6],[3,5]]
=> [[4,2,0,0,0,0],[3,2,0,0,0],[3,1,0,0],[2,1,0],[2,0],[1]]
=> ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12}
[[1,2,3,6],[4,5]]
=> [[4,2,0,0,0,0],[3,2,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12}
[[1,3,4,5],[2,6]]
=> [[4,2,0,0,0,0],[4,1,0,0,0],[3,1,0,0],[2,1,0],[1,1],[1]]
=> ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12}
[[1,2,4,5],[3,6]]
=> [[4,2,0,0,0,0],[4,1,0,0,0],[3,1,0,0],[2,1,0],[2,0],[1]]
=> ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12}
[[1,2,3,5],[4,6]]
=> [[4,2,0,0,0,0],[4,1,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12}
[[1,2,3,4],[5,6]]
=> [[4,2,0,0,0,0],[4,1,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12}
[[1,4,5,6],[2],[3]]
=> [[4,1,1,0,0,0],[3,1,1,0,0],[2,1,1,0],[1,1,1],[1,1],[1]]
=> ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12}
[[1,3,5,6],[2],[4]]
=> [[4,1,1,0,0,0],[3,1,1,0,0],[2,1,1,0],[2,1,0],[1,1],[1]]
=> ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12}
[[1,2,5,6],[3],[4]]
=> [[4,1,1,0,0,0],[3,1,1,0,0],[2,1,1,0],[2,1,0],[2,0],[1]]
=> ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12}
[[1,3,4,6],[2],[5]]
=> [[4,1,1,0,0,0],[3,1,1,0,0],[3,1,0,0],[2,1,0],[1,1],[1]]
=> ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12}
[[1,2,4,6],[3],[5]]
=> [[4,1,1,0,0,0],[3,1,1,0,0],[3,1,0,0],[2,1,0],[2,0],[1]]
=> ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12}
[[1,2,3,6],[4],[5]]
=> [[4,1,1,0,0,0],[3,1,1,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12}
[[1,3,4,5],[2],[6]]
=> [[4,1,1,0,0,0],[4,1,0,0,0],[3,1,0,0],[2,1,0],[1,1],[1]]
=> ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12}
[[1,2,4,5],[3],[6]]
=> [[4,1,1,0,0,0],[4,1,0,0,0],[3,1,0,0],[2,1,0],[2,0],[1]]
=> ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12}
Description
The order of promotion on a Gelfand-Tsetlin pattern.
Matching statistic: St001207
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00106: Standard tableaux —catabolism⟶ Standard tableaux
Mp00284: Standard tableaux —rows⟶ Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
St001207: Permutations ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 50%
Mp00284: Standard tableaux —rows⟶ Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
St001207: Permutations ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 50%
Values
[[1]]
=> [[1]]
=> {{1}}
=> [1] => ? = 1
[[1,2]]
=> [[1,2]]
=> {{1,2}}
=> [2,1] => 1
[[1],[2]]
=> [[1,2]]
=> {{1,2}}
=> [2,1] => 1
[[1,2,3]]
=> [[1,2,3]]
=> {{1,2,3}}
=> [2,3,1] => 2
[[1,3],[2]]
=> [[1,2],[3]]
=> {{1,2},{3}}
=> [2,1,3] => 1
[[1,2],[3]]
=> [[1,2,3]]
=> {{1,2,3}}
=> [2,3,1] => 2
[[1],[2],[3]]
=> [[1,2],[3]]
=> {{1,2},{3}}
=> [2,1,3] => 1
[[1,2,3,4]]
=> [[1,2,3,4]]
=> {{1,2,3,4}}
=> [2,3,4,1] => 3
[[1,3,4],[2]]
=> [[1,2,4],[3]]
=> {{1,2,4},{3}}
=> [2,4,3,1] => 3
[[1,2,4],[3]]
=> [[1,2,3],[4]]
=> {{1,2,3},{4}}
=> [2,3,1,4] => 2
[[1,2,3],[4]]
=> [[1,2,3,4]]
=> {{1,2,3,4}}
=> [2,3,4,1] => 3
[[1,3],[2,4]]
=> [[1,2,4],[3]]
=> {{1,2,4},{3}}
=> [2,4,3,1] => 3
[[1,2],[3,4]]
=> [[1,2,3,4]]
=> {{1,2,3,4}}
=> [2,3,4,1] => 3
[[1,4],[2],[3]]
=> [[1,2],[3],[4]]
=> {{1,2},{3},{4}}
=> [2,1,3,4] => 1
[[1,3],[2],[4]]
=> [[1,2,4],[3]]
=> {{1,2,4},{3}}
=> [2,4,3,1] => 3
[[1,2],[3],[4]]
=> [[1,2,3],[4]]
=> {{1,2,3},{4}}
=> [2,3,1,4] => 2
[[1],[2],[3],[4]]
=> [[1,2],[3],[4]]
=> {{1,2},{3},{4}}
=> [2,1,3,4] => 1
[[1,2,3,4,5]]
=> [[1,2,3,4,5]]
=> {{1,2,3,4,5}}
=> [2,3,4,5,1] => ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4}
[[1,3,4,5],[2]]
=> [[1,2,4,5],[3]]
=> {{1,2,4,5},{3}}
=> [2,4,3,5,1] => ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4}
[[1,2,4,5],[3]]
=> [[1,2,3,5],[4]]
=> {{1,2,3,5},{4}}
=> [2,3,5,4,1] => ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4}
[[1,2,3,5],[4]]
=> [[1,2,3,4],[5]]
=> {{1,2,3,4},{5}}
=> [2,3,4,1,5] => ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4}
[[1,2,3,4],[5]]
=> [[1,2,3,4,5]]
=> {{1,2,3,4,5}}
=> [2,3,4,5,1] => ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4}
[[1,3,5],[2,4]]
=> [[1,2,4],[3,5]]
=> {{1,2,4},{3,5}}
=> [2,4,5,1,3] => ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4}
[[1,2,5],[3,4]]
=> [[1,2,3,4],[5]]
=> {{1,2,3,4},{5}}
=> [2,3,4,1,5] => ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4}
[[1,3,4],[2,5]]
=> [[1,2,4,5],[3]]
=> {{1,2,4,5},{3}}
=> [2,4,3,5,1] => ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4}
[[1,2,4],[3,5]]
=> [[1,2,3,5],[4]]
=> {{1,2,3,5},{4}}
=> [2,3,5,4,1] => ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4}
[[1,2,3],[4,5]]
=> [[1,2,3,4,5]]
=> {{1,2,3,4,5}}
=> [2,3,4,5,1] => ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4}
[[1,4,5],[2],[3]]
=> [[1,2,5],[3],[4]]
=> {{1,2,5},{3},{4}}
=> [2,5,3,4,1] => ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4}
[[1,3,5],[2],[4]]
=> [[1,2,4],[3],[5]]
=> {{1,2,4},{3},{5}}
=> [2,4,3,1,5] => ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4}
[[1,2,5],[3],[4]]
=> [[1,2,3],[4],[5]]
=> {{1,2,3},{4},{5}}
=> [2,3,1,4,5] => ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4}
[[1,3,4],[2],[5]]
=> [[1,2,4,5],[3]]
=> {{1,2,4,5},{3}}
=> [2,4,3,5,1] => ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4}
[[1,2,4],[3],[5]]
=> [[1,2,3,5],[4]]
=> {{1,2,3,5},{4}}
=> [2,3,5,4,1] => ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4}
[[1,2,3],[4],[5]]
=> [[1,2,3,4],[5]]
=> {{1,2,3,4},{5}}
=> [2,3,4,1,5] => ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4}
[[1,4],[2,5],[3]]
=> [[1,2,5],[3],[4]]
=> {{1,2,5},{3},{4}}
=> [2,5,3,4,1] => ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4}
[[1,3],[2,5],[4]]
=> [[1,2,4,5],[3]]
=> {{1,2,4,5},{3}}
=> [2,4,3,5,1] => ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4}
[[1,2],[3,5],[4]]
=> [[1,2,3,5],[4]]
=> {{1,2,3,5},{4}}
=> [2,3,5,4,1] => ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4}
[[1,3],[2,4],[5]]
=> [[1,2,4],[3,5]]
=> {{1,2,4},{3,5}}
=> [2,4,5,1,3] => ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4}
[[1,2],[3,4],[5]]
=> [[1,2,3,4],[5]]
=> {{1,2,3,4},{5}}
=> [2,3,4,1,5] => ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4}
[[1,5],[2],[3],[4]]
=> [[1,2],[3],[4],[5]]
=> {{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4}
[[1,4],[2],[3],[5]]
=> [[1,2,5],[3],[4]]
=> {{1,2,5},{3},{4}}
=> [2,5,3,4,1] => ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4}
[[1,3],[2],[4],[5]]
=> [[1,2,4],[3],[5]]
=> {{1,2,4},{3},{5}}
=> [2,4,3,1,5] => ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4}
[[1,2],[3],[4],[5]]
=> [[1,2,3],[4],[5]]
=> {{1,2,3},{4},{5}}
=> [2,3,1,4,5] => ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4}
[[1],[2],[3],[4],[5]]
=> [[1,2],[3],[4],[5]]
=> {{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => ? ∊ {1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4}
[[1,2,3,4,5,6]]
=> [[1,2,3,4,5,6]]
=> {{1,2,3,4,5,6}}
=> [2,3,4,5,6,1] => ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12}
[[1,3,4,5,6],[2]]
=> [[1,2,4,5,6],[3]]
=> {{1,2,4,5,6},{3}}
=> [2,4,3,5,6,1] => ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12}
[[1,2,4,5,6],[3]]
=> [[1,2,3,5,6],[4]]
=> {{1,2,3,5,6},{4}}
=> [2,3,5,4,6,1] => ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12}
[[1,2,3,5,6],[4]]
=> [[1,2,3,4,6],[5]]
=> {{1,2,3,4,6},{5}}
=> [2,3,4,6,5,1] => ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12}
[[1,2,3,4,6],[5]]
=> [[1,2,3,4,5],[6]]
=> {{1,2,3,4,5},{6}}
=> [2,3,4,5,1,6] => ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12}
[[1,2,3,4,5],[6]]
=> [[1,2,3,4,5,6]]
=> {{1,2,3,4,5,6}}
=> [2,3,4,5,6,1] => ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12}
[[1,3,5,6],[2,4]]
=> [[1,2,4,6],[3,5]]
=> {{1,2,4,6},{3,5}}
=> [2,4,5,6,3,1] => ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12}
[[1,2,5,6],[3,4]]
=> [[1,2,3,4],[5,6]]
=> {{1,2,3,4},{5,6}}
=> [2,3,4,1,6,5] => ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12}
[[1,3,4,6],[2,5]]
=> [[1,2,4,5],[3,6]]
=> {{1,2,4,5},{3,6}}
=> [2,4,6,5,1,3] => ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12}
[[1,2,4,6],[3,5]]
=> [[1,2,3,5],[4,6]]
=> {{1,2,3,5},{4,6}}
=> [2,3,5,6,1,4] => ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12}
[[1,2,3,6],[4,5]]
=> [[1,2,3,4,5],[6]]
=> {{1,2,3,4,5},{6}}
=> [2,3,4,5,1,6] => ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12}
[[1,3,4,5],[2,6]]
=> [[1,2,4,5,6],[3]]
=> {{1,2,4,5,6},{3}}
=> [2,4,3,5,6,1] => ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12}
[[1,2,4,5],[3,6]]
=> [[1,2,3,5,6],[4]]
=> {{1,2,3,5,6},{4}}
=> [2,3,5,4,6,1] => ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12}
[[1,2,3,5],[4,6]]
=> [[1,2,3,4,6],[5]]
=> {{1,2,3,4,6},{5}}
=> [2,3,4,6,5,1] => ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12}
[[1,2,3,4],[5,6]]
=> [[1,2,3,4,5,6]]
=> {{1,2,3,4,5,6}}
=> [2,3,4,5,6,1] => ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12}
[[1,4,5,6],[2],[3]]
=> [[1,2,5,6],[3],[4]]
=> {{1,2,5,6},{3},{4}}
=> [2,5,3,4,6,1] => ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12}
[[1,3,5,6],[2],[4]]
=> [[1,2,4,6],[3],[5]]
=> {{1,2,4,6},{3},{5}}
=> [2,4,3,6,5,1] => ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12}
[[1,2,5,6],[3],[4]]
=> [[1,2,3,6],[4],[5]]
=> {{1,2,3,6},{4},{5}}
=> [2,3,6,4,5,1] => ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12}
[[1,3,4,6],[2],[5]]
=> [[1,2,4,5],[3],[6]]
=> {{1,2,4,5},{3},{6}}
=> [2,4,3,5,1,6] => ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12}
[[1,2,4,6],[3],[5]]
=> [[1,2,3,5],[4],[6]]
=> {{1,2,3,5},{4},{6}}
=> [2,3,5,4,1,6] => ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12}
[[1,2,3,6],[4],[5]]
=> [[1,2,3,4],[5],[6]]
=> {{1,2,3,4},{5},{6}}
=> [2,3,4,1,5,6] => ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12}
[[1,3,4,5],[2],[6]]
=> [[1,2,4,5,6],[3]]
=> {{1,2,4,5,6},{3}}
=> [2,4,3,5,6,1] => ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12}
[[1,2,4,5],[3],[6]]
=> [[1,2,3,5,6],[4]]
=> {{1,2,3,5,6},{4}}
=> [2,3,5,4,6,1] => ? ∊ {1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,12,12,12,12,12,12,12,12,12,12,12,12}
Description
The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$.
The following 11 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St000264The girth of a graph, which is not a tree. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000100The number of linear extensions of a poset. St000633The size of the automorphism group of a poset. St000640The rank of the largest boolean interval in a poset. St000910The number of maximal chains of minimal length in a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice.
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