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Your data matches 5 different statistics following compositions of up to 3 maps.
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Matching statistic: St001486
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Mp00106: Standard tableaux —catabolism⟶ Standard tableaux
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
St001486: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
St001486: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> [[1]]
=> [1] => 1
[[1,2]]
=> [[1,2]]
=> [2] => 2
[[1],[2]]
=> [[1,2]]
=> [2] => 2
[[1,2,3]]
=> [[1,2,3]]
=> [3] => 2
[[1,3],[2]]
=> [[1,2],[3]]
=> [3] => 2
[[1,2],[3]]
=> [[1,2,3]]
=> [3] => 2
[[1],[2],[3]]
=> [[1,2],[3]]
=> [3] => 2
[[1,2,3,4]]
=> [[1,2,3,4]]
=> [4] => 2
[[1,3,4],[2]]
=> [[1,2,4],[3]]
=> [3,1] => 3
[[1,2,4],[3]]
=> [[1,2,3],[4]]
=> [4] => 2
[[1,2,3],[4]]
=> [[1,2,3,4]]
=> [4] => 2
[[1,3],[2,4]]
=> [[1,2,4],[3]]
=> [3,1] => 3
[[1,2],[3,4]]
=> [[1,2,3,4]]
=> [4] => 2
[[1,4],[2],[3]]
=> [[1,2],[3],[4]]
=> [4] => 2
[[1,3],[2],[4]]
=> [[1,2,4],[3]]
=> [3,1] => 3
[[1,2],[3],[4]]
=> [[1,2,3],[4]]
=> [4] => 2
[[1],[2],[3],[4]]
=> [[1,2],[3],[4]]
=> [4] => 2
[[1,2,3,4,5]]
=> [[1,2,3,4,5]]
=> [5] => 2
[[1,3,4,5],[2]]
=> [[1,2,4,5],[3]]
=> [3,2] => 4
[[1,2,4,5],[3]]
=> [[1,2,3,5],[4]]
=> [4,1] => 3
[[1,2,3,5],[4]]
=> [[1,2,3,4],[5]]
=> [5] => 2
[[1,2,3,4],[5]]
=> [[1,2,3,4,5]]
=> [5] => 2
[[1,3,5],[2,4]]
=> [[1,2,4],[3,5]]
=> [3,2] => 4
[[1,2,5],[3,4]]
=> [[1,2,3,4],[5]]
=> [5] => 2
[[1,3,4],[2,5]]
=> [[1,2,4,5],[3]]
=> [3,2] => 4
[[1,2,4],[3,5]]
=> [[1,2,3,5],[4]]
=> [4,1] => 3
[[1,2,3],[4,5]]
=> [[1,2,3,4,5]]
=> [5] => 2
[[1,4,5],[2],[3]]
=> [[1,2,5],[3],[4]]
=> [4,1] => 3
[[1,3,5],[2],[4]]
=> [[1,2,4],[3],[5]]
=> [3,2] => 4
[[1,2,5],[3],[4]]
=> [[1,2,3],[4],[5]]
=> [5] => 2
[[1,3,4],[2],[5]]
=> [[1,2,4,5],[3]]
=> [3,2] => 4
[[1,2,4],[3],[5]]
=> [[1,2,3,5],[4]]
=> [4,1] => 3
[[1,2,3],[4],[5]]
=> [[1,2,3,4],[5]]
=> [5] => 2
[[1,4],[2,5],[3]]
=> [[1,2,5],[3],[4]]
=> [4,1] => 3
[[1,3],[2,5],[4]]
=> [[1,2,4,5],[3]]
=> [3,2] => 4
[[1,2],[3,5],[4]]
=> [[1,2,3,5],[4]]
=> [4,1] => 3
[[1,3],[2,4],[5]]
=> [[1,2,4],[3,5]]
=> [3,2] => 4
[[1,2],[3,4],[5]]
=> [[1,2,3,4],[5]]
=> [5] => 2
[[1,5],[2],[3],[4]]
=> [[1,2],[3],[4],[5]]
=> [5] => 2
[[1,4],[2],[3],[5]]
=> [[1,2,5],[3],[4]]
=> [4,1] => 3
[[1,3],[2],[4],[5]]
=> [[1,2,4],[3],[5]]
=> [3,2] => 4
[[1,2],[3],[4],[5]]
=> [[1,2,3],[4],[5]]
=> [5] => 2
[[1],[2],[3],[4],[5]]
=> [[1,2],[3],[4],[5]]
=> [5] => 2
[[1,2,3,4,5,6]]
=> [[1,2,3,4,5,6]]
=> [6] => 2
[[1,3,4,5,6],[2]]
=> [[1,2,4,5,6],[3]]
=> [3,3] => 4
[[1,2,4,5,6],[3]]
=> [[1,2,3,5,6],[4]]
=> [4,2] => 4
[[1,2,3,5,6],[4]]
=> [[1,2,3,4,6],[5]]
=> [5,1] => 3
[[1,2,3,4,6],[5]]
=> [[1,2,3,4,5],[6]]
=> [6] => 2
[[1,2,3,4,5],[6]]
=> [[1,2,3,4,5,6]]
=> [6] => 2
[[1,3,5,6],[2,4]]
=> [[1,2,4,6],[3,5]]
=> [3,2,1] => 5
Description
The number of corners of the ribbon associated with an integer composition.
We associate a ribbon shape to a composition $c=(c_1,\dots,c_n)$ with $c_i$ cells in the $i$-th row from bottom to top, such that the cells in two rows overlap in precisely one cell.
This statistic records the total number of corners of the ribbon shape.
Matching statistic: St000691
Mp00106: Standard tableaux —catabolism⟶ Standard tableaux
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000691: Binary words ⟶ ℤResult quality: 78% ●values known / values provided: 81%●distinct values known / distinct values provided: 78%
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000691: Binary words ⟶ ℤResult quality: 78% ●values known / values provided: 81%●distinct values known / distinct values provided: 78%
Values
[[1]]
=> [[1]]
=> [1] => 1 => 0 = 1 - 1
[[1,2]]
=> [[1,2]]
=> [2] => 10 => 1 = 2 - 1
[[1],[2]]
=> [[1,2]]
=> [2] => 10 => 1 = 2 - 1
[[1,2,3]]
=> [[1,2,3]]
=> [3] => 100 => 1 = 2 - 1
[[1,3],[2]]
=> [[1,2],[3]]
=> [3] => 100 => 1 = 2 - 1
[[1,2],[3]]
=> [[1,2,3]]
=> [3] => 100 => 1 = 2 - 1
[[1],[2],[3]]
=> [[1,2],[3]]
=> [3] => 100 => 1 = 2 - 1
[[1,2,3,4]]
=> [[1,2,3,4]]
=> [4] => 1000 => 1 = 2 - 1
[[1,3,4],[2]]
=> [[1,2,4],[3]]
=> [3,1] => 1001 => 2 = 3 - 1
[[1,2,4],[3]]
=> [[1,2,3],[4]]
=> [4] => 1000 => 1 = 2 - 1
[[1,2,3],[4]]
=> [[1,2,3,4]]
=> [4] => 1000 => 1 = 2 - 1
[[1,3],[2,4]]
=> [[1,2,4],[3]]
=> [3,1] => 1001 => 2 = 3 - 1
[[1,2],[3,4]]
=> [[1,2,3,4]]
=> [4] => 1000 => 1 = 2 - 1
[[1,4],[2],[3]]
=> [[1,2],[3],[4]]
=> [4] => 1000 => 1 = 2 - 1
[[1,3],[2],[4]]
=> [[1,2,4],[3]]
=> [3,1] => 1001 => 2 = 3 - 1
[[1,2],[3],[4]]
=> [[1,2,3],[4]]
=> [4] => 1000 => 1 = 2 - 1
[[1],[2],[3],[4]]
=> [[1,2],[3],[4]]
=> [4] => 1000 => 1 = 2 - 1
[[1,2,3,4,5]]
=> [[1,2,3,4,5]]
=> [5] => 10000 => 1 = 2 - 1
[[1,3,4,5],[2]]
=> [[1,2,4,5],[3]]
=> [3,2] => 10010 => 3 = 4 - 1
[[1,2,4,5],[3]]
=> [[1,2,3,5],[4]]
=> [4,1] => 10001 => 2 = 3 - 1
[[1,2,3,5],[4]]
=> [[1,2,3,4],[5]]
=> [5] => 10000 => 1 = 2 - 1
[[1,2,3,4],[5]]
=> [[1,2,3,4,5]]
=> [5] => 10000 => 1 = 2 - 1
[[1,3,5],[2,4]]
=> [[1,2,4],[3,5]]
=> [3,2] => 10010 => 3 = 4 - 1
[[1,2,5],[3,4]]
=> [[1,2,3,4],[5]]
=> [5] => 10000 => 1 = 2 - 1
[[1,3,4],[2,5]]
=> [[1,2,4,5],[3]]
=> [3,2] => 10010 => 3 = 4 - 1
[[1,2,4],[3,5]]
=> [[1,2,3,5],[4]]
=> [4,1] => 10001 => 2 = 3 - 1
[[1,2,3],[4,5]]
=> [[1,2,3,4,5]]
=> [5] => 10000 => 1 = 2 - 1
[[1,4,5],[2],[3]]
=> [[1,2,5],[3],[4]]
=> [4,1] => 10001 => 2 = 3 - 1
[[1,3,5],[2],[4]]
=> [[1,2,4],[3],[5]]
=> [3,2] => 10010 => 3 = 4 - 1
[[1,2,5],[3],[4]]
=> [[1,2,3],[4],[5]]
=> [5] => 10000 => 1 = 2 - 1
[[1,3,4],[2],[5]]
=> [[1,2,4,5],[3]]
=> [3,2] => 10010 => 3 = 4 - 1
[[1,2,4],[3],[5]]
=> [[1,2,3,5],[4]]
=> [4,1] => 10001 => 2 = 3 - 1
[[1,2,3],[4],[5]]
=> [[1,2,3,4],[5]]
=> [5] => 10000 => 1 = 2 - 1
[[1,4],[2,5],[3]]
=> [[1,2,5],[3],[4]]
=> [4,1] => 10001 => 2 = 3 - 1
[[1,3],[2,5],[4]]
=> [[1,2,4,5],[3]]
=> [3,2] => 10010 => 3 = 4 - 1
[[1,2],[3,5],[4]]
=> [[1,2,3,5],[4]]
=> [4,1] => 10001 => 2 = 3 - 1
[[1,3],[2,4],[5]]
=> [[1,2,4],[3,5]]
=> [3,2] => 10010 => 3 = 4 - 1
[[1,2],[3,4],[5]]
=> [[1,2,3,4],[5]]
=> [5] => 10000 => 1 = 2 - 1
[[1,5],[2],[3],[4]]
=> [[1,2],[3],[4],[5]]
=> [5] => 10000 => 1 = 2 - 1
[[1,4],[2],[3],[5]]
=> [[1,2,5],[3],[4]]
=> [4,1] => 10001 => 2 = 3 - 1
[[1,3],[2],[4],[5]]
=> [[1,2,4],[3],[5]]
=> [3,2] => 10010 => 3 = 4 - 1
[[1,2],[3],[4],[5]]
=> [[1,2,3],[4],[5]]
=> [5] => 10000 => 1 = 2 - 1
[[1],[2],[3],[4],[5]]
=> [[1,2],[3],[4],[5]]
=> [5] => 10000 => 1 = 2 - 1
[[1,2,3,4,5,6]]
=> [[1,2,3,4,5,6]]
=> [6] => 100000 => 1 = 2 - 1
[[1,3,4,5,6],[2]]
=> [[1,2,4,5,6],[3]]
=> [3,3] => 100100 => 3 = 4 - 1
[[1,2,4,5,6],[3]]
=> [[1,2,3,5,6],[4]]
=> [4,2] => 100010 => 3 = 4 - 1
[[1,2,3,5,6],[4]]
=> [[1,2,3,4,6],[5]]
=> [5,1] => 100001 => 2 = 3 - 1
[[1,2,3,4,6],[5]]
=> [[1,2,3,4,5],[6]]
=> [6] => 100000 => 1 = 2 - 1
[[1,2,3,4,5],[6]]
=> [[1,2,3,4,5,6]]
=> [6] => 100000 => 1 = 2 - 1
[[1,3,5,6],[2,4]]
=> [[1,2,4,6],[3,5]]
=> [3,2,1] => 100101 => 4 = 5 - 1
[[1,3,5,7,9],[2,4,6,8,10]]
=> [[1,2,4,6,8,10],[3,5,7,9]]
=> [3,2,2,2,1] => 1001010101 => ? = 9 - 1
[[1,3,5,7,8],[2,4,6,9,10]]
=> [[1,2,4,6,8,9,10],[3,5,7]]
=> [3,2,2,3] => 1001010100 => ? = 8 - 1
[[1,3,5,6,9],[2,4,7,8,10]]
=> [[1,2,4,6,7,8,10],[3,5,9]]
=> [3,2,4,1] => 1001010001 => ? = 7 - 1
[[1,3,5,6,8],[2,4,7,9,10]]
=> [[1,2,4,6,7,9,10],[3,5,8]]
=> [3,2,3,2] => 1001010010 => ? = 8 - 1
[[1,3,5,6,7],[2,4,8,9,10]]
=> [[1,2,4,6,7,8,9,10],[3,5]]
=> [3,2,5] => 1001010000 => ? = 6 - 1
[[1,3,4,7,9],[2,5,6,8,10]]
=> [[1,2,4,5,6,8,10],[3,7,9]]
=> [3,4,2,1] => 1001000101 => ? = 7 - 1
[[1,3,4,7,8],[2,5,6,9,10]]
=> [[1,2,4,5,6,9,10],[3,7,8]]
=> [3,4,3] => 1001000100 => ? = 6 - 1
[[1,3,4,6,9],[2,5,7,8,10]]
=> [[1,2,4,5,7,8,10],[3,6,9]]
=> [3,3,3,1] => 1001001001 => ? = 7 - 1
[[1,3,4,6,8],[2,5,7,9,10]]
=> [[1,2,4,5,7,9,10],[3,6,8]]
=> [3,3,2,2] => 1001001010 => ? = 8 - 1
[[1,3,4,6,7],[2,5,8,9,10]]
=> [[1,2,4,5,7,8,9,10],[3,6]]
=> [3,3,4] => 1001001000 => ? = 6 - 1
[[1,3,4,5,9],[2,6,7,8,10]]
=> [[1,2,4,5,6,7,8,10],[3,9]]
=> [3,6,1] => 1001000001 => ? = 5 - 1
[[1,3,4,5,8],[2,6,7,9,10]]
=> [[1,2,4,5,6,7,9,10],[3,8]]
=> [3,5,2] => 1001000010 => ? = 6 - 1
[[1,3,4,5,7],[2,6,8,9,10]]
=> [[1,2,4,5,6,8,9,10],[3,7]]
=> [3,4,3] => 1001000100 => ? = 6 - 1
[[1,3,4,5,6],[2,7,8,9,10]]
=> [[1,2,4,5,6,7,8,9,10],[3]]
=> [3,7] => 1001000000 => ? = 4 - 1
[[1,2,5,7,9],[3,4,6,8,10]]
=> [[1,2,3,4,6,8,10],[5,7,9]]
=> [5,2,2,1] => 1000010101 => ? = 7 - 1
[[1,2,5,7,8],[3,4,6,9,10]]
=> [[1,2,3,4,6,9,10],[5,7,8]]
=> [5,2,3] => 1000010100 => ? = 6 - 1
[[1,2,5,6,9],[3,4,7,8,10]]
=> [[1,2,3,4,7,8,10],[5,6,9]]
=> [5,4,1] => 1000010001 => ? = 5 - 1
[[1,2,5,6,8],[3,4,7,9,10]]
=> [[1,2,3,4,7,9,10],[5,6,8]]
=> [5,3,2] => 1000010010 => ? = 6 - 1
[[1,2,5,6,7],[3,4,8,9,10]]
=> [[1,2,3,4,7,8,9,10],[5,6]]
=> [5,5] => 1000010000 => ? = 4 - 1
[[1,2,4,7,9],[3,5,6,8,10]]
=> [[1,2,3,5,6,8,10],[4,7,9]]
=> [4,3,2,1] => 1000100101 => ? = 7 - 1
[[1,2,4,7,8],[3,5,6,9,10]]
=> [[1,2,3,5,6,9,10],[4,7,8]]
=> [4,3,3] => 1000100100 => ? = 6 - 1
[[1,2,4,6,9],[3,5,7,8,10]]
=> [[1,2,3,5,7,8,10],[4,6,9]]
=> [4,2,3,1] => 1000101001 => ? = 7 - 1
[[1,2,4,6,8],[3,5,7,9,10]]
=> [[1,2,3,5,7,9,10],[4,6,8]]
=> [4,2,2,2] => 1000101010 => ? = 8 - 1
[[1,2,4,6,7],[3,5,8,9,10]]
=> [[1,2,3,5,7,8,9,10],[4,6]]
=> [4,2,4] => 1000101000 => ? = 6 - 1
[[1,2,4,5,9],[3,6,7,8,10]]
=> [[1,2,3,5,6,7,8,10],[4,9]]
=> [4,5,1] => 1000100001 => ? = 5 - 1
[[1,2,4,5,8],[3,6,7,9,10]]
=> [[1,2,3,5,6,7,9,10],[4,8]]
=> [4,4,2] => 1000100010 => ? = 6 - 1
[[1,2,4,5,7],[3,6,8,9,10]]
=> [[1,2,3,5,6,8,9,10],[4,7]]
=> [4,3,3] => 1000100100 => ? = 6 - 1
[[1,2,4,5,6],[3,7,8,9,10]]
=> [[1,2,3,5,6,7,8,9,10],[4]]
=> [4,6] => 1000100000 => ? = 4 - 1
[[1,2,3,7,9],[4,5,6,8,10]]
=> [[1,2,3,4,5,6,8,10],[7,9]]
=> [7,2,1] => 1000000101 => ? = 5 - 1
[[1,2,3,7,8],[4,5,6,9,10]]
=> [[1,2,3,4,5,6,9,10],[7,8]]
=> [7,3] => 1000000100 => ? = 4 - 1
[[1,2,3,6,9],[4,5,7,8,10]]
=> [[1,2,3,4,5,7,8,10],[6,9]]
=> [6,3,1] => 1000001001 => ? = 5 - 1
[[1,2,3,6,8],[4,5,7,9,10]]
=> [[1,2,3,4,5,7,9,10],[6,8]]
=> [6,2,2] => 1000001010 => ? = 6 - 1
[[1,2,3,6,7],[4,5,8,9,10]]
=> [[1,2,3,4,5,8,9,10],[6,7]]
=> [6,4] => 1000001000 => ? = 4 - 1
[[1,2,3,5,9],[4,6,7,8,10]]
=> [[1,2,3,4,6,7,8,10],[5,9]]
=> [5,4,1] => 1000010001 => ? = 5 - 1
[[1,2,3,5,8],[4,6,7,9,10]]
=> [[1,2,3,4,6,7,9,10],[5,8]]
=> [5,3,2] => 1000010010 => ? = 6 - 1
[[1,2,3,5,7],[4,6,8,9,10]]
=> [[1,2,3,4,6,8,9,10],[5,7]]
=> [5,2,3] => 1000010100 => ? = 6 - 1
[[1,2,3,5,6],[4,7,8,9,10]]
=> [[1,2,3,4,6,7,8,9,10],[5]]
=> [5,5] => 1000010000 => ? = 4 - 1
[[1,2,3,4,9],[5,6,7,8,10]]
=> [[1,2,3,4,5,6,7,8,10],[9]]
=> [9,1] => 1000000001 => ? = 3 - 1
[[1,2,3,4,8],[5,6,7,9,10]]
=> [[1,2,3,4,5,6,7,9,10],[8]]
=> [8,2] => 1000000010 => ? = 4 - 1
[[1,2,3,4,7],[5,6,8,9,10]]
=> [[1,2,3,4,5,6,8,9,10],[7]]
=> [7,3] => 1000000100 => ? = 4 - 1
[[1,2,3,4,6],[5,7,8,9,10]]
=> [[1,2,3,4,5,7,8,9,10],[6]]
=> [6,4] => 1000001000 => ? = 4 - 1
[[1,2,3,4,5],[6,7,8,9,10]]
=> [[1,2,3,4,5,6,7,8,9,10]]
=> [10] => 1000000000 => ? = 2 - 1
[[1,2,3,4,5,6,7,8,9,10]]
=> [[1,2,3,4,5,6,7,8,9,10]]
=> [10] => 1000000000 => ? = 2 - 1
[[1,2,3,4,5,6,7,8,9],[10]]
=> [[1,2,3,4,5,6,7,8,9,10]]
=> [10] => 1000000000 => ? = 2 - 1
[[1,2,3,4,5,6,7,8],[9,10]]
=> [[1,2,3,4,5,6,7,8,9,10]]
=> [10] => 1000000000 => ? = 2 - 1
[[1,2,3,4,5,6,7,8],[9],[10]]
=> [[1,2,3,4,5,6,7,8,9],[10]]
=> [10] => 1000000000 => ? = 2 - 1
[[1,2,3,4,5,6,7],[8,9,10]]
=> [[1,2,3,4,5,6,7,8,9,10]]
=> [10] => 1000000000 => ? = 2 - 1
[[1,2,3,4,5,6,7],[8,9],[10]]
=> [[1,2,3,4,5,6,7,8,9],[10]]
=> [10] => 1000000000 => ? = 2 - 1
[[1,2,3,4,5,6,7],[8],[9],[10]]
=> [[1,2,3,4,5,6,7,8],[9],[10]]
=> [10] => 1000000000 => ? = 2 - 1
[[1,2,3,4,5,6],[7,8,9,10]]
=> [[1,2,3,4,5,6,7,8,9,10]]
=> [10] => 1000000000 => ? = 2 - 1
Description
The number of changes of a binary word.
This is the number of indices $i$ such that $w_i \neq w_{i+1}$.
Matching statistic: St000777
Mp00106: Standard tableaux —catabolism⟶ Standard tableaux
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000777: Graphs ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 33%
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000777: Graphs ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 33%
Values
[[1]]
=> [[1]]
=> [1] => ([],1)
=> 1
[[1,2]]
=> [[1,2]]
=> [2] => ([],2)
=> ? = 2
[[1],[2]]
=> [[1,2]]
=> [2] => ([],2)
=> ? = 2
[[1,2,3]]
=> [[1,2,3]]
=> [3] => ([],3)
=> ? = 2
[[1,3],[2]]
=> [[1,2],[3]]
=> [3] => ([],3)
=> ? = 2
[[1,2],[3]]
=> [[1,2,3]]
=> [3] => ([],3)
=> ? = 2
[[1],[2],[3]]
=> [[1,2],[3]]
=> [3] => ([],3)
=> ? = 2
[[1,2,3,4]]
=> [[1,2,3,4]]
=> [4] => ([],4)
=> ? = 2
[[1,3,4],[2]]
=> [[1,2,4],[3]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[[1,2,4],[3]]
=> [[1,2,3],[4]]
=> [4] => ([],4)
=> ? = 2
[[1,2,3],[4]]
=> [[1,2,3,4]]
=> [4] => ([],4)
=> ? = 2
[[1,3],[2,4]]
=> [[1,2,4],[3]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[[1,2],[3,4]]
=> [[1,2,3,4]]
=> [4] => ([],4)
=> ? = 2
[[1,4],[2],[3]]
=> [[1,2],[3],[4]]
=> [4] => ([],4)
=> ? = 2
[[1,3],[2],[4]]
=> [[1,2,4],[3]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[[1,2],[3],[4]]
=> [[1,2,3],[4]]
=> [4] => ([],4)
=> ? = 2
[[1],[2],[3],[4]]
=> [[1,2],[3],[4]]
=> [4] => ([],4)
=> ? = 2
[[1,2,3,4,5]]
=> [[1,2,3,4,5]]
=> [5] => ([],5)
=> ? = 2
[[1,3,4,5],[2]]
=> [[1,2,4,5],[3]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 4
[[1,2,4,5],[3]]
=> [[1,2,3,5],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[[1,2,3,5],[4]]
=> [[1,2,3,4],[5]]
=> [5] => ([],5)
=> ? = 2
[[1,2,3,4],[5]]
=> [[1,2,3,4,5]]
=> [5] => ([],5)
=> ? = 2
[[1,3,5],[2,4]]
=> [[1,2,4],[3,5]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 4
[[1,2,5],[3,4]]
=> [[1,2,3,4],[5]]
=> [5] => ([],5)
=> ? = 2
[[1,3,4],[2,5]]
=> [[1,2,4,5],[3]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 4
[[1,2,4],[3,5]]
=> [[1,2,3,5],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[[1,2,3],[4,5]]
=> [[1,2,3,4,5]]
=> [5] => ([],5)
=> ? = 2
[[1,4,5],[2],[3]]
=> [[1,2,5],[3],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[[1,3,5],[2],[4]]
=> [[1,2,4],[3],[5]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 4
[[1,2,5],[3],[4]]
=> [[1,2,3],[4],[5]]
=> [5] => ([],5)
=> ? = 2
[[1,3,4],[2],[5]]
=> [[1,2,4,5],[3]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 4
[[1,2,4],[3],[5]]
=> [[1,2,3,5],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[[1,2,3],[4],[5]]
=> [[1,2,3,4],[5]]
=> [5] => ([],5)
=> ? = 2
[[1,4],[2,5],[3]]
=> [[1,2,5],[3],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[[1,3],[2,5],[4]]
=> [[1,2,4,5],[3]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 4
[[1,2],[3,5],[4]]
=> [[1,2,3,5],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[[1,3],[2,4],[5]]
=> [[1,2,4],[3,5]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 4
[[1,2],[3,4],[5]]
=> [[1,2,3,4],[5]]
=> [5] => ([],5)
=> ? = 2
[[1,5],[2],[3],[4]]
=> [[1,2],[3],[4],[5]]
=> [5] => ([],5)
=> ? = 2
[[1,4],[2],[3],[5]]
=> [[1,2,5],[3],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[[1,3],[2],[4],[5]]
=> [[1,2,4],[3],[5]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 4
[[1,2],[3],[4],[5]]
=> [[1,2,3],[4],[5]]
=> [5] => ([],5)
=> ? = 2
[[1],[2],[3],[4],[5]]
=> [[1,2],[3],[4],[5]]
=> [5] => ([],5)
=> ? = 2
[[1,2,3,4,5,6]]
=> [[1,2,3,4,5,6]]
=> [6] => ([],6)
=> ? = 2
[[1,3,4,5,6],[2]]
=> [[1,2,4,5,6],[3]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? = 4
[[1,2,4,5,6],[3]]
=> [[1,2,3,5,6],[4]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 4
[[1,2,3,5,6],[4]]
=> [[1,2,3,4,6],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 3
[[1,2,3,4,6],[5]]
=> [[1,2,3,4,5],[6]]
=> [6] => ([],6)
=> ? = 2
[[1,2,3,4,5],[6]]
=> [[1,2,3,4,5,6]]
=> [6] => ([],6)
=> ? = 2
[[1,3,5,6],[2,4]]
=> [[1,2,4,6],[3,5]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
[[1,2,5,6],[3,4]]
=> [[1,2,3,4],[5,6]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 3
[[1,3,4,6],[2,5]]
=> [[1,2,4,5],[3,6]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? = 4
[[1,2,4,6],[3,5]]
=> [[1,2,3,5],[4,6]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 4
[[1,2,3,6],[4,5]]
=> [[1,2,3,4,5],[6]]
=> [6] => ([],6)
=> ? = 2
[[1,3,4,5],[2,6]]
=> [[1,2,4,5,6],[3]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? = 4
[[1,2,4,5],[3,6]]
=> [[1,2,3,5,6],[4]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 4
[[1,2,3,5],[4,6]]
=> [[1,2,3,4,6],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 3
[[1,2,3,4],[5,6]]
=> [[1,2,3,4,5,6]]
=> [6] => ([],6)
=> ? = 2
[[1,4,5,6],[2],[3]]
=> [[1,2,5,6],[3],[4]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 4
[[1,3,5,6],[2],[4]]
=> [[1,2,4,6],[3],[5]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
[[1,2,5,6],[3],[4]]
=> [[1,2,3,6],[4],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 3
[[1,3,4,6],[2],[5]]
=> [[1,2,4,5],[3],[6]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? = 4
[[1,2,4,6],[3],[5]]
=> [[1,2,3,5],[4],[6]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 4
[[1,2,3,6],[4],[5]]
=> [[1,2,3,4],[5],[6]]
=> [6] => ([],6)
=> ? = 2
[[1,3,4,5],[2],[6]]
=> [[1,2,4,5,6],[3]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? = 4
[[1,2,4,5],[3],[6]]
=> [[1,2,3,5,6],[4]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 4
[[1,2,3,5],[4],[6]]
=> [[1,2,3,4,6],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 3
[[1,2,3,4],[5],[6]]
=> [[1,2,3,4,5],[6]]
=> [6] => ([],6)
=> ? = 2
[[1,3,5],[2,4,6]]
=> [[1,2,4,6],[3,5]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
[[1,2,5],[3,4,6]]
=> [[1,2,3,4,6],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 3
[[1,3,5],[2,6],[4]]
=> [[1,2,4,6],[3],[5]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
[[1,2,5],[3,6],[4]]
=> [[1,2,3,6],[4],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 3
[[1,2,3],[4,6],[5]]
=> [[1,2,3,4,6],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 3
[[1,3,5],[2,4],[6]]
=> [[1,2,4,6],[3,5]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
[[1,2,5],[3,4],[6]]
=> [[1,2,3,4],[5,6]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 3
[[1,5,6],[2],[3],[4]]
=> [[1,2,6],[3],[4],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 3
[[1,3,5],[2],[4],[6]]
=> [[1,2,4,6],[3],[5]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
[[1,2,5],[3],[4],[6]]
=> [[1,2,3,6],[4],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 3
[[1,3],[2,4],[5,6]]
=> [[1,2,4,6],[3,5]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
[[1,2],[3,4],[5,6]]
=> [[1,2,3,4],[5,6]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 3
[[1,5],[2,6],[3],[4]]
=> [[1,2,6],[3],[4],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 3
[[1,3],[2,6],[4],[5]]
=> [[1,2,4,6],[3],[5]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
[[1,2],[3,6],[4],[5]]
=> [[1,2,3,6],[4],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 3
[[1,5],[2],[3],[4],[6]]
=> [[1,2,6],[3],[4],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 3
[[1,2,3,4,6,7],[5]]
=> [[1,2,3,4,5,7],[6]]
=> [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 3
[[1,3,4,6,7],[2,5]]
=> [[1,2,4,5,7],[3,6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 5
[[1,2,4,6,7],[3,5]]
=> [[1,2,3,5,7],[4,6]]
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 5
[[1,2,3,6,7],[4,5]]
=> [[1,2,3,4,5],[6,7]]
=> [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 3
[[1,2,3,4,6],[5,7]]
=> [[1,2,3,4,5,7],[6]]
=> [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 3
[[1,3,4,6,7],[2],[5]]
=> [[1,2,4,5,7],[3],[6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 5
[[1,2,4,6,7],[3],[5]]
=> [[1,2,3,5,7],[4],[6]]
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 5
[[1,2,3,6,7],[4],[5]]
=> [[1,2,3,4,7],[5],[6]]
=> [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 3
[[1,2,3,4,6],[5],[7]]
=> [[1,2,3,4,5,7],[6]]
=> [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 3
[[1,3,4,6],[2,5,7]]
=> [[1,2,4,5,7],[3,6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 5
[[1,2,4,6],[3,5,7]]
=> [[1,2,3,5,7],[4,6]]
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 5
[[1,2,3,6],[4,5,7]]
=> [[1,2,3,4,5,7],[6]]
=> [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 3
[[1,4,6,7],[2,5],[3]]
=> [[1,2,5,7],[3,6],[4]]
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 5
[[1,3,6,7],[2,5],[4]]
=> [[1,2,4,5],[3,7],[6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 5
[[1,2,6,7],[3,5],[4]]
=> [[1,2,3,5],[4,7],[6]]
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 5
[[1,3,6,7],[2,4],[5]]
=> [[1,2,4,7],[3,5],[6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 5
Description
The number of distinct eigenvalues of the distance Laplacian of a connected graph.
Matching statistic: St000741
Mp00106: Standard tableaux —catabolism⟶ Standard tableaux
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000741: Graphs ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 33%
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000741: Graphs ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 33%
Values
[[1]]
=> [[1]]
=> [1] => ([],1)
=> 0 = 1 - 1
[[1,2]]
=> [[1,2]]
=> [2] => ([],2)
=> 1 = 2 - 1
[[1],[2]]
=> [[1,2]]
=> [2] => ([],2)
=> 1 = 2 - 1
[[1,2,3]]
=> [[1,2,3]]
=> [3] => ([],3)
=> 1 = 2 - 1
[[1,3],[2]]
=> [[1,2],[3]]
=> [3] => ([],3)
=> 1 = 2 - 1
[[1,2],[3]]
=> [[1,2,3]]
=> [3] => ([],3)
=> 1 = 2 - 1
[[1],[2],[3]]
=> [[1,2],[3]]
=> [3] => ([],3)
=> 1 = 2 - 1
[[1,2,3,4]]
=> [[1,2,3,4]]
=> [4] => ([],4)
=> 1 = 2 - 1
[[1,3,4],[2]]
=> [[1,2,4],[3]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 3 - 1
[[1,2,4],[3]]
=> [[1,2,3],[4]]
=> [4] => ([],4)
=> 1 = 2 - 1
[[1,2,3],[4]]
=> [[1,2,3,4]]
=> [4] => ([],4)
=> 1 = 2 - 1
[[1,3],[2,4]]
=> [[1,2,4],[3]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 3 - 1
[[1,2],[3,4]]
=> [[1,2,3,4]]
=> [4] => ([],4)
=> 1 = 2 - 1
[[1,4],[2],[3]]
=> [[1,2],[3],[4]]
=> [4] => ([],4)
=> 1 = 2 - 1
[[1,3],[2],[4]]
=> [[1,2,4],[3]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 3 - 1
[[1,2],[3],[4]]
=> [[1,2,3],[4]]
=> [4] => ([],4)
=> 1 = 2 - 1
[[1],[2],[3],[4]]
=> [[1,2],[3],[4]]
=> [4] => ([],4)
=> 1 = 2 - 1
[[1,2,3,4,5]]
=> [[1,2,3,4,5]]
=> [5] => ([],5)
=> 1 = 2 - 1
[[1,3,4,5],[2]]
=> [[1,2,4,5],[3]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 4 - 1
[[1,2,4,5],[3]]
=> [[1,2,3,5],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 3 - 1
[[1,2,3,5],[4]]
=> [[1,2,3,4],[5]]
=> [5] => ([],5)
=> 1 = 2 - 1
[[1,2,3,4],[5]]
=> [[1,2,3,4,5]]
=> [5] => ([],5)
=> 1 = 2 - 1
[[1,3,5],[2,4]]
=> [[1,2,4],[3,5]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 4 - 1
[[1,2,5],[3,4]]
=> [[1,2,3,4],[5]]
=> [5] => ([],5)
=> 1 = 2 - 1
[[1,3,4],[2,5]]
=> [[1,2,4,5],[3]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 4 - 1
[[1,2,4],[3,5]]
=> [[1,2,3,5],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 3 - 1
[[1,2,3],[4,5]]
=> [[1,2,3,4,5]]
=> [5] => ([],5)
=> 1 = 2 - 1
[[1,4,5],[2],[3]]
=> [[1,2,5],[3],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 3 - 1
[[1,3,5],[2],[4]]
=> [[1,2,4],[3],[5]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 4 - 1
[[1,2,5],[3],[4]]
=> [[1,2,3],[4],[5]]
=> [5] => ([],5)
=> 1 = 2 - 1
[[1,3,4],[2],[5]]
=> [[1,2,4,5],[3]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 4 - 1
[[1,2,4],[3],[5]]
=> [[1,2,3,5],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 3 - 1
[[1,2,3],[4],[5]]
=> [[1,2,3,4],[5]]
=> [5] => ([],5)
=> 1 = 2 - 1
[[1,4],[2,5],[3]]
=> [[1,2,5],[3],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 3 - 1
[[1,3],[2,5],[4]]
=> [[1,2,4,5],[3]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 4 - 1
[[1,2],[3,5],[4]]
=> [[1,2,3,5],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 3 - 1
[[1,3],[2,4],[5]]
=> [[1,2,4],[3,5]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 4 - 1
[[1,2],[3,4],[5]]
=> [[1,2,3,4],[5]]
=> [5] => ([],5)
=> 1 = 2 - 1
[[1,5],[2],[3],[4]]
=> [[1,2],[3],[4],[5]]
=> [5] => ([],5)
=> 1 = 2 - 1
[[1,4],[2],[3],[5]]
=> [[1,2,5],[3],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 3 - 1
[[1,3],[2],[4],[5]]
=> [[1,2,4],[3],[5]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 4 - 1
[[1,2],[3],[4],[5]]
=> [[1,2,3],[4],[5]]
=> [5] => ([],5)
=> 1 = 2 - 1
[[1],[2],[3],[4],[5]]
=> [[1,2],[3],[4],[5]]
=> [5] => ([],5)
=> 1 = 2 - 1
[[1,2,3,4,5,6]]
=> [[1,2,3,4,5,6]]
=> [6] => ([],6)
=> 1 = 2 - 1
[[1,3,4,5,6],[2]]
=> [[1,2,4,5,6],[3]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? = 4 - 1
[[1,2,4,5,6],[3]]
=> [[1,2,3,5,6],[4]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 4 - 1
[[1,2,3,5,6],[4]]
=> [[1,2,3,4,6],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 3 - 1
[[1,2,3,4,6],[5]]
=> [[1,2,3,4,5],[6]]
=> [6] => ([],6)
=> 1 = 2 - 1
[[1,2,3,4,5],[6]]
=> [[1,2,3,4,5,6]]
=> [6] => ([],6)
=> 1 = 2 - 1
[[1,3,5,6],[2,4]]
=> [[1,2,4,6],[3,5]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 5 - 1
[[1,2,5,6],[3,4]]
=> [[1,2,3,4],[5,6]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 3 - 1
[[1,3,4,6],[2,5]]
=> [[1,2,4,5],[3,6]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? = 4 - 1
[[1,2,4,6],[3,5]]
=> [[1,2,3,5],[4,6]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 4 - 1
[[1,2,3,6],[4,5]]
=> [[1,2,3,4,5],[6]]
=> [6] => ([],6)
=> 1 = 2 - 1
[[1,3,4,5],[2,6]]
=> [[1,2,4,5,6],[3]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? = 4 - 1
[[1,2,4,5],[3,6]]
=> [[1,2,3,5,6],[4]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 4 - 1
[[1,2,3,5],[4,6]]
=> [[1,2,3,4,6],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 3 - 1
[[1,2,3,4],[5,6]]
=> [[1,2,3,4,5,6]]
=> [6] => ([],6)
=> 1 = 2 - 1
[[1,4,5,6],[2],[3]]
=> [[1,2,5,6],[3],[4]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 4 - 1
[[1,3,5,6],[2],[4]]
=> [[1,2,4,6],[3],[5]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 5 - 1
[[1,2,5,6],[3],[4]]
=> [[1,2,3,6],[4],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 3 - 1
[[1,3,4,6],[2],[5]]
=> [[1,2,4,5],[3],[6]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? = 4 - 1
[[1,2,4,6],[3],[5]]
=> [[1,2,3,5],[4],[6]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 4 - 1
[[1,2,3,6],[4],[5]]
=> [[1,2,3,4],[5],[6]]
=> [6] => ([],6)
=> 1 = 2 - 1
[[1,3,4,5],[2],[6]]
=> [[1,2,4,5,6],[3]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? = 4 - 1
[[1,2,4,5],[3],[6]]
=> [[1,2,3,5,6],[4]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 4 - 1
[[1,2,3,5],[4],[6]]
=> [[1,2,3,4,6],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 3 - 1
[[1,2,3,4],[5],[6]]
=> [[1,2,3,4,5],[6]]
=> [6] => ([],6)
=> 1 = 2 - 1
[[1,3,5],[2,4,6]]
=> [[1,2,4,6],[3,5]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 5 - 1
[[1,2,5],[3,4,6]]
=> [[1,2,3,4,6],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 3 - 1
[[1,3,4],[2,5,6]]
=> [[1,2,4,5,6],[3]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? = 4 - 1
[[1,2,4],[3,5,6]]
=> [[1,2,3,5,6],[4]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 4 - 1
[[1,2,3],[4,5,6]]
=> [[1,2,3,4,5,6]]
=> [6] => ([],6)
=> 1 = 2 - 1
[[1,4,6],[2,5],[3]]
=> [[1,2,5],[3,6],[4]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 4 - 1
[[1,3,6],[2,5],[4]]
=> [[1,2,4,5],[3],[6]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? = 4 - 1
[[1,2,6],[3,5],[4]]
=> [[1,2,3,5],[4],[6]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 4 - 1
[[1,3,6],[2,4],[5]]
=> [[1,2,4],[3,5],[6]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? = 4 - 1
[[1,2,6],[3,4],[5]]
=> [[1,2,3,4],[5],[6]]
=> [6] => ([],6)
=> 1 = 2 - 1
[[1,4,5],[2,6],[3]]
=> [[1,2,5,6],[3],[4]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 4 - 1
[[1,3,5],[2,6],[4]]
=> [[1,2,4,6],[3],[5]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 5 - 1
[[1,3,4],[2,6],[5]]
=> [[1,2,4,5,6],[3]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? = 4 - 1
[[1,2,4],[3,6],[5]]
=> [[1,2,3,5,6],[4]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 4 - 1
[[1,3,5],[2,4],[6]]
=> [[1,2,4,6],[3,5]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 5 - 1
[[1,3,4],[2,5],[6]]
=> [[1,2,4,5],[3,6]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? = 4 - 1
[[1,2,4],[3,5],[6]]
=> [[1,2,3,5],[4,6]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 4 - 1
[[1,4,6],[2],[3],[5]]
=> [[1,2,5],[3],[4],[6]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 4 - 1
[[1,3,6],[2],[4],[5]]
=> [[1,2,4],[3],[5],[6]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? = 4 - 1
[[1,4,5],[2],[3],[6]]
=> [[1,2,5,6],[3],[4]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 4 - 1
[[1,3,5],[2],[4],[6]]
=> [[1,2,4,6],[3],[5]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 5 - 1
[[1,3,4],[2],[5],[6]]
=> [[1,2,4,5],[3],[6]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? = 4 - 1
[[1,2,4],[3],[5],[6]]
=> [[1,2,3,5],[4],[6]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 4 - 1
[[1,4],[2,5],[3,6]]
=> [[1,2,5],[3,6],[4]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 4 - 1
[[1,3],[2,5],[4,6]]
=> [[1,2,4,5],[3,6]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? = 4 - 1
[[1,2],[3,5],[4,6]]
=> [[1,2,3,5],[4,6]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 4 - 1
[[1,3],[2,4],[5,6]]
=> [[1,2,4,6],[3,5]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 5 - 1
[[1,4],[2,6],[3],[5]]
=> [[1,2,5,6],[3],[4]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 4 - 1
[[1,3],[2,6],[4],[5]]
=> [[1,2,4,6],[3],[5]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 5 - 1
[[1,4],[2,5],[3],[6]]
=> [[1,2,5],[3,6],[4]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 4 - 1
[[1,3],[2,5],[4],[6]]
=> [[1,2,4,5],[3],[6]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? = 4 - 1
[[1,2],[3,5],[4],[6]]
=> [[1,2,3,5],[4],[6]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 4 - 1
Description
The Colin de Verdière graph invariant.
Matching statistic: St001488
Mp00106: Standard tableaux —catabolism⟶ Standard tableaux
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
St001488: Skew partitions ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 44%
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
St001488: Skew partitions ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 44%
Values
[[1]]
=> [[1]]
=> [1] => [[1],[]]
=> 1
[[1,2]]
=> [[1,2]]
=> [2] => [[2],[]]
=> 2
[[1],[2]]
=> [[1,2]]
=> [2] => [[2],[]]
=> 2
[[1,2,3]]
=> [[1,2,3]]
=> [3] => [[3],[]]
=> 2
[[1,3],[2]]
=> [[1,2],[3]]
=> [3] => [[3],[]]
=> 2
[[1,2],[3]]
=> [[1,2,3]]
=> [3] => [[3],[]]
=> 2
[[1],[2],[3]]
=> [[1,2],[3]]
=> [3] => [[3],[]]
=> 2
[[1,2,3,4]]
=> [[1,2,3,4]]
=> [4] => [[4],[]]
=> 2
[[1,3,4],[2]]
=> [[1,2,4],[3]]
=> [3,1] => [[3,3],[2]]
=> 3
[[1,2,4],[3]]
=> [[1,2,3],[4]]
=> [4] => [[4],[]]
=> 2
[[1,2,3],[4]]
=> [[1,2,3,4]]
=> [4] => [[4],[]]
=> 2
[[1,3],[2,4]]
=> [[1,2,4],[3]]
=> [3,1] => [[3,3],[2]]
=> 3
[[1,2],[3,4]]
=> [[1,2,3,4]]
=> [4] => [[4],[]]
=> 2
[[1,4],[2],[3]]
=> [[1,2],[3],[4]]
=> [4] => [[4],[]]
=> 2
[[1,3],[2],[4]]
=> [[1,2,4],[3]]
=> [3,1] => [[3,3],[2]]
=> 3
[[1,2],[3],[4]]
=> [[1,2,3],[4]]
=> [4] => [[4],[]]
=> 2
[[1],[2],[3],[4]]
=> [[1,2],[3],[4]]
=> [4] => [[4],[]]
=> 2
[[1,2,3,4,5]]
=> [[1,2,3,4,5]]
=> [5] => [[5],[]]
=> 2
[[1,3,4,5],[2]]
=> [[1,2,4,5],[3]]
=> [3,2] => [[4,3],[2]]
=> 4
[[1,2,4,5],[3]]
=> [[1,2,3,5],[4]]
=> [4,1] => [[4,4],[3]]
=> 3
[[1,2,3,5],[4]]
=> [[1,2,3,4],[5]]
=> [5] => [[5],[]]
=> 2
[[1,2,3,4],[5]]
=> [[1,2,3,4,5]]
=> [5] => [[5],[]]
=> 2
[[1,3,5],[2,4]]
=> [[1,2,4],[3,5]]
=> [3,2] => [[4,3],[2]]
=> 4
[[1,2,5],[3,4]]
=> [[1,2,3,4],[5]]
=> [5] => [[5],[]]
=> 2
[[1,3,4],[2,5]]
=> [[1,2,4,5],[3]]
=> [3,2] => [[4,3],[2]]
=> 4
[[1,2,4],[3,5]]
=> [[1,2,3,5],[4]]
=> [4,1] => [[4,4],[3]]
=> 3
[[1,2,3],[4,5]]
=> [[1,2,3,4,5]]
=> [5] => [[5],[]]
=> 2
[[1,4,5],[2],[3]]
=> [[1,2,5],[3],[4]]
=> [4,1] => [[4,4],[3]]
=> 3
[[1,3,5],[2],[4]]
=> [[1,2,4],[3],[5]]
=> [3,2] => [[4,3],[2]]
=> 4
[[1,2,5],[3],[4]]
=> [[1,2,3],[4],[5]]
=> [5] => [[5],[]]
=> 2
[[1,3,4],[2],[5]]
=> [[1,2,4,5],[3]]
=> [3,2] => [[4,3],[2]]
=> 4
[[1,2,4],[3],[5]]
=> [[1,2,3,5],[4]]
=> [4,1] => [[4,4],[3]]
=> 3
[[1,2,3],[4],[5]]
=> [[1,2,3,4],[5]]
=> [5] => [[5],[]]
=> 2
[[1,4],[2,5],[3]]
=> [[1,2,5],[3],[4]]
=> [4,1] => [[4,4],[3]]
=> 3
[[1,3],[2,5],[4]]
=> [[1,2,4,5],[3]]
=> [3,2] => [[4,3],[2]]
=> 4
[[1,2],[3,5],[4]]
=> [[1,2,3,5],[4]]
=> [4,1] => [[4,4],[3]]
=> 3
[[1,3],[2,4],[5]]
=> [[1,2,4],[3,5]]
=> [3,2] => [[4,3],[2]]
=> 4
[[1,2],[3,4],[5]]
=> [[1,2,3,4],[5]]
=> [5] => [[5],[]]
=> 2
[[1,5],[2],[3],[4]]
=> [[1,2],[3],[4],[5]]
=> [5] => [[5],[]]
=> 2
[[1,4],[2],[3],[5]]
=> [[1,2,5],[3],[4]]
=> [4,1] => [[4,4],[3]]
=> 3
[[1,3],[2],[4],[5]]
=> [[1,2,4],[3],[5]]
=> [3,2] => [[4,3],[2]]
=> 4
[[1,2],[3],[4],[5]]
=> [[1,2,3],[4],[5]]
=> [5] => [[5],[]]
=> 2
[[1],[2],[3],[4],[5]]
=> [[1,2],[3],[4],[5]]
=> [5] => [[5],[]]
=> 2
[[1,2,3,4,5,6]]
=> [[1,2,3,4,5,6]]
=> [6] => [[6],[]]
=> ? = 2
[[1,3,4,5,6],[2]]
=> [[1,2,4,5,6],[3]]
=> [3,3] => [[5,3],[2]]
=> ? = 4
[[1,2,4,5,6],[3]]
=> [[1,2,3,5,6],[4]]
=> [4,2] => [[5,4],[3]]
=> ? = 4
[[1,2,3,5,6],[4]]
=> [[1,2,3,4,6],[5]]
=> [5,1] => [[5,5],[4]]
=> ? = 3
[[1,2,3,4,6],[5]]
=> [[1,2,3,4,5],[6]]
=> [6] => [[6],[]]
=> ? = 2
[[1,2,3,4,5],[6]]
=> [[1,2,3,4,5,6]]
=> [6] => [[6],[]]
=> ? = 2
[[1,3,5,6],[2,4]]
=> [[1,2,4,6],[3,5]]
=> [3,2,1] => [[4,4,3],[3,2]]
=> ? = 5
[[1,2,5,6],[3,4]]
=> [[1,2,3,4],[5,6]]
=> [5,1] => [[5,5],[4]]
=> ? = 3
[[1,3,4,6],[2,5]]
=> [[1,2,4,5],[3,6]]
=> [3,3] => [[5,3],[2]]
=> ? = 4
[[1,2,4,6],[3,5]]
=> [[1,2,3,5],[4,6]]
=> [4,2] => [[5,4],[3]]
=> ? = 4
[[1,2,3,6],[4,5]]
=> [[1,2,3,4,5],[6]]
=> [6] => [[6],[]]
=> ? = 2
[[1,3,4,5],[2,6]]
=> [[1,2,4,5,6],[3]]
=> [3,3] => [[5,3],[2]]
=> ? = 4
[[1,2,4,5],[3,6]]
=> [[1,2,3,5,6],[4]]
=> [4,2] => [[5,4],[3]]
=> ? = 4
[[1,2,3,5],[4,6]]
=> [[1,2,3,4,6],[5]]
=> [5,1] => [[5,5],[4]]
=> ? = 3
[[1,2,3,4],[5,6]]
=> [[1,2,3,4,5,6]]
=> [6] => [[6],[]]
=> ? = 2
[[1,4,5,6],[2],[3]]
=> [[1,2,5,6],[3],[4]]
=> [4,2] => [[5,4],[3]]
=> ? = 4
[[1,3,5,6],[2],[4]]
=> [[1,2,4,6],[3],[5]]
=> [3,2,1] => [[4,4,3],[3,2]]
=> ? = 5
[[1,2,5,6],[3],[4]]
=> [[1,2,3,6],[4],[5]]
=> [5,1] => [[5,5],[4]]
=> ? = 3
[[1,3,4,6],[2],[5]]
=> [[1,2,4,5],[3],[6]]
=> [3,3] => [[5,3],[2]]
=> ? = 4
[[1,2,4,6],[3],[5]]
=> [[1,2,3,5],[4],[6]]
=> [4,2] => [[5,4],[3]]
=> ? = 4
[[1,2,3,6],[4],[5]]
=> [[1,2,3,4],[5],[6]]
=> [6] => [[6],[]]
=> ? = 2
[[1,3,4,5],[2],[6]]
=> [[1,2,4,5,6],[3]]
=> [3,3] => [[5,3],[2]]
=> ? = 4
[[1,2,4,5],[3],[6]]
=> [[1,2,3,5,6],[4]]
=> [4,2] => [[5,4],[3]]
=> ? = 4
[[1,2,3,5],[4],[6]]
=> [[1,2,3,4,6],[5]]
=> [5,1] => [[5,5],[4]]
=> ? = 3
[[1,2,3,4],[5],[6]]
=> [[1,2,3,4,5],[6]]
=> [6] => [[6],[]]
=> ? = 2
[[1,3,5],[2,4,6]]
=> [[1,2,4,6],[3,5]]
=> [3,2,1] => [[4,4,3],[3,2]]
=> ? = 5
[[1,2,5],[3,4,6]]
=> [[1,2,3,4,6],[5]]
=> [5,1] => [[5,5],[4]]
=> ? = 3
[[1,3,4],[2,5,6]]
=> [[1,2,4,5,6],[3]]
=> [3,3] => [[5,3],[2]]
=> ? = 4
[[1,2,4],[3,5,6]]
=> [[1,2,3,5,6],[4]]
=> [4,2] => [[5,4],[3]]
=> ? = 4
[[1,2,3],[4,5,6]]
=> [[1,2,3,4,5,6]]
=> [6] => [[6],[]]
=> ? = 2
[[1,4,6],[2,5],[3]]
=> [[1,2,5],[3,6],[4]]
=> [4,2] => [[5,4],[3]]
=> ? = 4
[[1,3,6],[2,5],[4]]
=> [[1,2,4,5],[3],[6]]
=> [3,3] => [[5,3],[2]]
=> ? = 4
[[1,2,6],[3,5],[4]]
=> [[1,2,3,5],[4],[6]]
=> [4,2] => [[5,4],[3]]
=> ? = 4
[[1,3,6],[2,4],[5]]
=> [[1,2,4],[3,5],[6]]
=> [3,3] => [[5,3],[2]]
=> ? = 4
[[1,2,6],[3,4],[5]]
=> [[1,2,3,4],[5],[6]]
=> [6] => [[6],[]]
=> ? = 2
[[1,4,5],[2,6],[3]]
=> [[1,2,5,6],[3],[4]]
=> [4,2] => [[5,4],[3]]
=> ? = 4
[[1,3,5],[2,6],[4]]
=> [[1,2,4,6],[3],[5]]
=> [3,2,1] => [[4,4,3],[3,2]]
=> ? = 5
[[1,2,5],[3,6],[4]]
=> [[1,2,3,6],[4],[5]]
=> [5,1] => [[5,5],[4]]
=> ? = 3
[[1,3,4],[2,6],[5]]
=> [[1,2,4,5,6],[3]]
=> [3,3] => [[5,3],[2]]
=> ? = 4
[[1,2,4],[3,6],[5]]
=> [[1,2,3,5,6],[4]]
=> [4,2] => [[5,4],[3]]
=> ? = 4
[[1,2,3],[4,6],[5]]
=> [[1,2,3,4,6],[5]]
=> [5,1] => [[5,5],[4]]
=> ? = 3
[[1,3,5],[2,4],[6]]
=> [[1,2,4,6],[3,5]]
=> [3,2,1] => [[4,4,3],[3,2]]
=> ? = 5
[[1,2,5],[3,4],[6]]
=> [[1,2,3,4],[5,6]]
=> [5,1] => [[5,5],[4]]
=> ? = 3
[[1,3,4],[2,5],[6]]
=> [[1,2,4,5],[3,6]]
=> [3,3] => [[5,3],[2]]
=> ? = 4
[[1,2,4],[3,5],[6]]
=> [[1,2,3,5],[4,6]]
=> [4,2] => [[5,4],[3]]
=> ? = 4
[[1,2,3],[4,5],[6]]
=> [[1,2,3,4,5],[6]]
=> [6] => [[6],[]]
=> ? = 2
[[1,5,6],[2],[3],[4]]
=> [[1,2,6],[3],[4],[5]]
=> [5,1] => [[5,5],[4]]
=> ? = 3
[[1,4,6],[2],[3],[5]]
=> [[1,2,5],[3],[4],[6]]
=> [4,2] => [[5,4],[3]]
=> ? = 4
[[1,3,6],[2],[4],[5]]
=> [[1,2,4],[3],[5],[6]]
=> [3,3] => [[5,3],[2]]
=> ? = 4
[[1,2,6],[3],[4],[5]]
=> [[1,2,3],[4],[5],[6]]
=> [6] => [[6],[]]
=> ? = 2
Description
The number of corners of a skew partition.
This is also known as the number of removable cells of the skew partition.
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