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Your data matches 18 different statistics following compositions of up to 3 maps.
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Matching statistic: St000734
St000734: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> 1
[[1,2]]
=> 2
[[1],[2]]
=> 1
[[1,2,3]]
=> 3
[[1,3],[2]]
=> 3
[[1,2],[3]]
=> 2
[[1],[2],[3]]
=> 1
[[1,2,3,4]]
=> 4
[[1,3,4],[2]]
=> 4
[[1,2,4],[3]]
=> 4
[[1,2,3],[4]]
=> 3
[[1,3],[2,4]]
=> 3
[[1,2],[3,4]]
=> 2
[[1,4],[2],[3]]
=> 4
[[1,3],[2],[4]]
=> 3
[[1,2],[3],[4]]
=> 2
[[1],[2],[3],[4]]
=> 1
[[1,2,3,4,5]]
=> 5
[[1,3,4,5],[2]]
=> 5
[[1,2,4,5],[3]]
=> 5
[[1,2,3,5],[4]]
=> 5
[[1,2,3,4],[5]]
=> 4
[[1,3,5],[2,4]]
=> 5
[[1,2,5],[3,4]]
=> 5
[[1,3,4],[2,5]]
=> 4
[[1,2,4],[3,5]]
=> 4
[[1,2,3],[4,5]]
=> 3
[[1,4,5],[2],[3]]
=> 5
[[1,3,5],[2],[4]]
=> 5
[[1,2,5],[3],[4]]
=> 5
[[1,3,4],[2],[5]]
=> 4
[[1,2,4],[3],[5]]
=> 4
[[1,2,3],[4],[5]]
=> 3
[[1,4],[2,5],[3]]
=> 4
[[1,3],[2,5],[4]]
=> 3
[[1,2],[3,5],[4]]
=> 2
[[1,3],[2,4],[5]]
=> 3
[[1,2],[3,4],[5]]
=> 2
[[1,5],[2],[3],[4]]
=> 5
[[1,4],[2],[3],[5]]
=> 4
[[1,3],[2],[4],[5]]
=> 3
[[1,2],[3],[4],[5]]
=> 2
[[1],[2],[3],[4],[5]]
=> 1
[[1,2,3,4,5,6]]
=> 6
[[1,3,4,5,6],[2]]
=> 6
[[1,2,4,5,6],[3]]
=> 6
[[1,2,3,5,6],[4]]
=> 6
[[1,2,3,4,6],[5]]
=> 6
[[1,2,3,4,5],[6]]
=> 5
[[1,3,5,6],[2,4]]
=> 6
Description
The last entry in the first row of a standard tableau.
Matching statistic: St000738
Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
St000738: Standard tableaux ⟶ ℤResult quality: 46% ●values known / values provided: 46%●distinct values known / distinct values provided: 55%
St000738: Standard tableaux ⟶ ℤResult quality: 46% ●values known / values provided: 46%●distinct values known / distinct values provided: 55%
Values
[[1]]
=> [[1]]
=> 1
[[1,2]]
=> [[1],[2]]
=> 2
[[1],[2]]
=> [[1,2]]
=> 1
[[1,2,3]]
=> [[1],[2],[3]]
=> 3
[[1,3],[2]]
=> [[1,2],[3]]
=> 3
[[1,2],[3]]
=> [[1,3],[2]]
=> 2
[[1],[2],[3]]
=> [[1,2,3]]
=> 1
[[1,2,3,4]]
=> [[1],[2],[3],[4]]
=> 4
[[1,3,4],[2]]
=> [[1,2],[3],[4]]
=> 4
[[1,2,4],[3]]
=> [[1,3],[2],[4]]
=> 4
[[1,2,3],[4]]
=> [[1,4],[2],[3]]
=> 3
[[1,3],[2,4]]
=> [[1,2],[3,4]]
=> 3
[[1,2],[3,4]]
=> [[1,3],[2,4]]
=> 2
[[1,4],[2],[3]]
=> [[1,2,3],[4]]
=> 4
[[1,3],[2],[4]]
=> [[1,2,4],[3]]
=> 3
[[1,2],[3],[4]]
=> [[1,3,4],[2]]
=> 2
[[1],[2],[3],[4]]
=> [[1,2,3,4]]
=> 1
[[1,2,3,4,5]]
=> [[1],[2],[3],[4],[5]]
=> 5
[[1,3,4,5],[2]]
=> [[1,2],[3],[4],[5]]
=> 5
[[1,2,4,5],[3]]
=> [[1,3],[2],[4],[5]]
=> 5
[[1,2,3,5],[4]]
=> [[1,4],[2],[3],[5]]
=> 5
[[1,2,3,4],[5]]
=> [[1,5],[2],[3],[4]]
=> 4
[[1,3,5],[2,4]]
=> [[1,2],[3,4],[5]]
=> 5
[[1,2,5],[3,4]]
=> [[1,3],[2,4],[5]]
=> 5
[[1,3,4],[2,5]]
=> [[1,2],[3,5],[4]]
=> 4
[[1,2,4],[3,5]]
=> [[1,3],[2,5],[4]]
=> 4
[[1,2,3],[4,5]]
=> [[1,4],[2,5],[3]]
=> 3
[[1,4,5],[2],[3]]
=> [[1,2,3],[4],[5]]
=> 5
[[1,3,5],[2],[4]]
=> [[1,2,4],[3],[5]]
=> 5
[[1,2,5],[3],[4]]
=> [[1,3,4],[2],[5]]
=> 5
[[1,3,4],[2],[5]]
=> [[1,2,5],[3],[4]]
=> 4
[[1,2,4],[3],[5]]
=> [[1,3,5],[2],[4]]
=> 4
[[1,2,3],[4],[5]]
=> [[1,4,5],[2],[3]]
=> 3
[[1,4],[2,5],[3]]
=> [[1,2,3],[4,5]]
=> 4
[[1,3],[2,5],[4]]
=> [[1,2,4],[3,5]]
=> 3
[[1,2],[3,5],[4]]
=> [[1,3,4],[2,5]]
=> 2
[[1,3],[2,4],[5]]
=> [[1,2,5],[3,4]]
=> 3
[[1,2],[3,4],[5]]
=> [[1,3,5],[2,4]]
=> 2
[[1,5],[2],[3],[4]]
=> [[1,2,3,4],[5]]
=> 5
[[1,4],[2],[3],[5]]
=> [[1,2,3,5],[4]]
=> 4
[[1,3],[2],[4],[5]]
=> [[1,2,4,5],[3]]
=> 3
[[1,2],[3],[4],[5]]
=> [[1,3,4,5],[2]]
=> 2
[[1],[2],[3],[4],[5]]
=> [[1,2,3,4,5]]
=> 1
[[1,2,3,4,5,6]]
=> [[1],[2],[3],[4],[5],[6]]
=> 6
[[1,3,4,5,6],[2]]
=> [[1,2],[3],[4],[5],[6]]
=> 6
[[1,2,4,5,6],[3]]
=> [[1,3],[2],[4],[5],[6]]
=> 6
[[1,2,3,5,6],[4]]
=> [[1,4],[2],[3],[5],[6]]
=> 6
[[1,2,3,4,6],[5]]
=> [[1,5],[2],[3],[4],[6]]
=> 6
[[1,2,3,4,5],[6]]
=> [[1,6],[2],[3],[4],[5]]
=> 5
[[1,3,5,6],[2,4]]
=> [[1,2],[3,4],[5],[6]]
=> 6
[[1,3,5,7,8],[2,4,6,9,10]]
=> [[1,2],[3,4],[5,6],[7,9],[8,10]]
=> ? = 8
[[1,3,5,6,9],[2,4,7,8,10]]
=> [[1,2],[3,4],[5,7],[6,8],[9,10]]
=> ? = 9
[[1,3,5,6,8],[2,4,7,9,10]]
=> [[1,2],[3,4],[5,7],[6,9],[8,10]]
=> ? = 8
[[1,3,5,6,7],[2,4,8,9,10]]
=> [[1,2],[3,4],[5,8],[6,9],[7,10]]
=> ? = 7
[[1,3,4,7,9],[2,5,6,8,10]]
=> [[1,2],[3,5],[4,6],[7,8],[9,10]]
=> ? = 9
[[1,3,4,7,8],[2,5,6,9,10]]
=> [[1,2],[3,5],[4,6],[7,9],[8,10]]
=> ? = 8
[[1,3,4,6,9],[2,5,7,8,10]]
=> [[1,2],[3,5],[4,7],[6,8],[9,10]]
=> ? = 9
[[1,3,4,6,8],[2,5,7,9,10]]
=> [[1,2],[3,5],[4,7],[6,9],[8,10]]
=> ? = 8
[[1,3,4,6,7],[2,5,8,9,10]]
=> [[1,2],[3,5],[4,8],[6,9],[7,10]]
=> ? = 7
[[1,3,4,5,9],[2,6,7,8,10]]
=> [[1,2],[3,6],[4,7],[5,8],[9,10]]
=> ? = 9
[[1,3,4,5,8],[2,6,7,9,10]]
=> [[1,2],[3,6],[4,7],[5,9],[8,10]]
=> ? = 8
[[1,3,4,5,7],[2,6,8,9,10]]
=> [[1,2],[3,6],[4,8],[5,9],[7,10]]
=> ? = 7
[[1,3,4,5,6],[2,7,8,9,10]]
=> [[1,2],[3,7],[4,8],[5,9],[6,10]]
=> ? = 6
[[1,2,5,7,9],[3,4,6,8,10]]
=> [[1,3],[2,4],[5,6],[7,8],[9,10]]
=> ? = 9
[[1,2,5,7,8],[3,4,6,9,10]]
=> [[1,3],[2,4],[5,6],[7,9],[8,10]]
=> ? = 8
[[1,2,5,6,9],[3,4,7,8,10]]
=> [[1,3],[2,4],[5,7],[6,8],[9,10]]
=> ? = 9
[[1,2,5,6,8],[3,4,7,9,10]]
=> [[1,3],[2,4],[5,7],[6,9],[8,10]]
=> ? = 8
[[1,2,5,6,7],[3,4,8,9,10]]
=> [[1,3],[2,4],[5,8],[6,9],[7,10]]
=> ? = 7
[[1,2,4,7,9],[3,5,6,8,10]]
=> [[1,3],[2,5],[4,6],[7,8],[9,10]]
=> ? = 9
[[1,2,4,7,8],[3,5,6,9,10]]
=> [[1,3],[2,5],[4,6],[7,9],[8,10]]
=> ? = 8
[[1,2,4,6,9],[3,5,7,8,10]]
=> [[1,3],[2,5],[4,7],[6,8],[9,10]]
=> ? = 9
[[1,2,4,6,8],[3,5,7,9,10]]
=> [[1,3],[2,5],[4,7],[6,9],[8,10]]
=> ? = 8
[[1,2,4,6,7],[3,5,8,9,10]]
=> [[1,3],[2,5],[4,8],[6,9],[7,10]]
=> ? = 7
[[1,2,4,5,9],[3,6,7,8,10]]
=> [[1,3],[2,6],[4,7],[5,8],[9,10]]
=> ? = 9
[[1,2,4,5,8],[3,6,7,9,10]]
=> [[1,3],[2,6],[4,7],[5,9],[8,10]]
=> ? = 8
[[1,2,4,5,7],[3,6,8,9,10]]
=> [[1,3],[2,6],[4,8],[5,9],[7,10]]
=> ? = 7
[[1,2,4,5,6],[3,7,8,9,10]]
=> [[1,3],[2,7],[4,8],[5,9],[6,10]]
=> ? = 6
[[1,2,3,7,9],[4,5,6,8,10]]
=> [[1,4],[2,5],[3,6],[7,8],[9,10]]
=> ? = 9
[[1,2,3,7,8],[4,5,6,9,10]]
=> [[1,4],[2,5],[3,6],[7,9],[8,10]]
=> ? = 8
[[1,2,3,6,9],[4,5,7,8,10]]
=> [[1,4],[2,5],[3,7],[6,8],[9,10]]
=> ? = 9
[[1,2,3,6,8],[4,5,7,9,10]]
=> [[1,4],[2,5],[3,7],[6,9],[8,10]]
=> ? = 8
[[1,2,3,6,7],[4,5,8,9,10]]
=> [[1,4],[2,5],[3,8],[6,9],[7,10]]
=> ? = 7
[[1,2,3,5,9],[4,6,7,8,10]]
=> [[1,4],[2,6],[3,7],[5,8],[9,10]]
=> ? = 9
[[1,2,3,5,8],[4,6,7,9,10]]
=> [[1,4],[2,6],[3,7],[5,9],[8,10]]
=> ? = 8
[[1,2,3,5,7],[4,6,8,9,10]]
=> [[1,4],[2,6],[3,8],[5,9],[7,10]]
=> ? = 7
[[1,2,3,5,6],[4,7,8,9,10]]
=> [[1,4],[2,7],[3,8],[5,9],[6,10]]
=> ? = 6
[[1,2,3,4,9],[5,6,7,8,10]]
=> [[1,5],[2,6],[3,7],[4,8],[9,10]]
=> ? = 9
[[1,2,3,4,8],[5,6,7,9,10]]
=> [[1,5],[2,6],[3,7],[4,9],[8,10]]
=> ? = 8
[[1,2,3,4,7],[5,6,8,9,10]]
=> [[1,5],[2,6],[3,8],[4,9],[7,10]]
=> ? = 7
[[1,2,3,4,6],[5,7,8,9,10]]
=> [[1,5],[2,7],[3,8],[4,9],[6,10]]
=> ? = 6
[[1,2,3,4,5],[6,7,8,9,10]]
=> [[1,6],[2,7],[3,8],[4,9],[5,10]]
=> ? = 5
[[1,3,5,7,9,10],[2,4,6,8,11,12]]
=> [[1,2],[3,4],[5,6],[7,8],[9,11],[10,12]]
=> ? = 10
[[1,3,5,7,8,11],[2,4,6,9,10,12]]
=> [[1,2],[3,4],[5,6],[7,9],[8,10],[11,12]]
=> ? = 11
[[1,3,5,7,8,10],[2,4,6,9,11,12]]
=> [[1,2],[3,4],[5,6],[7,9],[8,11],[10,12]]
=> ? = 10
[[1,3,5,7,8,9],[2,4,6,10,11,12]]
=> [[1,2],[3,4],[5,6],[7,10],[8,11],[9,12]]
=> ? = 9
[[1,3,5,6,9,11],[2,4,7,8,10,12]]
=> [[1,2],[3,4],[5,7],[6,8],[9,10],[11,12]]
=> ? = 11
[[1,3,5,6,9,10],[2,4,7,8,11,12]]
=> [[1,2],[3,4],[5,7],[6,8],[9,11],[10,12]]
=> ? = 10
[[1,3,5,6,8,11],[2,4,7,9,10,12]]
=> [[1,2],[3,4],[5,7],[6,9],[8,10],[11,12]]
=> ? = 11
[[1,3,5,6,8,10],[2,4,7,9,11,12]]
=> [[1,2],[3,4],[5,7],[6,9],[8,11],[10,12]]
=> ? = 10
[[1,3,5,6,8,9],[2,4,7,10,11,12]]
=> [[1,2],[3,4],[5,7],[6,10],[8,11],[9,12]]
=> ? = 9
Description
The first entry in the last row of a standard tableau.
For the last entry in the first row, see [[St000734]].
Matching statistic: St000645
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000645: Dyck paths ⟶ ℤResult quality: 37% ●values known / values provided: 37%●distinct values known / distinct values provided: 55%
Mp00066: Permutations —inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000645: Dyck paths ⟶ ℤResult quality: 37% ●values known / values provided: 37%●distinct values known / distinct values provided: 55%
Values
[[1]]
=> [1] => [1] => [1,0]
=> 0 = 1 - 1
[[1,2]]
=> [1,2] => [1,2] => [1,0,1,0]
=> 1 = 2 - 1
[[1],[2]]
=> [2,1] => [2,1] => [1,1,0,0]
=> 0 = 1 - 1
[[1,2,3]]
=> [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 2 = 3 - 1
[[1,3],[2]]
=> [2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> 2 = 3 - 1
[[1,2],[3]]
=> [3,1,2] => [2,3,1] => [1,1,0,1,0,0]
=> 1 = 2 - 1
[[1],[2],[3]]
=> [3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> 0 = 1 - 1
[[1,2,3,4]]
=> [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 3 = 4 - 1
[[1,3,4],[2]]
=> [2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 3 = 4 - 1
[[1,2,4],[3]]
=> [3,1,2,4] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 3 = 4 - 1
[[1,2,3],[4]]
=> [4,1,2,3] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 2 = 3 - 1
[[1,3],[2,4]]
=> [2,4,1,3] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[[1,2],[3,4]]
=> [3,4,1,2] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[[1,4],[2],[3]]
=> [3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 3 = 4 - 1
[[1,3],[2],[4]]
=> [4,2,1,3] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[[1,2],[3],[4]]
=> [4,3,1,2] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[[1],[2],[3],[4]]
=> [4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[[1,2,3,4,5]]
=> [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 4 = 5 - 1
[[1,3,4,5],[2]]
=> [2,1,3,4,5] => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> 4 = 5 - 1
[[1,2,4,5],[3]]
=> [3,1,2,4,5] => [2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
=> 4 = 5 - 1
[[1,2,3,5],[4]]
=> [4,1,2,3,5] => [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
=> 4 = 5 - 1
[[1,2,3,4],[5]]
=> [5,1,2,3,4] => [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> 3 = 4 - 1
[[1,3,5],[2,4]]
=> [2,4,1,3,5] => [3,1,4,2,5] => [1,1,1,0,0,1,0,0,1,0]
=> 4 = 5 - 1
[[1,2,5],[3,4]]
=> [3,4,1,2,5] => [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
=> 4 = 5 - 1
[[1,3,4],[2,5]]
=> [2,5,1,3,4] => [3,1,4,5,2] => [1,1,1,0,0,1,0,1,0,0]
=> 3 = 4 - 1
[[1,2,4],[3,5]]
=> [3,5,1,2,4] => [3,4,1,5,2] => [1,1,1,0,1,0,0,1,0,0]
=> 3 = 4 - 1
[[1,2,3],[4,5]]
=> [4,5,1,2,3] => [3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
=> 2 = 3 - 1
[[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> 4 = 5 - 1
[[1,3,5],[2],[4]]
=> [4,2,1,3,5] => [3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0]
=> 4 = 5 - 1
[[1,2,5],[3],[4]]
=> [4,3,1,2,5] => [3,4,2,1,5] => [1,1,1,0,1,0,0,0,1,0]
=> 4 = 5 - 1
[[1,3,4],[2],[5]]
=> [5,2,1,3,4] => [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
=> 3 = 4 - 1
[[1,2,4],[3],[5]]
=> [5,3,1,2,4] => [3,4,2,5,1] => [1,1,1,0,1,0,0,1,0,0]
=> 3 = 4 - 1
[[1,2,3],[4],[5]]
=> [5,4,1,2,3] => [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> 2 = 3 - 1
[[1,4],[2,5],[3]]
=> [3,2,5,1,4] => [4,2,1,5,3] => [1,1,1,1,0,0,0,1,0,0]
=> 3 = 4 - 1
[[1,3],[2,5],[4]]
=> [4,2,5,1,3] => [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> 2 = 3 - 1
[[1,2],[3,5],[4]]
=> [4,3,5,1,2] => [4,5,2,1,3] => [1,1,1,1,0,1,0,0,0,0]
=> 1 = 2 - 1
[[1,3],[2,4],[5]]
=> [5,2,4,1,3] => [4,2,5,3,1] => [1,1,1,1,0,0,1,0,0,0]
=> 2 = 3 - 1
[[1,2],[3,4],[5]]
=> [5,3,4,1,2] => [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> 1 = 2 - 1
[[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 4 = 5 - 1
[[1,4],[2],[3],[5]]
=> [5,3,2,1,4] => [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> 3 = 4 - 1
[[1,3],[2],[4],[5]]
=> [5,4,2,1,3] => [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
=> 2 = 3 - 1
[[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> 1 = 2 - 1
[[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> 5 = 6 - 1
[[1,3,4,5,6],[2]]
=> [2,1,3,4,5,6] => [2,1,3,4,5,6] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> 5 = 6 - 1
[[1,2,4,5,6],[3]]
=> [3,1,2,4,5,6] => [2,3,1,4,5,6] => [1,1,0,1,0,0,1,0,1,0,1,0]
=> 5 = 6 - 1
[[1,2,3,5,6],[4]]
=> [4,1,2,3,5,6] => [2,3,4,1,5,6] => [1,1,0,1,0,1,0,0,1,0,1,0]
=> 5 = 6 - 1
[[1,2,3,4,6],[5]]
=> [5,1,2,3,4,6] => [2,3,4,5,1,6] => [1,1,0,1,0,1,0,1,0,0,1,0]
=> 5 = 6 - 1
[[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => [2,3,4,5,6,1] => [1,1,0,1,0,1,0,1,0,1,0,0]
=> 4 = 5 - 1
[[1,3,5,6],[2,4]]
=> [2,4,1,3,5,6] => [3,1,4,2,5,6] => [1,1,1,0,0,1,0,0,1,0,1,0]
=> 5 = 6 - 1
[[1,2,4,6,7,8],[3,5]]
=> [3,5,1,2,4,6,7,8] => [3,4,1,5,2,6,7,8] => [1,1,1,0,1,0,0,1,0,0,1,0,1,0,1,0]
=> ? = 8 - 1
[[1,3,4,5,7,8],[2,6]]
=> [2,6,1,3,4,5,7,8] => [3,1,4,5,6,2,7,8] => ?
=> ? = 8 - 1
[[1,2,4,5,7,8],[3,6]]
=> [3,6,1,2,4,5,7,8] => [3,4,1,5,6,2,7,8] => [1,1,1,0,1,0,0,1,0,1,0,0,1,0,1,0]
=> ? = 8 - 1
[[1,2,3,5,6,8],[4,7]]
=> [4,7,1,2,3,5,6,8] => [3,4,5,1,6,7,2,8] => ?
=> ? = 8 - 1
[[1,2,3,4,6,8],[5,7]]
=> [5,7,1,2,3,4,6,8] => [3,4,5,6,1,7,2,8] => [1,1,1,0,1,0,1,0,1,0,0,1,0,0,1,0]
=> ? = 8 - 1
[[1,2,3,4,6,7],[5,8]]
=> [5,8,1,2,3,4,6,7] => [3,4,5,6,1,7,8,2] => [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 7 - 1
[[1,2,4,6,7,8],[3],[5]]
=> [5,3,1,2,4,6,7,8] => [3,4,2,5,1,6,7,8] => [1,1,1,0,1,0,0,1,0,0,1,0,1,0,1,0]
=> ? = 8 - 1
[[1,2,4,5,7,8],[3],[6]]
=> [6,3,1,2,4,5,7,8] => [3,4,2,5,6,1,7,8] => [1,1,1,0,1,0,0,1,0,1,0,0,1,0,1,0]
=> ? = 8 - 1
[[1,2,3,5,6,8],[4],[7]]
=> [7,4,1,2,3,5,6,8] => [3,4,5,2,6,7,1,8] => ?
=> ? = 8 - 1
[[1,2,3,4,6,8],[5],[7]]
=> [7,5,1,2,3,4,6,8] => [3,4,5,6,2,7,1,8] => [1,1,1,0,1,0,1,0,1,0,0,1,0,0,1,0]
=> ? = 8 - 1
[[1,2,3,4,6,7],[5],[8]]
=> [8,5,1,2,3,4,6,7] => [3,4,5,6,2,7,8,1] => [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 7 - 1
[[1,3,5,7,8],[2,4,6]]
=> [2,4,6,1,3,5,7,8] => [4,1,5,2,6,3,7,8] => [1,1,1,1,0,0,1,0,0,1,0,0,1,0,1,0]
=> ? = 8 - 1
[[1,2,5,7,8],[3,4,6]]
=> [3,4,6,1,2,5,7,8] => [4,5,1,2,6,3,7,8] => ?
=> ? = 8 - 1
[[1,3,4,7,8],[2,5,6]]
=> [2,5,6,1,3,4,7,8] => [4,1,5,6,2,3,7,8] => [1,1,1,1,0,0,1,0,1,0,0,0,1,0,1,0]
=> ? = 8 - 1
[[1,2,4,7,8],[3,5,6]]
=> [3,5,6,1,2,4,7,8] => [4,5,1,6,2,3,7,8] => [1,1,1,1,0,1,0,0,1,0,0,0,1,0,1,0]
=> ? = 8 - 1
[[1,2,3,7,8],[4,5,6]]
=> [4,5,6,1,2,3,7,8] => [4,5,6,1,2,3,7,8] => [1,1,1,1,0,1,0,1,0,0,0,0,1,0,1,0]
=> ? = 8 - 1
[[1,3,5,6,8],[2,4,7]]
=> [2,4,7,1,3,5,6,8] => [4,1,5,2,6,7,3,8] => ?
=> ? = 8 - 1
[[1,2,5,6,8],[3,4,7]]
=> [3,4,7,1,2,5,6,8] => [4,5,1,2,6,7,3,8] => [1,1,1,1,0,1,0,0,0,1,0,1,0,0,1,0]
=> ? = 8 - 1
[[1,3,4,6,8],[2,5,7]]
=> [2,5,7,1,3,4,6,8] => [4,1,5,6,2,7,3,8] => ?
=> ? = 8 - 1
[[1,2,3,6,8],[4,5,7]]
=> [4,5,7,1,2,3,6,8] => [4,5,6,1,2,7,3,8] => [1,1,1,1,0,1,0,1,0,0,0,1,0,0,1,0]
=> ? = 8 - 1
[[1,3,4,5,8],[2,6,7]]
=> [2,6,7,1,3,4,5,8] => [4,1,5,6,7,2,3,8] => [1,1,1,1,0,0,1,0,1,0,1,0,0,0,1,0]
=> ? = 8 - 1
[[1,2,4,5,8],[3,6,7]]
=> [3,6,7,1,2,4,5,8] => [4,5,1,6,7,2,3,8] => [1,1,1,1,0,1,0,0,1,0,1,0,0,0,1,0]
=> ? = 8 - 1
[[1,2,3,5,8],[4,6,7]]
=> [4,6,7,1,2,3,5,8] => [4,5,6,1,7,2,3,8] => [1,1,1,1,0,1,0,1,0,0,1,0,0,0,1,0]
=> ? = 8 - 1
[[1,2,3,4,8],[5,6,7]]
=> [5,6,7,1,2,3,4,8] => [4,5,6,7,1,2,3,8] => [1,1,1,1,0,1,0,1,0,1,0,0,0,0,1,0]
=> ? = 8 - 1
[[1,3,5,6,7],[2,4,8]]
=> [2,4,8,1,3,5,6,7] => [4,1,5,2,6,7,8,3] => [1,1,1,1,0,0,1,0,0,1,0,1,0,1,0,0]
=> ? = 7 - 1
[[1,3,4,6,7],[2,5,8]]
=> [2,5,8,1,3,4,6,7] => [4,1,5,6,2,7,8,3] => ?
=> ? = 7 - 1
[[1,2,4,6,7],[3,5,8]]
=> [3,5,8,1,2,4,6,7] => [4,5,1,6,2,7,8,3] => [1,1,1,1,0,1,0,0,1,0,0,1,0,1,0,0]
=> ? = 7 - 1
[[1,2,4,5,7],[3,6,8]]
=> [3,6,8,1,2,4,5,7] => [4,5,1,6,7,2,8,3] => [1,1,1,1,0,1,0,0,1,0,1,0,0,1,0,0]
=> ? = 7 - 1
[[1,2,3,5,7],[4,6,8]]
=> [4,6,8,1,2,3,5,7] => [4,5,6,1,7,2,8,3] => [1,1,1,1,0,1,0,1,0,0,1,0,0,1,0,0]
=> ? = 7 - 1
[[1,2,6,7,8],[3,5],[4]]
=> [4,3,5,1,2,6,7,8] => [4,5,2,1,3,6,7,8] => [1,1,1,1,0,1,0,0,0,0,1,0,1,0,1,0]
=> ? = 8 - 1
[[1,2,6,7,8],[3,4],[5]]
=> [5,3,4,1,2,6,7,8] => [4,5,2,3,1,6,7,8] => [1,1,1,1,0,1,0,0,0,0,1,0,1,0,1,0]
=> ? = 8 - 1
[[1,3,5,7,8],[2,6],[4]]
=> [4,2,6,1,3,5,7,8] => [4,2,5,1,6,3,7,8] => [1,1,1,1,0,0,1,0,0,1,0,0,1,0,1,0]
=> ? = 8 - 1
[[1,2,5,7,8],[3,6],[4]]
=> [4,3,6,1,2,5,7,8] => [4,5,2,1,6,3,7,8] => ?
=> ? = 8 - 1
[[1,3,4,7,8],[2,6],[5]]
=> [5,2,6,1,3,4,7,8] => [4,2,5,6,1,3,7,8] => ?
=> ? = 8 - 1
[[1,2,4,7,8],[3,6],[5]]
=> [5,3,6,1,2,4,7,8] => [4,5,2,6,1,3,7,8] => [1,1,1,1,0,1,0,0,1,0,0,0,1,0,1,0]
=> ? = 8 - 1
[[1,2,3,7,8],[4,6],[5]]
=> [5,4,6,1,2,3,7,8] => [4,5,6,2,1,3,7,8] => [1,1,1,1,0,1,0,1,0,0,0,0,1,0,1,0]
=> ? = 8 - 1
[[1,3,5,7,8],[2,4],[6]]
=> [6,2,4,1,3,5,7,8] => [4,2,5,3,6,1,7,8] => [1,1,1,1,0,0,1,0,0,1,0,0,1,0,1,0]
=> ? = 8 - 1
[[1,2,5,7,8],[3,4],[6]]
=> [6,3,4,1,2,5,7,8] => [4,5,2,3,6,1,7,8] => ?
=> ? = 8 - 1
[[1,3,4,7,8],[2,5],[6]]
=> [6,2,5,1,3,4,7,8] => [4,2,5,6,3,1,7,8] => ?
=> ? = 8 - 1
[[1,2,4,7,8],[3,5],[6]]
=> [6,3,5,1,2,4,7,8] => [4,5,2,6,3,1,7,8] => [1,1,1,1,0,1,0,0,1,0,0,0,1,0,1,0]
=> ? = 8 - 1
[[1,2,3,7,8],[4,5],[6]]
=> [6,4,5,1,2,3,7,8] => [4,5,6,2,3,1,7,8] => ?
=> ? = 8 - 1
[[1,3,5,6,8],[2,7],[4]]
=> [4,2,7,1,3,5,6,8] => [4,2,5,1,6,7,3,8] => ?
=> ? = 8 - 1
[[1,2,5,6,8],[3,7],[4]]
=> [4,3,7,1,2,5,6,8] => [4,5,2,1,6,7,3,8] => [1,1,1,1,0,1,0,0,0,1,0,1,0,0,1,0]
=> ? = 8 - 1
[[1,3,4,6,8],[2,7],[5]]
=> [5,2,7,1,3,4,6,8] => [4,2,5,6,1,7,3,8] => [1,1,1,1,0,0,1,0,1,0,0,1,0,0,1,0]
=> ? = 8 - 1
[[1,2,3,6,8],[4,7],[5]]
=> [5,4,7,1,2,3,6,8] => [4,5,6,2,1,7,3,8] => [1,1,1,1,0,1,0,1,0,0,0,1,0,0,1,0]
=> ? = 8 - 1
[[1,3,4,5,8],[2,7],[6]]
=> [6,2,7,1,3,4,5,8] => [4,2,5,6,7,1,3,8] => [1,1,1,1,0,0,1,0,1,0,1,0,0,0,1,0]
=> ? = 8 - 1
[[1,2,4,5,8],[3,7],[6]]
=> [6,3,7,1,2,4,5,8] => [4,5,2,6,7,1,3,8] => [1,1,1,1,0,1,0,0,1,0,1,0,0,0,1,0]
=> ? = 8 - 1
[[1,2,3,5,8],[4,7],[6]]
=> [6,4,7,1,2,3,5,8] => [4,5,6,2,7,1,3,8] => [1,1,1,1,0,1,0,1,0,0,1,0,0,0,1,0]
=> ? = 8 - 1
[[1,2,3,4,8],[5,7],[6]]
=> [6,5,7,1,2,3,4,8] => [4,5,6,7,2,1,3,8] => [1,1,1,1,0,1,0,1,0,1,0,0,0,0,1,0]
=> ? = 8 - 1
[[1,3,5,6,8],[2,4],[7]]
=> [7,2,4,1,3,5,6,8] => [4,2,5,3,6,7,1,8] => ?
=> ? = 8 - 1
Description
The sum of the areas of the rectangles formed by two consecutive peaks and the valley in between.
For a Dyck path $D = D_1 \cdots D_{2n}$ with peaks in positions $i_1 < \ldots < i_k$ and valleys in positions $j_1 < \ldots < j_{k-1}$, this statistic is given by
$$
\sum_{a=1}^{k-1} (j_a-i_a)(i_{a+1}-j_a)
$$
Matching statistic: St000439
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000439: Dyck paths ⟶ ℤResult quality: 36% ●values known / values provided: 36%●distinct values known / distinct values provided: 50%
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000439: Dyck paths ⟶ ℤResult quality: 36% ●values known / values provided: 36%●distinct values known / distinct values provided: 50%
Values
[[1]]
=> [[1]]
=> [1] => [1,0]
=> 2 = 1 + 1
[[1,2]]
=> [[1],[2]]
=> [2,1] => [1,1,0,0]
=> 3 = 2 + 1
[[1],[2]]
=> [[1,2]]
=> [1,2] => [1,0,1,0]
=> 2 = 1 + 1
[[1,2,3]]
=> [[1],[2],[3]]
=> [3,2,1] => [1,1,1,0,0,0]
=> 4 = 3 + 1
[[1,3],[2]]
=> [[1,2],[3]]
=> [3,1,2] => [1,1,1,0,0,0]
=> 4 = 3 + 1
[[1,2],[3]]
=> [[1,3],[2]]
=> [2,1,3] => [1,1,0,0,1,0]
=> 3 = 2 + 1
[[1],[2],[3]]
=> [[1,2,3]]
=> [1,2,3] => [1,0,1,0,1,0]
=> 2 = 1 + 1
[[1,2,3,4]]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
[[1,3,4],[2]]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
[[1,2,4],[3]]
=> [[1,3],[2],[4]]
=> [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
[[1,2,3],[4]]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
[[1,3],[2,4]]
=> [[1,2],[3,4]]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 4 = 3 + 1
[[1,2],[3,4]]
=> [[1,3],[2,4]]
=> [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[[1,4],[2],[3]]
=> [[1,2,3],[4]]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
[[1,3],[2],[4]]
=> [[1,2,4],[3]]
=> [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
[[1,2],[3],[4]]
=> [[1,3,4],[2]]
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[[1],[2],[3],[4]]
=> [[1,2,3,4]]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[[1,2,3,4,5]]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 6 = 5 + 1
[[1,3,4,5],[2]]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> 6 = 5 + 1
[[1,2,4,5],[3]]
=> [[1,3],[2],[4],[5]]
=> [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> 6 = 5 + 1
[[1,2,3,5],[4]]
=> [[1,4],[2],[3],[5]]
=> [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> 6 = 5 + 1
[[1,2,3,4],[5]]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 5 = 4 + 1
[[1,3,5],[2,4]]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> 6 = 5 + 1
[[1,2,5],[3,4]]
=> [[1,3],[2,4],[5]]
=> [5,2,4,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> 6 = 5 + 1
[[1,3,4],[2,5]]
=> [[1,2],[3,5],[4]]
=> [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> 5 = 4 + 1
[[1,2,4],[3,5]]
=> [[1,3],[2,5],[4]]
=> [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> 5 = 4 + 1
[[1,2,3],[4,5]]
=> [[1,4],[2,5],[3]]
=> [3,2,5,1,4] => [1,1,1,0,0,1,1,0,0,0]
=> 4 = 3 + 1
[[1,4,5],[2],[3]]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> 6 = 5 + 1
[[1,3,5],[2],[4]]
=> [[1,2,4],[3],[5]]
=> [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> 6 = 5 + 1
[[1,2,5],[3],[4]]
=> [[1,3,4],[2],[5]]
=> [5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> 6 = 5 + 1
[[1,3,4],[2],[5]]
=> [[1,2,5],[3],[4]]
=> [4,3,1,2,5] => [1,1,1,1,0,0,0,0,1,0]
=> 5 = 4 + 1
[[1,2,4],[3],[5]]
=> [[1,3,5],[2],[4]]
=> [4,2,1,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> 5 = 4 + 1
[[1,2,3],[4],[5]]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> 4 = 3 + 1
[[1,4],[2,5],[3]]
=> [[1,2,3],[4,5]]
=> [4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
=> 5 = 4 + 1
[[1,3],[2,5],[4]]
=> [[1,2,4],[3,5]]
=> [3,5,1,2,4] => [1,1,1,0,1,1,0,0,0,0]
=> 4 = 3 + 1
[[1,2],[3,5],[4]]
=> [[1,3,4],[2,5]]
=> [2,5,1,3,4] => [1,1,0,1,1,1,0,0,0,0]
=> 3 = 2 + 1
[[1,3],[2,4],[5]]
=> [[1,2,5],[3,4]]
=> [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
=> 4 = 3 + 1
[[1,2],[3,4],[5]]
=> [[1,3,5],[2,4]]
=> [2,4,1,3,5] => [1,1,0,1,1,0,0,0,1,0]
=> 3 = 2 + 1
[[1,5],[2],[3],[4]]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> 6 = 5 + 1
[[1,4],[2],[3],[5]]
=> [[1,2,3,5],[4]]
=> [4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> 5 = 4 + 1
[[1,3],[2],[4],[5]]
=> [[1,2,4,5],[3]]
=> [3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> 4 = 3 + 1
[[1,2],[3],[4],[5]]
=> [[1,3,4,5],[2]]
=> [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> 3 = 2 + 1
[[1],[2],[3],[4],[5]]
=> [[1,2,3,4,5]]
=> [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[[1,2,3,4,5,6]]
=> [[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 7 = 6 + 1
[[1,3,4,5,6],[2]]
=> [[1,2],[3],[4],[5],[6]]
=> [6,5,4,3,1,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 7 = 6 + 1
[[1,2,4,5,6],[3]]
=> [[1,3],[2],[4],[5],[6]]
=> [6,5,4,2,1,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 7 = 6 + 1
[[1,2,3,5,6],[4]]
=> [[1,4],[2],[3],[5],[6]]
=> [6,5,3,2,1,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 7 = 6 + 1
[[1,2,3,4,6],[5]]
=> [[1,5],[2],[3],[4],[6]]
=> [6,4,3,2,1,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 7 = 6 + 1
[[1,2,3,4,5],[6]]
=> [[1,6],[2],[3],[4],[5]]
=> [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 6 = 5 + 1
[[1,3,5,6],[2,4]]
=> [[1,2],[3,4],[5],[6]]
=> [6,5,3,4,1,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 7 = 6 + 1
[[1,3,4,6,7,8],[2,5]]
=> [[1,2],[3,5],[4],[6],[7],[8]]
=> [8,7,6,4,3,5,1,2] => ?
=> ? = 8 + 1
[[1,3,4,5,7,8],[2,6]]
=> [[1,2],[3,6],[4],[5],[7],[8]]
=> [8,7,5,4,3,6,1,2] => ?
=> ? = 8 + 1
[[1,2,4,5,6,8],[3,7]]
=> [[1,3],[2,7],[4],[5],[6],[8]]
=> [8,6,5,4,2,7,1,3] => ?
=> ? = 8 + 1
[[1,2,3,4,6,8],[5,7]]
=> [[1,5],[2,7],[3],[4],[6],[8]]
=> [8,6,4,3,2,7,1,5] => ?
=> ? = 8 + 1
[[1,2,3,5,6,7],[4,8]]
=> [[1,4],[2,8],[3],[5],[6],[7]]
=> [7,6,5,3,2,8,1,4] => ?
=> ? = 7 + 1
[[1,2,3,4,5,6],[7,8]]
=> [[1,7],[2,8],[3],[4],[5],[6]]
=> [6,5,4,3,2,8,1,7] => [1,1,1,1,1,1,0,0,0,0,0,1,1,0,0,0]
=> ? = 6 + 1
[[1,2,4,5,7,8],[3],[6]]
=> [[1,3,6],[2],[4],[5],[7],[8]]
=> [8,7,5,4,2,1,3,6] => ?
=> ? = 8 + 1
[[1,2,3,4,5,6],[7],[8]]
=> [[1,7,8],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1,7,8] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 6 + 1
[[1,3,4,6,7],[2,5,8]]
=> [[1,2],[3,5],[4,8],[6],[7]]
=> [7,6,4,8,3,5,1,2] => ?
=> ? = 7 + 1
[[1,3,4,5,7],[2,6,8]]
=> [[1,2],[3,6],[4,8],[5],[7]]
=> [7,5,4,8,3,6,1,2] => ?
=> ? = 7 + 1
[[1,2,5,7,8],[3,4],[6]]
=> [[1,3,6],[2,4],[5],[7],[8]]
=> [8,7,5,2,4,1,3,6] => ?
=> ? = 8 + 1
[[1,3,4,7,8],[2,5],[6]]
=> [[1,2,6],[3,5],[4],[7],[8]]
=> [8,7,4,3,5,1,2,6] => ?
=> ? = 8 + 1
[[1,2,4,7,8],[3,5],[6]]
=> [[1,3,6],[2,5],[4],[7],[8]]
=> [8,7,4,2,5,1,3,6] => ?
=> ? = 8 + 1
[[1,2,5,6,8],[3,7],[4]]
=> [[1,3,4],[2,7],[5],[6],[8]]
=> [8,6,5,2,7,1,3,4] => ?
=> ? = 8 + 1
[[1,2,4,6,8],[3,7],[5]]
=> [[1,3,5],[2,7],[4],[6],[8]]
=> [8,6,4,2,7,1,3,5] => ?
=> ? = 8 + 1
[[1,3,4,5,8],[2,7],[6]]
=> [[1,2,6],[3,7],[4],[5],[8]]
=> [8,5,4,3,7,1,2,6] => ?
=> ? = 8 + 1
[[1,2,4,5,8],[3,7],[6]]
=> [[1,3,6],[2,7],[4],[5],[8]]
=> [8,5,4,2,7,1,3,6] => ?
=> ? = 8 + 1
[[1,2,3,4,8],[5,7],[6]]
=> [[1,5,6],[2,7],[3],[4],[8]]
=> [8,4,3,2,7,1,5,6] => ?
=> ? = 8 + 1
[[1,3,4,6,8],[2,5],[7]]
=> [[1,2,7],[3,5],[4],[6],[8]]
=> [8,6,4,3,5,1,2,7] => ?
=> ? = 8 + 1
[[1,2,4,6,8],[3,5],[7]]
=> [[1,3,7],[2,5],[4],[6],[8]]
=> [8,6,4,2,5,1,3,7] => ?
=> ? = 8 + 1
[[1,2,3,4,8],[5,6],[7]]
=> [[1,5,7],[2,6],[3],[4],[8]]
=> [8,4,3,2,6,1,5,7] => ?
=> ? = 8 + 1
[[1,2,4,6,7],[3,8],[5]]
=> [[1,3,5],[2,8],[4],[6],[7]]
=> [7,6,4,2,8,1,3,5] => ?
=> ? = 7 + 1
[[1,3,4,5,7],[2,8],[6]]
=> [[1,2,6],[3,8],[4],[5],[7]]
=> [7,5,4,3,8,1,2,6] => ?
=> ? = 7 + 1
[[1,2,3,4,7],[5,8],[6]]
=> [[1,5,6],[2,8],[3],[4],[7]]
=> [7,4,3,2,8,1,5,6] => ?
=> ? = 7 + 1
[[1,3,4,5,6],[2,8],[7]]
=> [[1,2,7],[3,8],[4],[5],[6]]
=> [6,5,4,3,8,1,2,7] => [1,1,1,1,1,1,0,0,0,0,1,1,0,0,0,0]
=> ? = 6 + 1
[[1,2,4,5,6],[3,8],[7]]
=> [[1,3,7],[2,8],[4],[5],[6]]
=> [6,5,4,2,8,1,3,7] => [1,1,1,1,1,1,0,0,0,0,1,1,0,0,0,0]
=> ? = 6 + 1
[[1,2,3,5,6],[4,8],[7]]
=> [[1,4,7],[2,8],[3],[5],[6]]
=> [6,5,3,2,8,1,4,7] => [1,1,1,1,1,1,0,0,0,0,1,1,0,0,0,0]
=> ? = 6 + 1
[[1,2,3,4,6],[5,8],[7]]
=> [[1,5,7],[2,8],[3],[4],[6]]
=> [6,4,3,2,8,1,5,7] => ?
=> ? = 6 + 1
[[1,2,3,4,5],[6,8],[7]]
=> [[1,6,7],[2,8],[3],[4],[5]]
=> [5,4,3,2,8,1,6,7] => [1,1,1,1,1,0,0,0,0,1,1,1,0,0,0,0]
=> ? = 5 + 1
[[1,2,5,6,7],[3,4],[8]]
=> [[1,3,8],[2,4],[5],[6],[7]]
=> [7,6,5,2,4,1,3,8] => ?
=> ? = 7 + 1
[[1,3,4,6,7],[2,5],[8]]
=> [[1,2,8],[3,5],[4],[6],[7]]
=> [7,6,4,3,5,1,2,8] => ?
=> ? = 7 + 1
[[1,2,4,6,7],[3,5],[8]]
=> [[1,3,8],[2,5],[4],[6],[7]]
=> [7,6,4,2,5,1,3,8] => ?
=> ? = 7 + 1
[[1,3,4,5,7],[2,6],[8]]
=> [[1,2,8],[3,6],[4],[5],[7]]
=> [7,5,4,3,6,1,2,8] => ?
=> ? = 7 + 1
[[1,2,3,4,7],[5,6],[8]]
=> [[1,5,8],[2,6],[3],[4],[7]]
=> [7,4,3,2,6,1,5,8] => ?
=> ? = 7 + 1
[[1,3,4,5,6],[2,7],[8]]
=> [[1,2,8],[3,7],[4],[5],[6]]
=> [6,5,4,3,7,1,2,8] => [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
=> ? = 6 + 1
[[1,2,4,5,6],[3,7],[8]]
=> [[1,3,8],[2,7],[4],[5],[6]]
=> [6,5,4,2,7,1,3,8] => [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
=> ? = 6 + 1
[[1,2,3,5,6],[4,7],[8]]
=> [[1,4,8],[2,7],[3],[5],[6]]
=> [6,5,3,2,7,1,4,8] => [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
=> ? = 6 + 1
[[1,2,3,4,6],[5,7],[8]]
=> [[1,5,8],[2,7],[3],[4],[6]]
=> [6,4,3,2,7,1,5,8] => [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
=> ? = 6 + 1
[[1,2,3,4,5],[6,7],[8]]
=> [[1,6,8],[2,7],[3],[4],[5]]
=> [5,4,3,2,7,1,6,8] => [1,1,1,1,1,0,0,0,0,1,1,0,0,0,1,0]
=> ? = 5 + 1
[[1,3,4,7,8],[2],[5],[6]]
=> [[1,2,5,6],[3],[4],[7],[8]]
=> [8,7,4,3,1,2,5,6] => ?
=> ? = 8 + 1
[[1,3,5,6,8],[2],[4],[7]]
=> [[1,2,4,7],[3],[5],[6],[8]]
=> [8,6,5,3,1,2,4,7] => ?
=> ? = 8 + 1
[[1,3,4,5,8],[2],[6],[7]]
=> [[1,2,6,7],[3],[4],[5],[8]]
=> [8,5,4,3,1,2,6,7] => ?
=> ? = 8 + 1
[[1,3,5,6,7],[2],[4],[8]]
=> [[1,2,4,8],[3],[5],[6],[7]]
=> [7,6,5,3,1,2,4,8] => ?
=> ? = 7 + 1
[[1,2,4,5,7],[3],[6],[8]]
=> [[1,3,6,8],[2],[4],[5],[7]]
=> [7,5,4,2,1,3,6,8] => ?
=> ? = 7 + 1
[[1,3,4,5,6],[2],[7],[8]]
=> [[1,2,7,8],[3],[4],[5],[6]]
=> [6,5,4,3,1,2,7,8] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 6 + 1
[[1,2,4,5,6],[3],[7],[8]]
=> [[1,3,7,8],[2],[4],[5],[6]]
=> [6,5,4,2,1,3,7,8] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 6 + 1
[[1,2,3,5,6],[4],[7],[8]]
=> [[1,4,7,8],[2],[3],[5],[6]]
=> [6,5,3,2,1,4,7,8] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 6 + 1
[[1,2,3,4,6],[5],[7],[8]]
=> [[1,5,7,8],[2],[3],[4],[6]]
=> [6,4,3,2,1,5,7,8] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 6 + 1
[[1,2,3,4,5],[6],[7],[8]]
=> [[1,6,7,8],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7,8] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 5 + 1
[[1,4,5,8],[2,6,7],[3]]
=> [[1,2,3],[4,6],[5,7],[8]]
=> [8,5,7,4,6,1,2,3] => ?
=> ? = 8 + 1
Description
The position of the first down step of a Dyck path.
Matching statistic: St000839
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
Mp00284: Standard tableaux —rows⟶ Set partitions
Mp00164: Set partitions —Chen Deng Du Stanley Yan⟶ Set partitions
St000839: Set partitions ⟶ ℤResult quality: 24% ●values known / values provided: 24%●distinct values known / distinct values provided: 55%
Mp00284: Standard tableaux —rows⟶ Set partitions
Mp00164: Set partitions —Chen Deng Du Stanley Yan⟶ Set partitions
St000839: Set partitions ⟶ ℤResult quality: 24% ●values known / values provided: 24%●distinct values known / distinct values provided: 55%
Values
[[1]]
=> [[1]]
=> {{1}}
=> {{1}}
=> 1
[[1,2]]
=> [[1],[2]]
=> {{1},{2}}
=> {{1},{2}}
=> 2
[[1],[2]]
=> [[1,2]]
=> {{1,2}}
=> {{1,2}}
=> 1
[[1,2,3]]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> {{1},{2},{3}}
=> 3
[[1,3],[2]]
=> [[1,2],[3]]
=> {{1,2},{3}}
=> {{1,2},{3}}
=> 3
[[1,2],[3]]
=> [[1,3],[2]]
=> {{1,3},{2}}
=> {{1,3},{2}}
=> 2
[[1],[2],[3]]
=> [[1,2,3]]
=> {{1,2,3}}
=> {{1,2,3}}
=> 1
[[1,2,3,4]]
=> [[1],[2],[3],[4]]
=> {{1},{2},{3},{4}}
=> {{1},{2},{3},{4}}
=> 4
[[1,3,4],[2]]
=> [[1,2],[3],[4]]
=> {{1,2},{3},{4}}
=> {{1,2},{3},{4}}
=> 4
[[1,2,4],[3]]
=> [[1,3],[2],[4]]
=> {{1,3},{2},{4}}
=> {{1,3},{2},{4}}
=> 4
[[1,2,3],[4]]
=> [[1,4],[2],[3]]
=> {{1,4},{2},{3}}
=> {{1,4},{2},{3}}
=> 3
[[1,3],[2,4]]
=> [[1,2],[3,4]]
=> {{1,2},{3,4}}
=> {{1,2},{3,4}}
=> 3
[[1,2],[3,4]]
=> [[1,3],[2,4]]
=> {{1,3},{2,4}}
=> {{1,4},{2,3}}
=> 2
[[1,4],[2],[3]]
=> [[1,2,3],[4]]
=> {{1,2,3},{4}}
=> {{1,2,3},{4}}
=> 4
[[1,3],[2],[4]]
=> [[1,2,4],[3]]
=> {{1,2,4},{3}}
=> {{1,2,4},{3}}
=> 3
[[1,2],[3],[4]]
=> [[1,3,4],[2]]
=> {{1,3,4},{2}}
=> {{1,3,4},{2}}
=> 2
[[1],[2],[3],[4]]
=> [[1,2,3,4]]
=> {{1,2,3,4}}
=> {{1,2,3,4}}
=> 1
[[1,2,3,4,5]]
=> [[1],[2],[3],[4],[5]]
=> {{1},{2},{3},{4},{5}}
=> {{1},{2},{3},{4},{5}}
=> 5
[[1,3,4,5],[2]]
=> [[1,2],[3],[4],[5]]
=> {{1,2},{3},{4},{5}}
=> {{1,2},{3},{4},{5}}
=> 5
[[1,2,4,5],[3]]
=> [[1,3],[2],[4],[5]]
=> {{1,3},{2},{4},{5}}
=> {{1,3},{2},{4},{5}}
=> 5
[[1,2,3,5],[4]]
=> [[1,4],[2],[3],[5]]
=> {{1,4},{2},{3},{5}}
=> {{1,4},{2},{3},{5}}
=> 5
[[1,2,3,4],[5]]
=> [[1,5],[2],[3],[4]]
=> {{1,5},{2},{3},{4}}
=> {{1,5},{2},{3},{4}}
=> 4
[[1,3,5],[2,4]]
=> [[1,2],[3,4],[5]]
=> {{1,2},{3,4},{5}}
=> {{1,2},{3,4},{5}}
=> 5
[[1,2,5],[3,4]]
=> [[1,3],[2,4],[5]]
=> {{1,3},{2,4},{5}}
=> {{1,4},{2,3},{5}}
=> 5
[[1,3,4],[2,5]]
=> [[1,2],[3,5],[4]]
=> {{1,2},{3,5},{4}}
=> {{1,2},{3,5},{4}}
=> 4
[[1,2,4],[3,5]]
=> [[1,3],[2,5],[4]]
=> {{1,3},{2,5},{4}}
=> {{1,5},{2,3},{4}}
=> 4
[[1,2,3],[4,5]]
=> [[1,4],[2,5],[3]]
=> {{1,4},{2,5},{3}}
=> {{1,5},{2,4},{3}}
=> 3
[[1,4,5],[2],[3]]
=> [[1,2,3],[4],[5]]
=> {{1,2,3},{4},{5}}
=> {{1,2,3},{4},{5}}
=> 5
[[1,3,5],[2],[4]]
=> [[1,2,4],[3],[5]]
=> {{1,2,4},{3},{5}}
=> {{1,2,4},{3},{5}}
=> 5
[[1,2,5],[3],[4]]
=> [[1,3,4],[2],[5]]
=> {{1,3,4},{2},{5}}
=> {{1,3,4},{2},{5}}
=> 5
[[1,3,4],[2],[5]]
=> [[1,2,5],[3],[4]]
=> {{1,2,5},{3},{4}}
=> {{1,2,5},{3},{4}}
=> 4
[[1,2,4],[3],[5]]
=> [[1,3,5],[2],[4]]
=> {{1,3,5},{2},{4}}
=> {{1,3,5},{2},{4}}
=> 4
[[1,2,3],[4],[5]]
=> [[1,4,5],[2],[3]]
=> {{1,4,5},{2},{3}}
=> {{1,4,5},{2},{3}}
=> 3
[[1,4],[2,5],[3]]
=> [[1,2,3],[4,5]]
=> {{1,2,3},{4,5}}
=> {{1,2,3},{4,5}}
=> 4
[[1,3],[2,5],[4]]
=> [[1,2,4],[3,5]]
=> {{1,2,4},{3,5}}
=> {{1,2,5},{3,4}}
=> 3
[[1,2],[3,5],[4]]
=> [[1,3,4],[2,5]]
=> {{1,3,4},{2,5}}
=> {{1,4},{2,3,5}}
=> 2
[[1,3],[2,4],[5]]
=> [[1,2,5],[3,4]]
=> {{1,2,5},{3,4}}
=> {{1,2,4},{3,5}}
=> 3
[[1,2],[3,4],[5]]
=> [[1,3,5],[2,4]]
=> {{1,3,5},{2,4}}
=> {{1,5},{2,3,4}}
=> 2
[[1,5],[2],[3],[4]]
=> [[1,2,3,4],[5]]
=> {{1,2,3,4},{5}}
=> {{1,2,3,4},{5}}
=> 5
[[1,4],[2],[3],[5]]
=> [[1,2,3,5],[4]]
=> {{1,2,3,5},{4}}
=> {{1,2,3,5},{4}}
=> 4
[[1,3],[2],[4],[5]]
=> [[1,2,4,5],[3]]
=> {{1,2,4,5},{3}}
=> {{1,2,4,5},{3}}
=> 3
[[1,2],[3],[4],[5]]
=> [[1,3,4,5],[2]]
=> {{1,3,4,5},{2}}
=> {{1,3,4,5},{2}}
=> 2
[[1],[2],[3],[4],[5]]
=> [[1,2,3,4,5]]
=> {{1,2,3,4,5}}
=> {{1,2,3,4,5}}
=> 1
[[1,2,3,4,5,6]]
=> [[1],[2],[3],[4],[5],[6]]
=> {{1},{2},{3},{4},{5},{6}}
=> {{1},{2},{3},{4},{5},{6}}
=> 6
[[1,3,4,5,6],[2]]
=> [[1,2],[3],[4],[5],[6]]
=> {{1,2},{3},{4},{5},{6}}
=> {{1,2},{3},{4},{5},{6}}
=> 6
[[1,2,4,5,6],[3]]
=> [[1,3],[2],[4],[5],[6]]
=> {{1,3},{2},{4},{5},{6}}
=> {{1,3},{2},{4},{5},{6}}
=> 6
[[1,2,3,5,6],[4]]
=> [[1,4],[2],[3],[5],[6]]
=> {{1,4},{2},{3},{5},{6}}
=> {{1,4},{2},{3},{5},{6}}
=> 6
[[1,2,3,4,6],[5]]
=> [[1,5],[2],[3],[4],[6]]
=> {{1,5},{2},{3},{4},{6}}
=> {{1,5},{2},{3},{4},{6}}
=> 6
[[1,2,3,4,5],[6]]
=> [[1,6],[2],[3],[4],[5]]
=> {{1,6},{2},{3},{4},{5}}
=> {{1,6},{2},{3},{4},{5}}
=> 5
[[1,3,5,6],[2,4]]
=> [[1,2],[3,4],[5],[6]]
=> {{1,2},{3,4},{5},{6}}
=> {{1,2},{3,4},{5},{6}}
=> 6
[[1,3,4,5,6,7,8],[2]]
=> [[1,2],[3],[4],[5],[6],[7],[8]]
=> {{1,2},{3},{4},{5},{6},{7},{8}}
=> {{1,2},{3},{4},{5},{6},{7},{8}}
=> ? = 8
[[1,2,4,5,6,7,8],[3]]
=> [[1,3],[2],[4],[5],[6],[7],[8]]
=> {{1,3},{2},{4},{5},{6},{7},{8}}
=> {{1,3},{2},{4},{5},{6},{7},{8}}
=> ? = 8
[[1,2,3,5,6,7,8],[4]]
=> [[1,4],[2],[3],[5],[6],[7],[8]]
=> {{1,4},{2},{3},{5},{6},{7},{8}}
=> {{1,4},{2},{3},{5},{6},{7},{8}}
=> ? = 8
[[1,2,3,4,6,7,8],[5]]
=> [[1,5],[2],[3],[4],[6],[7],[8]]
=> {{1,5},{2},{3},{4},{6},{7},{8}}
=> {{1,5},{2},{3},{4},{6},{7},{8}}
=> ? = 8
[[1,2,3,4,5,7,8],[6]]
=> [[1,6],[2],[3],[4],[5],[7],[8]]
=> {{1,6},{2},{3},{4},{5},{7},{8}}
=> {{1,6},{2},{3},{4},{5},{7},{8}}
=> ? = 8
[[1,2,3,4,5,6,8],[7]]
=> [[1,7],[2],[3],[4],[5],[6],[8]]
=> {{1,7},{2},{3},{4},{5},{6},{8}}
=> {{1,7},{2},{3},{4},{5},{6},{8}}
=> ? = 8
[[1,3,5,6,7,8],[2,4]]
=> [[1,2],[3,4],[5],[6],[7],[8]]
=> {{1,2},{3,4},{5},{6},{7},{8}}
=> {{1,2},{3,4},{5},{6},{7},{8}}
=> ? = 8
[[1,2,5,6,7,8],[3,4]]
=> [[1,3],[2,4],[5],[6],[7],[8]]
=> {{1,3},{2,4},{5},{6},{7},{8}}
=> {{1,4},{2,3},{5},{6},{7},{8}}
=> ? = 8
[[1,3,4,6,7,8],[2,5]]
=> [[1,2],[3,5],[4],[6],[7],[8]]
=> {{1,2},{3,5},{4},{6},{7},{8}}
=> ?
=> ? = 8
[[1,2,4,6,7,8],[3,5]]
=> [[1,3],[2,5],[4],[6],[7],[8]]
=> {{1,3},{2,5},{4},{6},{7},{8}}
=> ?
=> ? = 8
[[1,2,3,6,7,8],[4,5]]
=> [[1,4],[2,5],[3],[6],[7],[8]]
=> {{1,4},{2,5},{3},{6},{7},{8}}
=> {{1,5},{2,4},{3},{6},{7},{8}}
=> ? = 8
[[1,3,4,5,7,8],[2,6]]
=> [[1,2],[3,6],[4],[5],[7],[8]]
=> {{1,2},{3,6},{4},{5},{7},{8}}
=> ?
=> ? = 8
[[1,2,4,5,7,8],[3,6]]
=> [[1,3],[2,6],[4],[5],[7],[8]]
=> {{1,3},{2,6},{4},{5},{7},{8}}
=> ?
=> ? = 8
[[1,2,3,5,7,8],[4,6]]
=> [[1,4],[2,6],[3],[5],[7],[8]]
=> {{1,4},{2,6},{3},{5},{7},{8}}
=> {{1,6},{2,4},{3},{5},{7},{8}}
=> ? = 8
[[1,2,3,4,7,8],[5,6]]
=> [[1,5],[2,6],[3],[4],[7],[8]]
=> {{1,5},{2,6},{3},{4},{7},{8}}
=> {{1,6},{2,5},{3},{4},{7},{8}}
=> ? = 8
[[1,3,4,5,6,8],[2,7]]
=> [[1,2],[3,7],[4],[5],[6],[8]]
=> {{1,2},{3,7},{4},{5},{6},{8}}
=> ?
=> ? = 8
[[1,2,4,5,6,8],[3,7]]
=> [[1,3],[2,7],[4],[5],[6],[8]]
=> {{1,3},{2,7},{4},{5},{6},{8}}
=> ?
=> ? = 8
[[1,2,3,5,6,8],[4,7]]
=> [[1,4],[2,7],[3],[5],[6],[8]]
=> {{1,4},{2,7},{3},{5},{6},{8}}
=> ?
=> ? = 8
[[1,2,3,4,6,8],[5,7]]
=> [[1,5],[2,7],[3],[4],[6],[8]]
=> {{1,5},{2,7},{3},{4},{6},{8}}
=> ?
=> ? = 8
[[1,2,3,4,5,8],[6,7]]
=> [[1,6],[2,7],[3],[4],[5],[8]]
=> {{1,6},{2,7},{3},{4},{5},{8}}
=> ?
=> ? = 8
[[1,3,4,5,6,7],[2,8]]
=> [[1,2],[3,8],[4],[5],[6],[7]]
=> {{1,2},{3,8},{4},{5},{6},{7}}
=> ?
=> ? = 7
[[1,2,3,4,6,7],[5,8]]
=> [[1,5],[2,8],[3],[4],[6],[7]]
=> {{1,5},{2,8},{3},{4},{6},{7}}
=> ?
=> ? = 7
[[1,2,3,4,5,7],[6,8]]
=> [[1,6],[2,8],[3],[4],[5],[7]]
=> {{1,6},{2,8},{3},{4},{5},{7}}
=> {{1,8},{2,6},{3},{4},{5},{7}}
=> ? = 7
[[1,2,3,4,5,6],[7,8]]
=> [[1,7],[2,8],[3],[4],[5],[6]]
=> {{1,7},{2,8},{3},{4},{5},{6}}
=> {{1,8},{2,7},{3},{4},{5},{6}}
=> ? = 6
[[1,4,5,6,7,8],[2],[3]]
=> [[1,2,3],[4],[5],[6],[7],[8]]
=> {{1,2,3},{4},{5},{6},{7},{8}}
=> {{1,2,3},{4},{5},{6},{7},{8}}
=> ? = 8
[[1,3,5,6,7,8],[2],[4]]
=> [[1,2,4],[3],[5],[6],[7],[8]]
=> {{1,2,4},{3},{5},{6},{7},{8}}
=> {{1,2,4},{3},{5},{6},{7},{8}}
=> ? = 8
[[1,2,5,6,7,8],[3],[4]]
=> [[1,3,4],[2],[5],[6],[7],[8]]
=> {{1,3,4},{2},{5},{6},{7},{8}}
=> {{1,3,4},{2},{5},{6},{7},{8}}
=> ? = 8
[[1,3,4,6,7,8],[2],[5]]
=> [[1,2,5],[3],[4],[6],[7],[8]]
=> {{1,2,5},{3},{4},{6},{7},{8}}
=> ?
=> ? = 8
[[1,2,4,6,7,8],[3],[5]]
=> [[1,3,5],[2],[4],[6],[7],[8]]
=> {{1,3,5},{2},{4},{6},{7},{8}}
=> {{1,3,5},{2},{4},{6},{7},{8}}
=> ? = 8
[[1,2,3,6,7,8],[4],[5]]
=> [[1,4,5],[2],[3],[6],[7],[8]]
=> {{1,4,5},{2},{3},{6},{7},{8}}
=> ?
=> ? = 8
[[1,3,4,5,7,8],[2],[6]]
=> [[1,2,6],[3],[4],[5],[7],[8]]
=> {{1,2,6},{3},{4},{5},{7},{8}}
=> ?
=> ? = 8
[[1,2,4,5,7,8],[3],[6]]
=> [[1,3,6],[2],[4],[5],[7],[8]]
=> {{1,3,6},{2},{4},{5},{7},{8}}
=> ?
=> ? = 8
[[1,2,3,5,7,8],[4],[6]]
=> [[1,4,6],[2],[3],[5],[7],[8]]
=> {{1,4,6},{2},{3},{5},{7},{8}}
=> ?
=> ? = 8
[[1,2,3,4,7,8],[5],[6]]
=> [[1,5,6],[2],[3],[4],[7],[8]]
=> {{1,5,6},{2},{3},{4},{7},{8}}
=> ?
=> ? = 8
[[1,3,4,5,6,8],[2],[7]]
=> [[1,2,7],[3],[4],[5],[6],[8]]
=> {{1,2,7},{3},{4},{5},{6},{8}}
=> ?
=> ? = 8
[[1,2,4,5,6,8],[3],[7]]
=> [[1,3,7],[2],[4],[5],[6],[8]]
=> {{1,3,7},{2},{4},{5},{6},{8}}
=> {{1,3,7},{2},{4},{5},{6},{8}}
=> ? = 8
[[1,2,3,5,6,8],[4],[7]]
=> [[1,4,7],[2],[3],[5],[6],[8]]
=> {{1,4,7},{2},{3},{5},{6},{8}}
=> ?
=> ? = 8
[[1,2,3,4,6,8],[5],[7]]
=> [[1,5,7],[2],[3],[4],[6],[8]]
=> {{1,5,7},{2},{3},{4},{6},{8}}
=> {{1,5,7},{2},{3},{4},{6},{8}}
=> ? = 8
[[1,2,3,4,5,8],[6],[7]]
=> [[1,6,7],[2],[3],[4],[5],[8]]
=> {{1,6,7},{2},{3},{4},{5},{8}}
=> ?
=> ? = 8
[[1,3,4,5,6,7],[2],[8]]
=> [[1,2,8],[3],[4],[5],[6],[7]]
=> {{1,2,8},{3},{4},{5},{6},{7}}
=> {{1,2,8},{3},{4},{5},{6},{7}}
=> ? = 7
[[1,2,4,5,6,7],[3],[8]]
=> [[1,3,8],[2],[4],[5],[6],[7]]
=> {{1,3,8},{2},{4},{5},{6},{7}}
=> ?
=> ? = 7
[[1,2,3,5,6,7],[4],[8]]
=> [[1,4,8],[2],[3],[5],[6],[7]]
=> {{1,4,8},{2},{3},{5},{6},{7}}
=> ?
=> ? = 7
[[1,2,3,4,6,7],[5],[8]]
=> [[1,5,8],[2],[3],[4],[6],[7]]
=> {{1,5,8},{2},{3},{4},{6},{7}}
=> {{1,5,8},{2},{3},{4},{6},{7}}
=> ? = 7
[[1,2,3,4,5,7],[6],[8]]
=> [[1,6,8],[2],[3],[4],[5],[7]]
=> {{1,6,8},{2},{3},{4},{5},{7}}
=> ?
=> ? = 7
[[1,2,3,4,5,6],[7],[8]]
=> [[1,7,8],[2],[3],[4],[5],[6]]
=> {{1,7,8},{2},{3},{4},{5},{6}}
=> {{1,7,8},{2},{3},{4},{5},{6}}
=> ? = 6
[[1,3,5,7,8],[2,4,6]]
=> [[1,2],[3,4],[5,6],[7],[8]]
=> {{1,2},{3,4},{5,6},{7},{8}}
=> {{1,2},{3,4},{5,6},{7},{8}}
=> ? = 8
[[1,2,5,7,8],[3,4,6]]
=> [[1,3],[2,4],[5,6],[7],[8]]
=> {{1,3},{2,4},{5,6},{7},{8}}
=> ?
=> ? = 8
[[1,3,4,7,8],[2,5,6]]
=> [[1,2],[3,5],[4,6],[7],[8]]
=> {{1,2},{3,5},{4,6},{7},{8}}
=> ?
=> ? = 8
[[1,2,4,7,8],[3,5,6]]
=> [[1,3],[2,5],[4,6],[7],[8]]
=> {{1,3},{2,5},{4,6},{7},{8}}
=> {{1,6},{2,3},{4,5},{7},{8}}
=> ? = 8
[[1,2,3,7,8],[4,5,6]]
=> [[1,4],[2,5],[3,6],[7],[8]]
=> {{1,4},{2,5},{3,6},{7},{8}}
=> {{1,6},{2,5},{3,4},{7},{8}}
=> ? = 8
Description
The largest opener of a set partition.
An opener (or left hand endpoint) of a set partition is a number that is minimal in its block. For this statistic, singletons are considered as openers.
Matching statistic: St000505
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00284: Standard tableaux —rows⟶ Set partitions
St000505: Set partitions ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 45%
St000505: Set partitions ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 45%
Values
[[1]]
=> {{1}}
=> 1
[[1,2]]
=> {{1,2}}
=> 2
[[1],[2]]
=> {{1},{2}}
=> 1
[[1,2,3]]
=> {{1,2,3}}
=> 3
[[1,3],[2]]
=> {{1,3},{2}}
=> 3
[[1,2],[3]]
=> {{1,2},{3}}
=> 2
[[1],[2],[3]]
=> {{1},{2},{3}}
=> 1
[[1,2,3,4]]
=> {{1,2,3,4}}
=> 4
[[1,3,4],[2]]
=> {{1,3,4},{2}}
=> 4
[[1,2,4],[3]]
=> {{1,2,4},{3}}
=> 4
[[1,2,3],[4]]
=> {{1,2,3},{4}}
=> 3
[[1,3],[2,4]]
=> {{1,3},{2,4}}
=> 3
[[1,2],[3,4]]
=> {{1,2},{3,4}}
=> 2
[[1,4],[2],[3]]
=> {{1,4},{2},{3}}
=> 4
[[1,3],[2],[4]]
=> {{1,3},{2},{4}}
=> 3
[[1,2],[3],[4]]
=> {{1,2},{3},{4}}
=> 2
[[1],[2],[3],[4]]
=> {{1},{2},{3},{4}}
=> 1
[[1,2,3,4,5]]
=> {{1,2,3,4,5}}
=> 5
[[1,3,4,5],[2]]
=> {{1,3,4,5},{2}}
=> 5
[[1,2,4,5],[3]]
=> {{1,2,4,5},{3}}
=> 5
[[1,2,3,5],[4]]
=> {{1,2,3,5},{4}}
=> 5
[[1,2,3,4],[5]]
=> {{1,2,3,4},{5}}
=> 4
[[1,3,5],[2,4]]
=> {{1,3,5},{2,4}}
=> 5
[[1,2,5],[3,4]]
=> {{1,2,5},{3,4}}
=> 5
[[1,3,4],[2,5]]
=> {{1,3,4},{2,5}}
=> 4
[[1,2,4],[3,5]]
=> {{1,2,4},{3,5}}
=> 4
[[1,2,3],[4,5]]
=> {{1,2,3},{4,5}}
=> 3
[[1,4,5],[2],[3]]
=> {{1,4,5},{2},{3}}
=> 5
[[1,3,5],[2],[4]]
=> {{1,3,5},{2},{4}}
=> 5
[[1,2,5],[3],[4]]
=> {{1,2,5},{3},{4}}
=> 5
[[1,3,4],[2],[5]]
=> {{1,3,4},{2},{5}}
=> 4
[[1,2,4],[3],[5]]
=> {{1,2,4},{3},{5}}
=> 4
[[1,2,3],[4],[5]]
=> {{1,2,3},{4},{5}}
=> 3
[[1,4],[2,5],[3]]
=> {{1,4},{2,5},{3}}
=> 4
[[1,3],[2,5],[4]]
=> {{1,3},{2,5},{4}}
=> 3
[[1,2],[3,5],[4]]
=> {{1,2},{3,5},{4}}
=> 2
[[1,3],[2,4],[5]]
=> {{1,3},{2,4},{5}}
=> 3
[[1,2],[3,4],[5]]
=> {{1,2},{3,4},{5}}
=> 2
[[1,5],[2],[3],[4]]
=> {{1,5},{2},{3},{4}}
=> 5
[[1,4],[2],[3],[5]]
=> {{1,4},{2},{3},{5}}
=> 4
[[1,3],[2],[4],[5]]
=> {{1,3},{2},{4},{5}}
=> 3
[[1,2],[3],[4],[5]]
=> {{1,2},{3},{4},{5}}
=> 2
[[1],[2],[3],[4],[5]]
=> {{1},{2},{3},{4},{5}}
=> 1
[[1,2,3,4,5,6]]
=> {{1,2,3,4,5,6}}
=> 6
[[1,3,4,5,6],[2]]
=> {{1,3,4,5,6},{2}}
=> 6
[[1,2,4,5,6],[3]]
=> {{1,2,4,5,6},{3}}
=> 6
[[1,2,3,5,6],[4]]
=> {{1,2,3,5,6},{4}}
=> 6
[[1,2,3,4,6],[5]]
=> {{1,2,3,4,6},{5}}
=> 6
[[1,2,3,4,5],[6]]
=> {{1,2,3,4,5},{6}}
=> 5
[[1,3,5,6],[2,4]]
=> {{1,3,5,6},{2,4}}
=> 6
[[1,2,5,6,7,8],[3],[4]]
=> {{1,2,5,6,7,8},{3},{4}}
=> ? = 8
[[1,3,4,6,7,8],[2],[5]]
=> {{1,3,4,6,7,8},{2},{5}}
=> ? = 8
[[1,2,4,6,7,8],[3],[5]]
=> {{1,2,4,6,7,8},{3},{5}}
=> ? = 8
[[1,2,3,6,7,8],[4],[5]]
=> {{1,2,3,6,7,8},{4},{5}}
=> ? = 8
[[1,3,4,5,7,8],[2],[6]]
=> {{1,3,4,5,7,8},{2},{6}}
=> ? = 8
[[1,2,4,5,7,8],[3],[6]]
=> {{1,2,4,5,7,8},{3},{6}}
=> ? = 8
[[1,2,3,5,7,8],[4],[6]]
=> {{1,2,3,5,7,8},{4},{6}}
=> ? = 8
[[1,2,3,4,7,8],[5],[6]]
=> {{1,2,3,4,7,8},{5},{6}}
=> ? = 8
[[1,3,4,5,6,8],[2],[7]]
=> {{1,3,4,5,6,8},{2},{7}}
=> ? = 8
[[1,2,4,5,6,8],[3],[7]]
=> {{1,2,4,5,6,8},{3},{7}}
=> ? = 8
[[1,2,3,5,6,8],[4],[7]]
=> {{1,2,3,5,6,8},{4},{7}}
=> ? = 8
[[1,2,3,4,6,8],[5],[7]]
=> {{1,2,3,4,6,8},{5},{7}}
=> ? = 8
[[1,2,3,4,5,8],[6],[7]]
=> {{1,2,3,4,5,8},{6},{7}}
=> ? = 8
[[1,3,4,5,6,7],[2],[8]]
=> {{1,3,4,5,6,7},{2},{8}}
=> ? = 7
[[1,2,4,5,6,7],[3],[8]]
=> {{1,2,4,5,6,7},{3},{8}}
=> ? = 7
[[1,2,3,4,5,6],[7],[8]]
=> {{1,2,3,4,5,6},{7},{8}}
=> ? = 6
[[1,3,5,7,8],[2,4,6]]
=> {{1,3,5,7,8},{2,4,6}}
=> ? = 8
[[1,2,5,7,8],[3,4,6]]
=> {{1,2,5,7,8},{3,4,6}}
=> ? = 8
[[1,3,4,7,8],[2,5,6]]
=> {{1,3,4,7,8},{2,5,6}}
=> ? = 8
[[1,2,4,7,8],[3,5,6]]
=> {{1,2,4,7,8},{3,5,6}}
=> ? = 8
[[1,2,3,7,8],[4,5,6]]
=> {{1,2,3,7,8},{4,5,6}}
=> ? = 8
[[1,3,5,6,8],[2,4,7]]
=> {{1,3,5,6,8},{2,4,7}}
=> ? = 8
[[1,2,5,6,8],[3,4,7]]
=> {{1,2,5,6,8},{3,4,7}}
=> ? = 8
[[1,3,4,6,8],[2,5,7]]
=> {{1,3,4,6,8},{2,5,7}}
=> ? = 8
[[1,2,4,6,8],[3,5,7]]
=> {{1,2,4,6,8},{3,5,7}}
=> ? = 8
[[1,2,3,6,8],[4,5,7]]
=> {{1,2,3,6,8},{4,5,7}}
=> ? = 8
[[1,3,4,5,8],[2,6,7]]
=> {{1,3,4,5,8},{2,6,7}}
=> ? = 8
[[1,2,4,5,8],[3,6,7]]
=> {{1,2,4,5,8},{3,6,7}}
=> ? = 8
[[1,2,3,5,8],[4,6,7]]
=> {{1,2,3,5,8},{4,6,7}}
=> ? = 8
[[1,2,3,4,8],[5,6,7]]
=> {{1,2,3,4,8},{5,6,7}}
=> ? = 8
[[1,3,5,6,7],[2,4,8]]
=> {{1,3,5,6,7},{2,4,8}}
=> ? = 7
[[1,2,5,6,7],[3,4,8]]
=> {{1,2,5,6,7},{3,4,8}}
=> ? = 7
[[1,3,4,6,7],[2,5,8]]
=> {{1,3,4,6,7},{2,5,8}}
=> ? = 7
[[1,2,4,6,7],[3,5,8]]
=> {{1,2,4,6,7},{3,5,8}}
=> ? = 7
[[1,2,3,6,7],[4,5,8]]
=> {{1,2,3,6,7},{4,5,8}}
=> ? = 7
[[1,3,4,5,7],[2,6,8]]
=> {{1,3,4,5,7},{2,6,8}}
=> ? = 7
[[1,2,4,5,7],[3,6,8]]
=> {{1,2,4,5,7},{3,6,8}}
=> ? = 7
[[1,2,3,5,7],[4,6,8]]
=> {{1,2,3,5,7},{4,6,8}}
=> ? = 7
[[1,2,3,4,7],[5,6,8]]
=> {{1,2,3,4,7},{5,6,8}}
=> ? = 7
[[1,3,4,5,6],[2,7,8]]
=> {{1,3,4,5,6},{2,7,8}}
=> ? = 6
[[1,2,4,5,6],[3,7,8]]
=> {{1,2,4,5,6},{3,7,8}}
=> ? = 6
[[1,2,3,5,6],[4,7,8]]
=> {{1,2,3,5,6},{4,7,8}}
=> ? = 6
[[1,2,3,4,6],[5,7,8]]
=> {{1,2,3,4,6},{5,7,8}}
=> ? = 6
[[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 5
[[1,4,6,7,8],[2,5],[3]]
=> {{1,4,6,7,8},{2,5},{3}}
=> ? = 8
[[1,3,6,7,8],[2,5],[4]]
=> {{1,3,6,7,8},{2,5},{4}}
=> ? = 8
[[1,2,6,7,8],[3,5],[4]]
=> {{1,2,6,7,8},{3,5},{4}}
=> ? = 8
[[1,3,6,7,8],[2,4],[5]]
=> {{1,3,6,7,8},{2,4},{5}}
=> ? = 8
[[1,2,6,7,8],[3,4],[5]]
=> {{1,2,6,7,8},{3,4},{5}}
=> ? = 8
[[1,4,5,7,8],[2,6],[3]]
=> {{1,4,5,7,8},{2,6},{3}}
=> ? = 8
Description
The biggest entry in the block containing the 1.
Matching statistic: St000054
(load all 12 compositions to match this statistic)
(load all 12 compositions to match this statistic)
Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000054: Permutations ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 55%
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000054: Permutations ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 55%
Values
[[1]]
=> [[1]]
=> [1] => 1
[[1,2]]
=> [[1],[2]]
=> [2,1] => 2
[[1],[2]]
=> [[1,2]]
=> [1,2] => 1
[[1,2,3]]
=> [[1],[2],[3]]
=> [3,2,1] => 3
[[1,3],[2]]
=> [[1,2],[3]]
=> [3,1,2] => 3
[[1,2],[3]]
=> [[1,3],[2]]
=> [2,1,3] => 2
[[1],[2],[3]]
=> [[1,2,3]]
=> [1,2,3] => 1
[[1,2,3,4]]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 4
[[1,3,4],[2]]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 4
[[1,2,4],[3]]
=> [[1,3],[2],[4]]
=> [4,2,1,3] => 4
[[1,2,3],[4]]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 3
[[1,3],[2,4]]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 3
[[1,2],[3,4]]
=> [[1,3],[2,4]]
=> [2,4,1,3] => 2
[[1,4],[2],[3]]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 4
[[1,3],[2],[4]]
=> [[1,2,4],[3]]
=> [3,1,2,4] => 3
[[1,2],[3],[4]]
=> [[1,3,4],[2]]
=> [2,1,3,4] => 2
[[1],[2],[3],[4]]
=> [[1,2,3,4]]
=> [1,2,3,4] => 1
[[1,2,3,4,5]]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => 5
[[1,3,4,5],[2]]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 5
[[1,2,4,5],[3]]
=> [[1,3],[2],[4],[5]]
=> [5,4,2,1,3] => 5
[[1,2,3,5],[4]]
=> [[1,4],[2],[3],[5]]
=> [5,3,2,1,4] => 5
[[1,2,3,4],[5]]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => 4
[[1,3,5],[2,4]]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 5
[[1,2,5],[3,4]]
=> [[1,3],[2,4],[5]]
=> [5,2,4,1,3] => 5
[[1,3,4],[2,5]]
=> [[1,2],[3,5],[4]]
=> [4,3,5,1,2] => 4
[[1,2,4],[3,5]]
=> [[1,3],[2,5],[4]]
=> [4,2,5,1,3] => 4
[[1,2,3],[4,5]]
=> [[1,4],[2,5],[3]]
=> [3,2,5,1,4] => 3
[[1,4,5],[2],[3]]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 5
[[1,3,5],[2],[4]]
=> [[1,2,4],[3],[5]]
=> [5,3,1,2,4] => 5
[[1,2,5],[3],[4]]
=> [[1,3,4],[2],[5]]
=> [5,2,1,3,4] => 5
[[1,3,4],[2],[5]]
=> [[1,2,5],[3],[4]]
=> [4,3,1,2,5] => 4
[[1,2,4],[3],[5]]
=> [[1,3,5],[2],[4]]
=> [4,2,1,3,5] => 4
[[1,2,3],[4],[5]]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 3
[[1,4],[2,5],[3]]
=> [[1,2,3],[4,5]]
=> [4,5,1,2,3] => 4
[[1,3],[2,5],[4]]
=> [[1,2,4],[3,5]]
=> [3,5,1,2,4] => 3
[[1,2],[3,5],[4]]
=> [[1,3,4],[2,5]]
=> [2,5,1,3,4] => 2
[[1,3],[2,4],[5]]
=> [[1,2,5],[3,4]]
=> [3,4,1,2,5] => 3
[[1,2],[3,4],[5]]
=> [[1,3,5],[2,4]]
=> [2,4,1,3,5] => 2
[[1,5],[2],[3],[4]]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 5
[[1,4],[2],[3],[5]]
=> [[1,2,3,5],[4]]
=> [4,1,2,3,5] => 4
[[1,3],[2],[4],[5]]
=> [[1,2,4,5],[3]]
=> [3,1,2,4,5] => 3
[[1,2],[3],[4],[5]]
=> [[1,3,4,5],[2]]
=> [2,1,3,4,5] => 2
[[1],[2],[3],[4],[5]]
=> [[1,2,3,4,5]]
=> [1,2,3,4,5] => 1
[[1,2,3,4,5,6]]
=> [[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => 6
[[1,3,4,5,6],[2]]
=> [[1,2],[3],[4],[5],[6]]
=> [6,5,4,3,1,2] => 6
[[1,2,4,5,6],[3]]
=> [[1,3],[2],[4],[5],[6]]
=> [6,5,4,2,1,3] => 6
[[1,2,3,5,6],[4]]
=> [[1,4],[2],[3],[5],[6]]
=> [6,5,3,2,1,4] => 6
[[1,2,3,4,6],[5]]
=> [[1,5],[2],[3],[4],[6]]
=> [6,4,3,2,1,5] => 6
[[1,2,3,4,5],[6]]
=> [[1,6],[2],[3],[4],[5]]
=> [5,4,3,2,1,6] => 5
[[1,3,5,6],[2,4]]
=> [[1,2],[3,4],[5],[6]]
=> [6,5,3,4,1,2] => 6
[[1,2,4,5,6,7],[3]]
=> [[1,3],[2],[4],[5],[6],[7]]
=> [7,6,5,4,2,1,3] => ? = 7
[[1,2,3,5,6,7],[4]]
=> [[1,4],[2],[3],[5],[6],[7]]
=> [7,6,5,3,2,1,4] => ? = 7
[[1,2,3,4,6,7],[5]]
=> [[1,5],[2],[3],[4],[6],[7]]
=> [7,6,4,3,2,1,5] => ? = 7
[[1,2,3,4,5,7],[6]]
=> [[1,6],[2],[3],[4],[5],[7]]
=> [7,5,4,3,2,1,6] => ? = 7
[[1,2,5,6,7],[3,4]]
=> [[1,3],[2,4],[5],[6],[7]]
=> [7,6,5,2,4,1,3] => ? = 7
[[1,3,4,6,7],[2,5]]
=> [[1,2],[3,5],[4],[6],[7]]
=> [7,6,4,3,5,1,2] => ? = 7
[[1,2,4,6,7],[3,5]]
=> [[1,3],[2,5],[4],[6],[7]]
=> [7,6,4,2,5,1,3] => ? = 7
[[1,2,3,6,7],[4,5]]
=> [[1,4],[2,5],[3],[6],[7]]
=> [7,6,3,2,5,1,4] => ? = 7
[[1,3,4,5,7],[2,6]]
=> [[1,2],[3,6],[4],[5],[7]]
=> [7,5,4,3,6,1,2] => ? = 7
[[1,2,4,5,7],[3,6]]
=> [[1,3],[2,6],[4],[5],[7]]
=> [7,5,4,2,6,1,3] => ? = 7
[[1,2,3,5,7],[4,6]]
=> [[1,4],[2,6],[3],[5],[7]]
=> [7,5,3,2,6,1,4] => ? = 7
[[1,2,3,4,7],[5,6]]
=> [[1,5],[2,6],[3],[4],[7]]
=> [7,4,3,2,6,1,5] => ? = 7
[[1,3,4,5,6],[2,7]]
=> [[1,2],[3,7],[4],[5],[6]]
=> [6,5,4,3,7,1,2] => ? = 6
[[1,2,4,5,6],[3,7]]
=> [[1,3],[2,7],[4],[5],[6]]
=> [6,5,4,2,7,1,3] => ? = 6
[[1,2,3,5,6],[4,7]]
=> [[1,4],[2,7],[3],[5],[6]]
=> [6,5,3,2,7,1,4] => ? = 6
[[1,2,3,4,5],[6,7]]
=> [[1,6],[2,7],[3],[4],[5]]
=> [5,4,3,2,7,1,6] => ? = 5
[[1,3,5,6,7],[2],[4]]
=> [[1,2,4],[3],[5],[6],[7]]
=> [7,6,5,3,1,2,4] => ? = 7
[[1,2,5,6,7],[3],[4]]
=> [[1,3,4],[2],[5],[6],[7]]
=> [7,6,5,2,1,3,4] => ? = 7
[[1,3,4,6,7],[2],[5]]
=> [[1,2,5],[3],[4],[6],[7]]
=> [7,6,4,3,1,2,5] => ? = 7
[[1,2,4,6,7],[3],[5]]
=> [[1,3,5],[2],[4],[6],[7]]
=> [7,6,4,2,1,3,5] => ? = 7
[[1,2,3,6,7],[4],[5]]
=> [[1,4,5],[2],[3],[6],[7]]
=> [7,6,3,2,1,4,5] => ? = 7
[[1,3,4,5,7],[2],[6]]
=> [[1,2,6],[3],[4],[5],[7]]
=> [7,5,4,3,1,2,6] => ? = 7
[[1,2,4,5,7],[3],[6]]
=> [[1,3,6],[2],[4],[5],[7]]
=> [7,5,4,2,1,3,6] => ? = 7
[[1,2,3,5,7],[4],[6]]
=> [[1,4,6],[2],[3],[5],[7]]
=> [7,5,3,2,1,4,6] => ? = 7
[[1,2,3,4,7],[5],[6]]
=> [[1,5,6],[2],[3],[4],[7]]
=> [7,4,3,2,1,5,6] => ? = 7
[[1,3,4,5,6],[2],[7]]
=> [[1,2,7],[3],[4],[5],[6]]
=> [6,5,4,3,1,2,7] => ? = 6
[[1,2,4,5,6],[3],[7]]
=> [[1,3,7],[2],[4],[5],[6]]
=> [6,5,4,2,1,3,7] => ? = 6
[[1,2,3,5,6],[4],[7]]
=> [[1,4,7],[2],[3],[5],[6]]
=> [6,5,3,2,1,4,7] => ? = 6
[[1,2,3,4,6],[5],[7]]
=> [[1,5,7],[2],[3],[4],[6]]
=> [6,4,3,2,1,5,7] => ? = 6
[[1,2,5,7],[3,4,6]]
=> [[1,3],[2,4],[5,6],[7]]
=> [7,5,6,2,4,1,3] => ? = 7
[[1,3,4,7],[2,5,6]]
=> [[1,2],[3,5],[4,6],[7]]
=> [7,4,6,3,5,1,2] => ? = 7
[[1,2,4,7],[3,5,6]]
=> [[1,3],[2,5],[4,6],[7]]
=> [7,4,6,2,5,1,3] => ? = 7
[[1,2,3,7],[4,5,6]]
=> [[1,4],[2,5],[3,6],[7]]
=> [7,3,6,2,5,1,4] => ? = 7
[[1,2,5,6],[3,4,7]]
=> [[1,3],[2,4],[5,7],[6]]
=> [6,5,7,2,4,1,3] => ? = 6
[[1,2,3,6],[4,5,7]]
=> [[1,4],[2,5],[3,7],[6]]
=> [6,3,7,2,5,1,4] => ? = 6
[[1,3,4,5],[2,6,7]]
=> [[1,2],[3,6],[4,7],[5]]
=> [5,4,7,3,6,1,2] => ? = 5
[[1,2,4,5],[3,6,7]]
=> [[1,3],[2,6],[4,7],[5]]
=> [5,4,7,2,6,1,3] => ? = 5
[[1,2,3,5],[4,6,7]]
=> [[1,4],[2,6],[3,7],[5]]
=> [5,3,7,2,6,1,4] => ? = 5
[[1,2,3,4],[5,6,7]]
=> [[1,5],[2,6],[3,7],[4]]
=> [4,3,7,2,6,1,5] => ? = 4
[[1,3,6,7],[2,5],[4]]
=> [[1,2,4],[3,5],[6],[7]]
=> [7,6,3,5,1,2,4] => ? = 7
[[1,2,6,7],[3,5],[4]]
=> [[1,3,4],[2,5],[6],[7]]
=> [7,6,2,5,1,3,4] => ? = 7
[[1,3,6,7],[2,4],[5]]
=> [[1,2,5],[3,4],[6],[7]]
=> [7,6,3,4,1,2,5] => ? = 7
[[1,2,6,7],[3,4],[5]]
=> [[1,3,5],[2,4],[6],[7]]
=> [7,6,2,4,1,3,5] => ? = 7
[[1,4,5,7],[2,6],[3]]
=> [[1,2,3],[4,6],[5],[7]]
=> [7,5,4,6,1,2,3] => ? = 7
[[1,3,5,7],[2,6],[4]]
=> [[1,2,4],[3,6],[5],[7]]
=> [7,5,3,6,1,2,4] => ? = 7
[[1,2,5,7],[3,6],[4]]
=> [[1,3,4],[2,6],[5],[7]]
=> [7,5,2,6,1,3,4] => ? = 7
[[1,3,4,7],[2,6],[5]]
=> [[1,2,5],[3,6],[4],[7]]
=> [7,4,3,6,1,2,5] => ? = 7
[[1,2,4,7],[3,6],[5]]
=> [[1,3,5],[2,6],[4],[7]]
=> [7,4,2,6,1,3,5] => ? = 7
[[1,2,3,7],[4,6],[5]]
=> [[1,4,5],[2,6],[3],[7]]
=> [7,3,2,6,1,4,5] => ? = 7
[[1,3,5,7],[2,4],[6]]
=> [[1,2,6],[3,4],[5],[7]]
=> [7,5,3,4,1,2,6] => ? = 7
Description
The first entry of the permutation.
This can be described as 1 plus the number of occurrences of the vincular pattern ([2,1], {(0,0),(0,1),(0,2)}), i.e., the first column is shaded, see [1].
This statistic is related to the number of deficiencies [[St000703]] as follows: consider the arc diagram of a permutation $\pi$ of $n$, together with its rotations, obtained by conjugating with the long cycle $(1,\dots,n)$. Drawing the labels $1$ to $n$ in this order on a circle, and the arcs $(i, \pi(i))$ as straight lines, the rotation of $\pi$ is obtained by replacing each number $i$ by $(i\bmod n) +1$. Then, $\pi(1)-1$ is the number of rotations of $\pi$ where the arc $(1, \pi(1))$ is a deficiency. In particular, if $O(\pi)$ is the orbit of rotations of $\pi$, then the number of deficiencies of $\pi$ equals
$$
\frac{1}{|O(\pi)|}\sum_{\sigma\in O(\pi)} (\sigma(1)-1).
$$
Matching statistic: St000025
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000025: Dyck paths ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 35%
Mp00064: Permutations —reverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000025: Dyck paths ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 35%
Values
[[1]]
=> [1] => [1] => [1,0]
=> 1
[[1,2]]
=> [1,2] => [2,1] => [1,1,0,0]
=> 2
[[1],[2]]
=> [2,1] => [1,2] => [1,0,1,0]
=> 1
[[1,2,3]]
=> [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> 3
[[1,3],[2]]
=> [2,1,3] => [3,1,2] => [1,1,1,0,0,0]
=> 3
[[1,2],[3]]
=> [3,1,2] => [2,1,3] => [1,1,0,0,1,0]
=> 2
[[1],[2],[3]]
=> [3,2,1] => [1,2,3] => [1,0,1,0,1,0]
=> 1
[[1,2,3,4]]
=> [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 4
[[1,3,4],[2]]
=> [2,1,3,4] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 4
[[1,2,4],[3]]
=> [3,1,2,4] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 4
[[1,2,3],[4]]
=> [4,1,2,3] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 3
[[1,3],[2,4]]
=> [2,4,1,3] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> 3
[[1,2],[3,4]]
=> [3,4,1,2] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 2
[[1,4],[2],[3]]
=> [3,2,1,4] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 4
[[1,3],[2],[4]]
=> [4,2,1,3] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 3
[[1,2],[3],[4]]
=> [4,3,1,2] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 2
[[1],[2],[3],[4]]
=> [4,3,2,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 1
[[1,2,3,4,5]]
=> [1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[[1,3,4,5],[2]]
=> [2,1,3,4,5] => [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[[1,2,4,5],[3]]
=> [3,1,2,4,5] => [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[[1,2,3,5],[4]]
=> [4,1,2,3,5] => [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[[1,2,3,4],[5]]
=> [5,1,2,3,4] => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 4
[[1,3,5],[2,4]]
=> [2,4,1,3,5] => [5,3,1,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[[1,2,5],[3,4]]
=> [3,4,1,2,5] => [5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[[1,3,4],[2,5]]
=> [2,5,1,3,4] => [4,3,1,5,2] => [1,1,1,1,0,0,0,1,0,0]
=> 4
[[1,2,4],[3,5]]
=> [3,5,1,2,4] => [4,2,1,5,3] => [1,1,1,1,0,0,0,1,0,0]
=> 4
[[1,2,3],[4,5]]
=> [4,5,1,2,3] => [3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> 3
[[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[[1,3,5],[2],[4]]
=> [4,2,1,3,5] => [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[[1,2,5],[3],[4]]
=> [4,3,1,2,5] => [5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[[1,3,4],[2],[5]]
=> [5,2,1,3,4] => [4,3,1,2,5] => [1,1,1,1,0,0,0,0,1,0]
=> 4
[[1,2,4],[3],[5]]
=> [5,3,1,2,4] => [4,2,1,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> 4
[[1,2,3],[4],[5]]
=> [5,4,1,2,3] => [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> 3
[[1,4],[2,5],[3]]
=> [3,2,5,1,4] => [4,1,5,2,3] => [1,1,1,1,0,0,1,0,0,0]
=> 4
[[1,3],[2,5],[4]]
=> [4,2,5,1,3] => [3,1,5,2,4] => [1,1,1,0,0,1,1,0,0,0]
=> 3
[[1,2],[3,5],[4]]
=> [4,3,5,1,2] => [2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0]
=> 2
[[1,3],[2,4],[5]]
=> [5,2,4,1,3] => [3,1,4,2,5] => [1,1,1,0,0,1,0,0,1,0]
=> 3
[[1,2],[3,4],[5]]
=> [5,3,4,1,2] => [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> 2
[[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[[1,4],[2],[3],[5]]
=> [5,3,2,1,4] => [4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> 4
[[1,3],[2],[4],[5]]
=> [5,4,2,1,3] => [3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> 3
[[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> 2
[[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 1
[[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => [6,5,4,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 6
[[1,3,4,5,6],[2]]
=> [2,1,3,4,5,6] => [6,5,4,3,1,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 6
[[1,2,4,5,6],[3]]
=> [3,1,2,4,5,6] => [6,5,4,2,1,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 6
[[1,2,3,5,6],[4]]
=> [4,1,2,3,5,6] => [6,5,3,2,1,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 6
[[1,2,3,4,6],[5]]
=> [5,1,2,3,4,6] => [6,4,3,2,1,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 6
[[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 5
[[1,3,5,6],[2,4]]
=> [2,4,1,3,5,6] => [6,5,3,1,4,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 6
[[1,2,3,4,5,6,7,8]]
=> [1,2,3,4,5,6,7,8] => [8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[[1,3,4,5,6,7,8],[2]]
=> [2,1,3,4,5,6,7,8] => [8,7,6,5,4,3,1,2] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[[1,2,4,5,6,7,8],[3]]
=> [3,1,2,4,5,6,7,8] => [8,7,6,5,4,2,1,3] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[[1,2,3,5,6,7,8],[4]]
=> [4,1,2,3,5,6,7,8] => [8,7,6,5,3,2,1,4] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[[1,2,3,4,6,7,8],[5]]
=> [5,1,2,3,4,6,7,8] => [8,7,6,4,3,2,1,5] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[[1,2,3,4,5,7,8],[6]]
=> [6,1,2,3,4,5,7,8] => [8,7,5,4,3,2,1,6] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[[1,2,3,4,5,6,8],[7]]
=> [7,1,2,3,4,5,6,8] => [8,6,5,4,3,2,1,7] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[[1,2,3,4,5,6,7],[8]]
=> [8,1,2,3,4,5,6,7] => [7,6,5,4,3,2,1,8] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 7
[[1,3,5,6,7,8],[2,4]]
=> [2,4,1,3,5,6,7,8] => [8,7,6,5,3,1,4,2] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[[1,2,5,6,7,8],[3,4]]
=> [3,4,1,2,5,6,7,8] => [8,7,6,5,2,1,4,3] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[[1,3,4,6,7,8],[2,5]]
=> [2,5,1,3,4,6,7,8] => [8,7,6,4,3,1,5,2] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[[1,2,4,6,7,8],[3,5]]
=> [3,5,1,2,4,6,7,8] => [8,7,6,4,2,1,5,3] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[[1,2,3,6,7,8],[4,5]]
=> [4,5,1,2,3,6,7,8] => [8,7,6,3,2,1,5,4] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[[1,3,4,5,7,8],[2,6]]
=> [2,6,1,3,4,5,7,8] => [8,7,5,4,3,1,6,2] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[[1,2,4,5,7,8],[3,6]]
=> [3,6,1,2,4,5,7,8] => [8,7,5,4,2,1,6,3] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[[1,2,3,5,7,8],[4,6]]
=> [4,6,1,2,3,5,7,8] => [8,7,5,3,2,1,6,4] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[[1,2,3,4,7,8],[5,6]]
=> [5,6,1,2,3,4,7,8] => [8,7,4,3,2,1,6,5] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[[1,3,4,5,6,8],[2,7]]
=> [2,7,1,3,4,5,6,8] => [8,6,5,4,3,1,7,2] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[[1,2,4,5,6,8],[3,7]]
=> [3,7,1,2,4,5,6,8] => [8,6,5,4,2,1,7,3] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[[1,2,3,5,6,8],[4,7]]
=> [4,7,1,2,3,5,6,8] => [8,6,5,3,2,1,7,4] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[[1,2,3,4,6,8],[5,7]]
=> [5,7,1,2,3,4,6,8] => [8,6,4,3,2,1,7,5] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[[1,2,3,4,5,8],[6,7]]
=> [6,7,1,2,3,4,5,8] => [8,5,4,3,2,1,7,6] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[[1,3,4,5,6,7],[2,8]]
=> [2,8,1,3,4,5,6,7] => [7,6,5,4,3,1,8,2] => [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? = 7
[[1,2,4,5,6,7],[3,8]]
=> [3,8,1,2,4,5,6,7] => [7,6,5,4,2,1,8,3] => [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? = 7
[[1,2,3,5,6,7],[4,8]]
=> [4,8,1,2,3,5,6,7] => [7,6,5,3,2,1,8,4] => [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? = 7
[[1,2,3,4,6,7],[5,8]]
=> [5,8,1,2,3,4,6,7] => [7,6,4,3,2,1,8,5] => [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? = 7
[[1,2,3,4,5,7],[6,8]]
=> [6,8,1,2,3,4,5,7] => [7,5,4,3,2,1,8,6] => [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? = 7
[[1,2,3,4,5,6],[7,8]]
=> [7,8,1,2,3,4,5,6] => [6,5,4,3,2,1,8,7] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 6
[[1,4,5,6,7,8],[2],[3]]
=> [3,2,1,4,5,6,7,8] => [8,7,6,5,4,1,2,3] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[[1,3,5,6,7,8],[2],[4]]
=> [4,2,1,3,5,6,7,8] => [8,7,6,5,3,1,2,4] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[[1,2,5,6,7,8],[3],[4]]
=> [4,3,1,2,5,6,7,8] => [8,7,6,5,2,1,3,4] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[[1,3,4,6,7,8],[2],[5]]
=> [5,2,1,3,4,6,7,8] => [8,7,6,4,3,1,2,5] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[[1,2,4,6,7,8],[3],[5]]
=> [5,3,1,2,4,6,7,8] => [8,7,6,4,2,1,3,5] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[[1,2,3,6,7,8],[4],[5]]
=> [5,4,1,2,3,6,7,8] => [8,7,6,3,2,1,4,5] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[[1,3,4,5,7,8],[2],[6]]
=> [6,2,1,3,4,5,7,8] => [8,7,5,4,3,1,2,6] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[[1,2,4,5,7,8],[3],[6]]
=> [6,3,1,2,4,5,7,8] => [8,7,5,4,2,1,3,6] => ?
=> ? = 8
[[1,2,3,5,7,8],[4],[6]]
=> [6,4,1,2,3,5,7,8] => [8,7,5,3,2,1,4,6] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[[1,2,3,4,7,8],[5],[6]]
=> [6,5,1,2,3,4,7,8] => [8,7,4,3,2,1,5,6] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[[1,3,4,5,6,8],[2],[7]]
=> [7,2,1,3,4,5,6,8] => [8,6,5,4,3,1,2,7] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[[1,2,4,5,6,8],[3],[7]]
=> [7,3,1,2,4,5,6,8] => [8,6,5,4,2,1,3,7] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[[1,2,3,5,6,8],[4],[7]]
=> [7,4,1,2,3,5,6,8] => [8,6,5,3,2,1,4,7] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[[1,2,3,4,6,8],[5],[7]]
=> [7,5,1,2,3,4,6,8] => [8,6,4,3,2,1,5,7] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[[1,2,3,4,5,8],[6],[7]]
=> [7,6,1,2,3,4,5,8] => [8,5,4,3,2,1,6,7] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[[1,3,4,5,6,7],[2],[8]]
=> [8,2,1,3,4,5,6,7] => [7,6,5,4,3,1,2,8] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 7
[[1,2,4,5,6,7],[3],[8]]
=> [8,3,1,2,4,5,6,7] => [7,6,5,4,2,1,3,8] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 7
[[1,2,3,5,6,7],[4],[8]]
=> [8,4,1,2,3,5,6,7] => [7,6,5,3,2,1,4,8] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 7
[[1,2,3,4,6,7],[5],[8]]
=> [8,5,1,2,3,4,6,7] => [7,6,4,3,2,1,5,8] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 7
[[1,2,3,4,5,7],[6],[8]]
=> [8,6,1,2,3,4,5,7] => [7,5,4,3,2,1,6,8] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 7
[[1,2,3,4,5,6],[7],[8]]
=> [8,7,1,2,3,4,5,6] => [6,5,4,3,2,1,7,8] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 6
[[1,3,5,7,8],[2,4,6]]
=> [2,4,6,1,3,5,7,8] => [8,7,5,3,1,6,4,2] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
Description
The number of initial rises of a Dyck path.
In other words, this is the height of the first peak of $D$.
Matching statistic: St000740
(load all 39 compositions to match this statistic)
(load all 39 compositions to match this statistic)
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000740: Permutations ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 35%
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000740: Permutations ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 35%
Values
[[1]]
=> [1] => [1] => [1] => 1
[[1,2]]
=> [1,2] => [1,2] => [1,2] => 2
[[1],[2]]
=> [2,1] => [2,1] => [2,1] => 1
[[1,2,3]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 3
[[1,3],[2]]
=> [2,1,3] => [2,1,3] => [2,1,3] => 3
[[1,2],[3]]
=> [3,1,2] => [1,3,2] => [1,3,2] => 2
[[1],[2],[3]]
=> [3,2,1] => [3,2,1] => [3,2,1] => 1
[[1,2,3,4]]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 4
[[1,3,4],[2]]
=> [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 4
[[1,2,4],[3]]
=> [3,1,2,4] => [1,3,2,4] => [1,3,2,4] => 4
[[1,2,3],[4]]
=> [4,1,2,3] => [1,2,4,3] => [1,2,4,3] => 3
[[1,3],[2,4]]
=> [2,4,1,3] => [1,3,4,2] => [1,4,2,3] => 3
[[1,2],[3,4]]
=> [3,4,1,2] => [1,4,2,3] => [1,3,4,2] => 2
[[1,4],[2],[3]]
=> [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 4
[[1,3],[2],[4]]
=> [4,2,1,3] => [3,1,4,2] => [2,4,1,3] => 3
[[1,2],[3],[4]]
=> [4,3,1,2] => [1,4,3,2] => [1,4,3,2] => 2
[[1],[2],[3],[4]]
=> [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 1
[[1,2,3,4,5]]
=> [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 5
[[1,3,4,5],[2]]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => 5
[[1,2,4,5],[3]]
=> [3,1,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 5
[[1,2,3,5],[4]]
=> [4,1,2,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 5
[[1,2,3,4],[5]]
=> [5,1,2,3,4] => [1,2,3,5,4] => [1,2,3,5,4] => 4
[[1,3,5],[2,4]]
=> [2,4,1,3,5] => [1,3,4,2,5] => [1,4,2,3,5] => 5
[[1,2,5],[3,4]]
=> [3,4,1,2,5] => [1,4,2,3,5] => [1,3,4,2,5] => 5
[[1,3,4],[2,5]]
=> [2,5,1,3,4] => [1,3,2,5,4] => [1,3,2,5,4] => 4
[[1,2,4],[3,5]]
=> [3,5,1,2,4] => [1,2,4,5,3] => [1,2,5,3,4] => 4
[[1,2,3],[4,5]]
=> [4,5,1,2,3] => [1,2,5,3,4] => [1,2,4,5,3] => 3
[[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => 5
[[1,3,5],[2],[4]]
=> [4,2,1,3,5] => [3,1,4,2,5] => [2,4,1,3,5] => 5
[[1,2,5],[3],[4]]
=> [4,3,1,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => 5
[[1,3,4],[2],[5]]
=> [5,2,1,3,4] => [3,1,2,5,4] => [2,3,1,5,4] => 4
[[1,2,4],[3],[5]]
=> [5,3,1,2,4] => [1,4,2,5,3] => [1,3,5,2,4] => 4
[[1,2,3],[4],[5]]
=> [5,4,1,2,3] => [1,2,5,4,3] => [1,2,5,4,3] => 3
[[1,4],[2,5],[3]]
=> [3,2,5,1,4] => [1,4,3,5,2] => [1,5,3,2,4] => 4
[[1,3],[2,5],[4]]
=> [4,2,5,1,3] => [1,4,5,2,3] => [1,4,5,2,3] => 3
[[1,2],[3,5],[4]]
=> [4,3,5,1,2] => [1,5,3,2,4] => [1,4,3,5,2] => 2
[[1,3],[2,4],[5]]
=> [5,2,4,1,3] => [1,3,5,4,2] => [1,5,2,4,3] => 3
[[1,2],[3,4],[5]]
=> [5,3,4,1,2] => [1,5,2,4,3] => [1,3,5,4,2] => 2
[[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => [4,3,2,1,5] => [4,3,2,1,5] => 5
[[1,4],[2],[3],[5]]
=> [5,3,2,1,4] => [4,3,1,5,2] => [3,5,2,1,4] => 4
[[1,3],[2],[4],[5]]
=> [5,4,2,1,3] => [4,1,5,3,2] => [2,5,4,1,3] => 3
[[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => [1,5,4,3,2] => [1,5,4,3,2] => 2
[[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [5,4,3,2,1] => [5,4,3,2,1] => 1
[[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 6
[[1,3,4,5,6],[2]]
=> [2,1,3,4,5,6] => [2,1,3,4,5,6] => [2,1,3,4,5,6] => 6
[[1,2,4,5,6],[3]]
=> [3,1,2,4,5,6] => [1,3,2,4,5,6] => [1,3,2,4,5,6] => 6
[[1,2,3,5,6],[4]]
=> [4,1,2,3,5,6] => [1,2,4,3,5,6] => [1,2,4,3,5,6] => 6
[[1,2,3,4,6],[5]]
=> [5,1,2,3,4,6] => [1,2,3,5,4,6] => [1,2,3,5,4,6] => 6
[[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => [1,2,3,4,6,5] => [1,2,3,4,6,5] => 5
[[1,3,5,6],[2,4]]
=> [2,4,1,3,5,6] => [1,3,4,2,5,6] => [1,4,2,3,5,6] => 6
[[1,3,4,5,6,7],[2]]
=> [2,1,3,4,5,6,7] => [2,1,3,4,5,6,7] => [2,1,3,4,5,6,7] => ? = 7
[[1,4,5,6,7],[2],[3]]
=> [3,2,1,4,5,6,7] => [3,2,1,4,5,6,7] => [3,2,1,4,5,6,7] => ? = 7
[[1,3,5,6,7],[2],[4]]
=> [4,2,1,3,5,6,7] => [3,1,4,2,5,6,7] => [2,4,1,3,5,6,7] => ? = 7
[[1,3,4,6,7],[2],[5]]
=> [5,2,1,3,4,6,7] => [3,1,2,5,4,6,7] => [2,3,1,5,4,6,7] => ? = 7
[[1,3,4,5,7],[2],[6]]
=> [6,2,1,3,4,5,7] => [3,1,2,4,6,5,7] => [2,3,1,4,6,5,7] => ? = 7
[[1,3,4,5,6],[2],[7]]
=> [7,2,1,3,4,5,6] => [3,1,2,4,5,7,6] => [2,3,1,4,5,7,6] => ? = 6
[[1,4,6,7],[2,5],[3]]
=> [3,2,5,1,4,6,7] => [1,4,3,5,2,6,7] => [1,5,3,2,4,6,7] => ? = 7
[[1,3,6,7],[2,4],[5]]
=> [5,2,4,1,3,6,7] => [1,3,5,4,2,6,7] => [1,5,2,4,3,6,7] => ? = 7
[[1,5,6,7],[2],[3],[4]]
=> [4,3,2,1,5,6,7] => [4,3,2,1,5,6,7] => [4,3,2,1,5,6,7] => ? = 7
[[1,4,6,7],[2],[3],[5]]
=> [5,3,2,1,4,6,7] => [4,3,1,5,2,6,7] => [3,5,2,1,4,6,7] => ? = 7
[[1,3,6,7],[2],[4],[5]]
=> [5,4,2,1,3,6,7] => [4,1,5,3,2,6,7] => [2,5,4,1,3,6,7] => ? = 7
[[1,2,6,7],[3],[4],[5]]
=> [5,4,3,1,2,6,7] => [1,5,4,3,2,6,7] => [1,5,4,3,2,6,7] => ? = 7
[[1,4,5,7],[2],[3],[6]]
=> [6,3,2,1,4,5,7] => [4,3,1,2,6,5,7] => [3,4,2,1,6,5,7] => ? = 7
[[1,3,5,7],[2],[4],[6]]
=> [6,4,2,1,3,5,7] => [4,1,5,2,6,3,7] => [2,4,6,1,3,5,7] => ? = 7
[[1,3,4,7],[2],[5],[6]]
=> [6,5,2,1,3,4,7] => [4,1,2,6,5,3,7] => [2,3,6,1,5,4,7] => ? = 7
[[1,4,5,6],[2],[3],[7]]
=> [7,3,2,1,4,5,6] => [4,3,1,2,5,7,6] => [3,4,2,1,5,7,6] => ? = 6
[[1,3,5,6],[2],[4],[7]]
=> [7,4,2,1,3,5,6] => [4,1,5,2,3,7,6] => [2,4,5,1,3,7,6] => ? = 6
[[1,3,4,6],[2],[5],[7]]
=> [7,5,2,1,3,4,6] => [4,1,2,6,3,7,5] => [2,3,5,1,7,4,6] => ? = 6
[[1,3,4,5],[2],[6],[7]]
=> [7,6,2,1,3,4,5] => [4,1,2,3,7,6,5] => [2,3,4,1,7,6,5] => ? = 5
[[1,4,7],[2,5],[3,6]]
=> [3,6,2,5,1,4,7] => [1,3,5,6,4,2,7] => [1,6,2,5,3,4,7] => ? = 7
[[1,3,7],[2,5],[4,6]]
=> [4,6,2,5,1,3,7] => [1,3,6,4,5,2,7] => [1,6,2,4,5,3,7] => ? = 7
[[1,3,7],[2,4],[5,6]]
=> [5,6,2,4,1,3,7] => [1,3,6,5,2,4,7] => [1,5,2,6,4,3,7] => ? = 7
[[1,4,6],[2,5],[3,7]]
=> [3,7,2,5,1,4,6] => [1,3,5,6,2,7,4] => [1,5,2,7,3,4,6] => ? = 6
[[1,3,6],[2,5],[4,7]]
=> [4,7,2,5,1,3,6] => [1,3,6,4,2,7,5] => [1,5,2,4,7,3,6] => ? = 6
[[1,5,7],[2,6],[3],[4]]
=> [4,3,2,6,1,5,7] => [1,5,4,3,6,2,7] => [1,6,4,3,2,5,7] => ? = 7
[[1,4,7],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7] => [1,5,4,6,2,3,7] => [1,5,6,3,2,4,7] => ? = 7
[[1,3,7],[2,6],[4],[5]]
=> [5,4,2,6,1,3,7] => [1,5,6,3,2,4,7] => [1,5,4,6,2,3,7] => ? = 7
[[1,2,7],[3,6],[4],[5]]
=> [5,4,3,6,1,2,7] => [1,6,4,3,2,5,7] => [1,5,4,3,6,2,7] => ? = 7
[[1,4,7],[2,5],[3],[6]]
=> [6,3,2,5,1,4,7] => [1,5,3,6,4,2,7] => [1,6,3,5,2,4,7] => ? = 7
[[1,3,7],[2,4],[5],[6]]
=> [6,5,2,4,1,3,7] => [1,3,6,5,4,2,7] => [1,6,2,5,4,3,7] => ? = 7
[[1,5,6],[2,7],[3],[4]]
=> [4,3,2,7,1,5,6] => [1,5,4,3,2,7,6] => [1,5,4,3,2,7,6] => ? = 6
[[1,4,6],[2,7],[3],[5]]
=> [5,3,2,7,1,4,6] => [1,5,4,2,6,7,3] => [1,4,7,3,2,5,6] => ? = 6
[[1,4,6],[2,5],[3],[7]]
=> [7,3,2,5,1,4,6] => [1,5,3,6,2,7,4] => [1,5,3,7,2,4,6] => ? = 6
[[1,3,6],[2,4],[5],[7]]
=> [7,5,2,4,1,3,6] => [1,3,6,5,2,7,4] => [1,5,2,7,4,3,6] => ? = 6
[[1,6,7],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7] => [5,4,3,2,1,6,7] => [5,4,3,2,1,6,7] => ? = 7
[[1,5,7],[2],[3],[4],[6]]
=> [6,4,3,2,1,5,7] => [5,4,3,1,6,2,7] => [4,6,3,2,1,5,7] => ? = 7
[[1,4,7],[2],[3],[5],[6]]
=> [6,5,3,2,1,4,7] => [5,4,1,6,3,2,7] => [3,6,5,2,1,4,7] => ? = 7
[[1,3,7],[2],[4],[5],[6]]
=> [6,5,4,2,1,3,7] => [5,1,6,4,3,2,7] => [2,6,5,4,1,3,7] => ? = 7
[[1,2,7],[3],[4],[5],[6]]
=> [6,5,4,3,1,2,7] => [1,6,5,4,3,2,7] => [1,6,5,4,3,2,7] => ? = 7
[[1,5,6],[2],[3],[4],[7]]
=> [7,4,3,2,1,5,6] => [5,4,3,1,2,7,6] => [4,5,3,2,1,7,6] => ? = 6
[[1,4,6],[2],[3],[5],[7]]
=> [7,5,3,2,1,4,6] => [5,4,1,6,2,7,3] => [3,5,7,2,1,4,6] => ? = 6
[[1,3,6],[2],[4],[5],[7]]
=> [7,5,4,2,1,3,6] => [5,1,6,4,2,7,3] => [2,5,7,4,1,3,6] => ? = 6
[[1,2,6],[3],[4],[5],[7]]
=> [7,5,4,3,1,2,6] => [1,6,5,4,2,7,3] => [1,5,7,4,3,2,6] => ? = 6
[[1,4,5],[2],[3],[6],[7]]
=> [7,6,3,2,1,4,5] => [5,4,1,2,7,6,3] => [3,4,7,2,1,6,5] => ? = 5
[[1,3,5],[2],[4],[6],[7]]
=> [7,6,4,2,1,3,5] => [5,1,6,2,7,4,3] => [2,4,7,6,1,3,5] => ? = 5
[[1,2,5],[3],[4],[6],[7]]
=> [7,6,4,3,1,2,5] => [1,6,5,2,7,4,3] => [1,4,7,6,3,2,5] => ? = 5
[[1,3,4],[2],[5],[6],[7]]
=> [7,6,5,2,1,3,4] => [5,1,2,7,6,4,3] => [2,3,7,6,1,5,4] => ? = 4
[[1,5],[2,6],[3,7],[4]]
=> [4,3,7,2,6,1,5] => [1,3,6,5,7,4,2] => [1,7,2,6,4,3,5] => ? = 5
[[1,4],[2,6],[3,7],[5]]
=> [5,3,7,2,6,1,4] => [1,3,6,7,4,5,2] => [1,7,2,5,6,3,4] => ? = 4
[[1,3],[2,6],[4,7],[5]]
=> [5,4,7,2,6,1,3] => [1,3,7,5,4,6,2] => [1,7,2,5,4,6,3] => ? = 3
Description
The last entry of a permutation.
This statistic is undefined for the empty permutation.
Matching statistic: St000066
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00284: Standard tableaux —rows⟶ Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
St000066: Alternating sign matrices ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 45%
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
St000066: Alternating sign matrices ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 45%
Values
[[1]]
=> {{1}}
=> [1] => [[1]]
=> 1
[[1,2]]
=> {{1,2}}
=> [2,1] => [[0,1],[1,0]]
=> 2
[[1],[2]]
=> {{1},{2}}
=> [1,2] => [[1,0],[0,1]]
=> 1
[[1,2,3]]
=> {{1,2,3}}
=> [2,3,1] => [[0,0,1],[1,0,0],[0,1,0]]
=> 3
[[1,3],[2]]
=> {{1,3},{2}}
=> [3,2,1] => [[0,0,1],[0,1,0],[1,0,0]]
=> 3
[[1,2],[3]]
=> {{1,2},{3}}
=> [2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
=> 2
[[1],[2],[3]]
=> {{1},{2},{3}}
=> [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> 1
[[1,2,3,4]]
=> {{1,2,3,4}}
=> [2,3,4,1] => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> 4
[[1,3,4],[2]]
=> {{1,3,4},{2}}
=> [3,2,4,1] => [[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
=> 4
[[1,2,4],[3]]
=> {{1,2,4},{3}}
=> [2,4,3,1] => [[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
=> 4
[[1,2,3],[4]]
=> {{1,2,3},{4}}
=> [2,3,1,4] => [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> 3
[[1,3],[2,4]]
=> {{1,3},{2,4}}
=> [3,4,1,2] => [[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> 3
[[1,2],[3,4]]
=> {{1,2},{3,4}}
=> [2,1,4,3] => [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> 2
[[1,4],[2],[3]]
=> {{1,4},{2},{3}}
=> [4,2,3,1] => [[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
=> 4
[[1,3],[2],[4]]
=> {{1,3},{2},{4}}
=> [3,2,1,4] => [[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> 3
[[1,2],[3],[4]]
=> {{1,2},{3},{4}}
=> [2,1,3,4] => [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> 2
[[1],[2],[3],[4]]
=> {{1},{2},{3},{4}}
=> [1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> 1
[[1,2,3,4,5]]
=> {{1,2,3,4,5}}
=> [2,3,4,5,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> 5
[[1,3,4,5],[2]]
=> {{1,3,4,5},{2}}
=> [3,2,4,5,1] => [[0,0,0,0,1],[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> 5
[[1,2,4,5],[3]]
=> {{1,2,4,5},{3}}
=> [2,4,3,5,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> 5
[[1,2,3,5],[4]]
=> {{1,2,3,5},{4}}
=> [2,3,5,4,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]
=> 5
[[1,2,3,4],[5]]
=> {{1,2,3,4},{5}}
=> [2,3,4,1,5] => [[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> 4
[[1,3,5],[2,4]]
=> {{1,3,5},{2,4}}
=> [3,4,5,2,1] => [[0,0,0,0,1],[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0]]
=> 5
[[1,2,5],[3,4]]
=> {{1,2,5},{3,4}}
=> [2,5,4,3,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0]]
=> 5
[[1,3,4],[2,5]]
=> {{1,3,4},{2,5}}
=> [3,5,4,1,2] => [[0,0,0,1,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0]]
=> 4
[[1,2,4],[3,5]]
=> {{1,2,4},{3,5}}
=> [2,4,5,1,3] => [[0,0,0,1,0],[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0]]
=> 4
[[1,2,3],[4,5]]
=> {{1,2,3},{4,5}}
=> [2,3,1,5,4] => [[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> 3
[[1,4,5],[2],[3]]
=> {{1,4,5},{2},{3}}
=> [4,2,3,5,1] => [[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0]]
=> 5
[[1,3,5],[2],[4]]
=> {{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [[0,0,0,0,1],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]
=> 5
[[1,2,5],[3],[4]]
=> {{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0]]
=> 5
[[1,3,4],[2],[5]]
=> {{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> 4
[[1,2,4],[3],[5]]
=> {{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> 4
[[1,2,3],[4],[5]]
=> {{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> 3
[[1,4],[2,5],[3]]
=> {{1,4},{2,5},{3}}
=> [4,5,3,1,2] => [[0,0,0,1,0],[0,0,0,0,1],[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0]]
=> 4
[[1,3],[2,5],[4]]
=> {{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [[0,0,1,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0]]
=> 3
[[1,2],[3,5],[4]]
=> {{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> 2
[[1,3],[2,4],[5]]
=> {{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> 3
[[1,2],[3,4],[5]]
=> {{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> 2
[[1,5],[2],[3],[4]]
=> {{1,5},{2},{3},{4}}
=> [5,2,3,4,1] => [[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0]]
=> 5
[[1,4],[2],[3],[5]]
=> {{1,4},{2},{3},{5}}
=> [4,2,3,1,5] => [[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1]]
=> 4
[[1,3],[2],[4],[5]]
=> {{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> 3
[[1,2],[3],[4],[5]]
=> {{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> 2
[[1],[2],[3],[4],[5]]
=> {{1},{2},{3},{4},{5}}
=> [1,2,3,4,5] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> 1
[[1,2,3,4,5,6]]
=> {{1,2,3,4,5,6}}
=> [2,3,4,5,6,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> 6
[[1,3,4,5,6],[2]]
=> {{1,3,4,5,6},{2}}
=> [3,2,4,5,6,1] => [[0,0,0,0,0,1],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> 6
[[1,2,4,5,6],[3]]
=> {{1,2,4,5,6},{3}}
=> [2,4,3,5,6,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> 6
[[1,2,3,5,6],[4]]
=> {{1,2,3,5,6},{4}}
=> [2,3,5,4,6,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> 6
[[1,2,3,4,6],[5]]
=> {{1,2,3,4,6},{5}}
=> [2,3,4,6,5,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0]]
=> 6
[[1,2,3,4,5],[6]]
=> {{1,2,3,4,5},{6}}
=> [2,3,4,5,1,6] => [[0,0,0,0,1,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> 5
[[1,3,5,6],[2,4]]
=> {{1,3,5,6},{2,4}}
=> [3,4,5,2,6,1] => [[0,0,0,0,0,1],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> 6
[[1,3,4,5],[2,6]]
=> {{1,3,4,5},{2,6}}
=> [3,6,4,5,1,2] => [[0,0,0,0,1,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0]]
=> ? = 5
[[1,2,4,5],[3,6]]
=> {{1,2,4,5},{3,6}}
=> [2,4,6,5,1,3] => [[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
=> ? = 5
[[1,2,5],[3,4,6]]
=> {{1,2,5},{3,4,6}}
=> [2,5,4,6,1,3] => [[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0]]
=> ? = 5
[[1,3,4],[2,5,6]]
=> {{1,3,4},{2,5,6}}
=> [3,5,4,1,6,2] => [[0,0,0,1,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0]]
=> ? = 4
[[1,3,6],[2,4],[5]]
=> {{1,3,6},{2,4},{5}}
=> [3,4,6,2,5,1] => [[0,0,0,0,0,1],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0]]
=> ? = 6
[[1,2,5],[3,6],[4]]
=> {{1,2,5},{3,6},{4}}
=> [2,5,6,4,1,3] => [[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0]]
=> ? = 5
[[1,3,4],[2,6],[5]]
=> {{1,3,4},{2,6},{5}}
=> [3,6,4,1,5,2] => [[0,0,0,1,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0]]
=> ? = 4
[[1,2,4],[3,6],[5]]
=> {{1,2,4},{3,6},{5}}
=> [2,4,6,1,5,3] => [[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0]]
=> ? = 4
[[1,3,4],[2,5],[6]]
=> {{1,3,4},{2,5},{6}}
=> [3,5,4,1,2,6] => [[0,0,0,1,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1]]
=> ? = 4
[[1,4],[2,6],[3],[5]]
=> {{1,4},{2,6},{3},{5}}
=> [4,6,3,1,5,2] => [[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0]]
=> ? = 4
[[1,3],[2,6],[4],[5]]
=> {{1,3},{2,6},{4},{5}}
=> [3,6,1,4,5,2] => [[0,0,1,0,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0]]
=> ? = 3
[[1,3],[2,5],[4],[6]]
=> {{1,3},{2,5},{4},{6}}
=> [3,5,1,4,2,6] => [[0,0,1,0,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1]]
=> ? = 3
[[1,2,3,4,5,6,7]]
=> {{1,2,3,4,5,6,7}}
=> [2,3,4,5,6,7,1] => [[0,0,0,0,0,0,1],[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 7
[[1,3,4,5,6,7],[2]]
=> {{1,3,4,5,6,7},{2}}
=> [3,2,4,5,6,7,1] => [[0,0,0,0,0,0,1],[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 7
[[1,2,4,5,6,7],[3]]
=> {{1,2,4,5,6,7},{3}}
=> [2,4,3,5,6,7,1] => [[0,0,0,0,0,0,1],[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 7
[[1,2,3,5,6,7],[4]]
=> {{1,2,3,5,6,7},{4}}
=> [2,3,5,4,6,7,1] => [[0,0,0,0,0,0,1],[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 7
[[1,2,3,4,6,7],[5]]
=> {{1,2,3,4,6,7},{5}}
=> [2,3,4,6,5,7,1] => [[0,0,0,0,0,0,1],[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0]]
=> ? = 7
[[1,2,3,4,5,7],[6]]
=> {{1,2,3,4,5,7},{6}}
=> [2,3,4,5,7,6,1] => [[0,0,0,0,0,0,1],[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0]]
=> ? = 7
[[1,3,5,6,7],[2,4]]
=> {{1,3,5,6,7},{2,4}}
=> [3,4,5,2,6,7,1] => [[0,0,0,0,0,0,1],[0,0,0,1,0,0,0],[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 7
[[1,2,5,6,7],[3,4]]
=> {{1,2,5,6,7},{3,4}}
=> [2,5,4,3,6,7,1] => [[0,0,0,0,0,0,1],[1,0,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 7
[[1,3,4,6,7],[2,5]]
=> {{1,3,4,6,7},{2,5}}
=> [3,5,4,6,2,7,1] => [[0,0,0,0,0,0,1],[0,0,0,0,1,0,0],[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0]]
=> ? = 7
[[1,2,4,6,7],[3,5]]
=> {{1,2,4,6,7},{3,5}}
=> [2,4,5,6,3,7,1] => [[0,0,0,0,0,0,1],[1,0,0,0,0,0,0],[0,0,0,0,1,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0]]
=> ? = 7
[[1,2,3,6,7],[4,5]]
=> {{1,2,3,6,7},{4,5}}
=> [2,3,6,5,4,7,1] => [[0,0,0,0,0,0,1],[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,1,0]]
=> ? = 7
[[1,3,4,5,7],[2,6]]
=> {{1,3,4,5,7},{2,6}}
=> [3,6,4,5,7,2,1] => [[0,0,0,0,0,0,1],[0,0,0,0,0,1,0],[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,1,0,0]]
=> ? = 7
[[1,2,4,5,7],[3,6]]
=> {{1,2,4,5,7},{3,6}}
=> [2,4,6,5,7,3,1] => [[0,0,0,0,0,0,1],[1,0,0,0,0,0,0],[0,0,0,0,0,1,0],[0,1,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0]]
=> ? = 7
[[1,2,3,5,7],[4,6]]
=> {{1,2,3,5,7},{4,6}}
=> [2,3,5,6,7,4,1] => [[0,0,0,0,0,0,1],[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0]]
=> ? = 7
[[1,2,3,4,7],[5,6]]
=> {{1,2,3,4,7},{5,6}}
=> [2,3,4,7,6,5,1] => [[0,0,0,0,0,0,1],[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0]]
=> ? = 7
[[1,3,4,5,6],[2,7]]
=> {{1,3,4,5,6},{2,7}}
=> [3,7,4,5,6,1,2] => [[0,0,0,0,0,1,0],[0,0,0,0,0,0,1],[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,1,0,0,0,0,0]]
=> ? = 6
[[1,2,4,5,6],[3,7]]
=> {{1,2,4,5,6},{3,7}}
=> [2,4,7,5,6,1,3] => [[0,0,0,0,0,1,0],[1,0,0,0,0,0,0],[0,0,0,0,0,0,1],[0,1,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,1,0,0,0,0]]
=> ? = 6
[[1,2,3,5,6],[4,7]]
=> {{1,2,3,5,6},{4,7}}
=> [2,3,5,7,6,1,4] => [[0,0,0,0,0,1,0],[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,0,0,1],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0]]
=> ? = 6
[[1,2,3,4,6],[5,7]]
=> {{1,2,3,4,6},{5,7}}
=> [2,3,4,6,7,1,5] => [[0,0,0,0,0,1,0],[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,0,1],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0]]
=> ? = 6
[[1,4,5,6,7],[2],[3]]
=> {{1,4,5,6,7},{2},{3}}
=> [4,2,3,5,6,7,1] => [[0,0,0,0,0,0,1],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 7
[[1,3,5,6,7],[2],[4]]
=> {{1,3,5,6,7},{2},{4}}
=> [3,2,5,4,6,7,1] => [[0,0,0,0,0,0,1],[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 7
[[1,2,5,6,7],[3],[4]]
=> {{1,2,5,6,7},{3},{4}}
=> [2,5,3,4,6,7,1] => [[0,0,0,0,0,0,1],[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 7
[[1,3,4,6,7],[2],[5]]
=> {{1,3,4,6,7},{2},{5}}
=> [3,2,4,6,5,7,1] => [[0,0,0,0,0,0,1],[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0]]
=> ? = 7
[[1,2,4,6,7],[3],[5]]
=> {{1,2,4,6,7},{3},{5}}
=> [2,4,3,6,5,7,1] => [[0,0,0,0,0,0,1],[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0]]
=> ? = 7
[[1,2,3,6,7],[4],[5]]
=> {{1,2,3,6,7},{4},{5}}
=> [2,3,6,4,5,7,1] => [[0,0,0,0,0,0,1],[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,1,0]]
=> ? = 7
[[1,3,4,5,7],[2],[6]]
=> {{1,3,4,5,7},{2},{6}}
=> [3,2,4,5,7,6,1] => [[0,0,0,0,0,0,1],[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0]]
=> ? = 7
[[1,2,4,5,7],[3],[6]]
=> {{1,2,4,5,7},{3},{6}}
=> [2,4,3,5,7,6,1] => [[0,0,0,0,0,0,1],[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0]]
=> ? = 7
[[1,2,3,5,7],[4],[6]]
=> {{1,2,3,5,7},{4},{6}}
=> [2,3,5,4,7,6,1] => [[0,0,0,0,0,0,1],[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0]]
=> ? = 7
[[1,2,3,4,7],[5],[6]]
=> {{1,2,3,4,7},{5},{6}}
=> [2,3,4,7,5,6,1] => [[0,0,0,0,0,0,1],[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,1,0,0,0]]
=> ? = 7
[[1,3,4,5,6],[2],[7]]
=> {{1,3,4,5,6},{2},{7}}
=> [3,2,4,5,6,1,7] => [[0,0,0,0,0,1,0],[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> ? = 6
[[1,2,4,5,6],[3],[7]]
=> {{1,2,4,5,6},{3},{7}}
=> [2,4,3,5,6,1,7] => [[0,0,0,0,0,1,0],[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> ? = 6
[[1,2,3,5,6],[4],[7]]
=> {{1,2,3,5,6},{4},{7}}
=> [2,3,5,4,6,1,7] => [[0,0,0,0,0,1,0],[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> ? = 6
[[1,2,3,4,6],[5],[7]]
=> {{1,2,3,4,6},{5},{7}}
=> [2,3,4,6,5,1,7] => [[0,0,0,0,0,1,0],[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,0,1]]
=> ? = 6
[[1,2,3,4,5],[6],[7]]
=> {{1,2,3,4,5},{6},{7}}
=> [2,3,4,5,1,6,7] => [[0,0,0,0,1,0,0],[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> ? = 5
[[1,3,5,7],[2,4,6]]
=> {{1,3,5,7},{2,4,6}}
=> [3,4,5,6,7,2,1] => [[0,0,0,0,0,0,1],[0,0,0,0,0,1,0],[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0]]
=> ? = 7
[[1,2,5,7],[3,4,6]]
=> {{1,2,5,7},{3,4,6}}
=> [2,5,4,6,7,3,1] => [[0,0,0,0,0,0,1],[1,0,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0]]
=> ? = 7
[[1,3,4,7],[2,5,6]]
=> {{1,3,4,7},{2,5,6}}
=> [3,5,4,7,6,2,1] => [[0,0,0,0,0,0,1],[0,0,0,0,0,1,0],[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0]]
=> ? = 7
[[1,2,4,7],[3,5,6]]
=> {{1,2,4,7},{3,5,6}}
=> [2,4,5,7,6,3,1] => [[0,0,0,0,0,0,1],[1,0,0,0,0,0,0],[0,0,0,0,0,1,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0]]
=> ? = 7
Description
The column of the unique '1' in the first row of the alternating sign matrix.
The generating function of this statistic is given by
$$\binom{n+k-2}{k-1}\frac{(2n-k-1)!}{(n-k)!}\;\prod_{j=0}^{n-2}\frac{(3j+1)!}{(n+j)!},$$
see [2].
The following 8 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St000051The size of the left subtree of a binary tree. St000736The last entry in the first row of a semistandard tableau. St000739The first entry in the last row of a semistandard tableau. St000200The row of the unique '1' in the last column of the alternating sign matrix. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000199The column of the unique '1' in the last row of the alternating sign matrix. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
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