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Matching statistic: St001699
St001699: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> 0
[[1,2]]
=> 0
[[1],[2]]
=> 0
[[1,2,3]]
=> 0
[[1,3],[2]]
=> 0
[[1,2],[3]]
=> 1
[[1],[2],[3]]
=> 0
[[1,2,3,4]]
=> 0
[[1,3,4],[2]]
=> 0
[[1,2,4],[3]]
=> 1
[[1,2,3],[4]]
=> 2
[[1,3],[2,4]]
=> 2
[[1,2],[3,4]]
=> 0
[[1,4],[2],[3]]
=> 0
[[1,3],[2],[4]]
=> 1
[[1,2],[3],[4]]
=> 2
[[1],[2],[3],[4]]
=> 0
[[1,2,3,4,5]]
=> 0
[[1,3,4,5],[2]]
=> 0
[[1,2,4,5],[3]]
=> 1
[[1,2,3,5],[4]]
=> 2
[[1,2,3,4],[5]]
=> 3
[[1,3,5],[2,4]]
=> 2
[[1,2,5],[3,4]]
=> 0
[[1,3,4],[2,5]]
=> 3
[[1,2,4],[3,5]]
=> 4
[[1,2,3],[4,5]]
=> 1
[[1,4,5],[2],[3]]
=> 0
[[1,3,5],[2],[4]]
=> 1
[[1,2,5],[3],[4]]
=> 2
[[1,3,4],[2],[5]]
=> 2
[[1,2,4],[3],[5]]
=> 3
[[1,2,3],[4],[5]]
=> 4
[[1,4],[2,5],[3]]
=> 3
[[1,3],[2,5],[4]]
=> 0
[[1,2],[3,5],[4]]
=> 1
[[1,3],[2,4],[5]]
=> 4
[[1,2],[3,4],[5]]
=> 2
[[1,5],[2],[3],[4]]
=> 0
[[1,4],[2],[3],[5]]
=> 1
[[1,3],[2],[4],[5]]
=> 2
[[1,2],[3],[4],[5]]
=> 3
[[1],[2],[3],[4],[5]]
=> 0
[[1,2,3,4,5,6]]
=> 0
[[1,3,4,5,6],[2]]
=> 0
[[1,2,4,5,6],[3]]
=> 1
[[1,2,3,5,6],[4]]
=> 2
[[1,2,3,4,6],[5]]
=> 3
[[1,2,3,4,5],[6]]
=> 4
[[1,3,5,6],[2,4]]
=> 2
Description
The major index of a standard tableau minus the weighted size of its shape.
Matching statistic: St001698
Mp00085: Standard tableaux —Schützenberger involution⟶ Standard tableaux
St001698: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001698: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> [[1]]
=> 0
[[1,2]]
=> [[1,2]]
=> 0
[[1],[2]]
=> [[1],[2]]
=> 0
[[1,2,3]]
=> [[1,2,3]]
=> 0
[[1,3],[2]]
=> [[1,2],[3]]
=> 0
[[1,2],[3]]
=> [[1,3],[2]]
=> 1
[[1],[2],[3]]
=> [[1],[2],[3]]
=> 0
[[1,2,3,4]]
=> [[1,2,3,4]]
=> 0
[[1,3,4],[2]]
=> [[1,2,3],[4]]
=> 0
[[1,2,4],[3]]
=> [[1,2,4],[3]]
=> 1
[[1,2,3],[4]]
=> [[1,3,4],[2]]
=> 2
[[1,3],[2,4]]
=> [[1,3],[2,4]]
=> 2
[[1,2],[3,4]]
=> [[1,2],[3,4]]
=> 0
[[1,4],[2],[3]]
=> [[1,2],[3],[4]]
=> 0
[[1,3],[2],[4]]
=> [[1,3],[2],[4]]
=> 1
[[1,2],[3],[4]]
=> [[1,4],[2],[3]]
=> 2
[[1],[2],[3],[4]]
=> [[1],[2],[3],[4]]
=> 0
[[1,2,3,4,5]]
=> [[1,2,3,4,5]]
=> 0
[[1,3,4,5],[2]]
=> [[1,2,3,4],[5]]
=> 0
[[1,2,4,5],[3]]
=> [[1,2,3,5],[4]]
=> 1
[[1,2,3,5],[4]]
=> [[1,2,4,5],[3]]
=> 2
[[1,2,3,4],[5]]
=> [[1,3,4,5],[2]]
=> 3
[[1,3,5],[2,4]]
=> [[1,2,4],[3,5]]
=> 2
[[1,2,5],[3,4]]
=> [[1,2,3],[4,5]]
=> 0
[[1,3,4],[2,5]]
=> [[1,3,4],[2,5]]
=> 3
[[1,2,4],[3,5]]
=> [[1,3,5],[2,4]]
=> 4
[[1,2,3],[4,5]]
=> [[1,2,5],[3,4]]
=> 1
[[1,4,5],[2],[3]]
=> [[1,2,3],[4],[5]]
=> 0
[[1,3,5],[2],[4]]
=> [[1,2,4],[3],[5]]
=> 1
[[1,2,5],[3],[4]]
=> [[1,2,5],[3],[4]]
=> 2
[[1,3,4],[2],[5]]
=> [[1,3,4],[2],[5]]
=> 2
[[1,2,4],[3],[5]]
=> [[1,3,5],[2],[4]]
=> 3
[[1,2,3],[4],[5]]
=> [[1,4,5],[2],[3]]
=> 4
[[1,4],[2,5],[3]]
=> [[1,3],[2,4],[5]]
=> 3
[[1,3],[2,5],[4]]
=> [[1,2],[3,4],[5]]
=> 0
[[1,2],[3,5],[4]]
=> [[1,2],[3,5],[4]]
=> 1
[[1,3],[2,4],[5]]
=> [[1,4],[2,5],[3]]
=> 4
[[1,2],[3,4],[5]]
=> [[1,3],[2,5],[4]]
=> 2
[[1,5],[2],[3],[4]]
=> [[1,2],[3],[4],[5]]
=> 0
[[1,4],[2],[3],[5]]
=> [[1,3],[2],[4],[5]]
=> 1
[[1,3],[2],[4],[5]]
=> [[1,4],[2],[3],[5]]
=> 2
[[1,2],[3],[4],[5]]
=> [[1,5],[2],[3],[4]]
=> 3
[[1],[2],[3],[4],[5]]
=> [[1],[2],[3],[4],[5]]
=> 0
[[1,2,3,4,5,6]]
=> [[1,2,3,4,5,6]]
=> 0
[[1,3,4,5,6],[2]]
=> [[1,2,3,4,5],[6]]
=> 0
[[1,2,4,5,6],[3]]
=> [[1,2,3,4,6],[5]]
=> 1
[[1,2,3,5,6],[4]]
=> [[1,2,3,5,6],[4]]
=> 2
[[1,2,3,4,6],[5]]
=> [[1,2,4,5,6],[3]]
=> 3
[[1,2,3,4,5],[6]]
=> [[1,3,4,5,6],[2]]
=> 4
[[1,3,5,6],[2,4]]
=> [[1,2,3,5],[4,6]]
=> 2
Description
The comajor index of a standard tableau minus the weighted size of its shape.
Matching statistic: St001377
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St001377: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St001377: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> [1] => [1] => [1] => 0
[[1,2]]
=> [1,2] => [1,2] => [1,2] => 0
[[1],[2]]
=> [2,1] => [2,1] => [2,1] => 0
[[1,2,3]]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0
[[1,3],[2]]
=> [2,1,3] => [2,1,3] => [2,1,3] => 0
[[1,2],[3]]
=> [3,1,2] => [1,3,2] => [1,3,2] => 1
[[1],[2],[3]]
=> [3,2,1] => [3,2,1] => [3,2,1] => 0
[[1,2,3,4]]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[[1,3,4],[2]]
=> [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 0
[[1,2,4],[3]]
=> [3,1,2,4] => [1,3,2,4] => [1,3,2,4] => 1
[[1,2,3],[4]]
=> [4,1,2,3] => [1,2,4,3] => [1,2,4,3] => 2
[[1,3],[2,4]]
=> [2,4,1,3] => [2,1,4,3] => [2,1,4,3] => 2
[[1,2],[3,4]]
=> [3,4,1,2] => [1,3,4,2] => [1,4,2,3] => 0
[[1,4],[2],[3]]
=> [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 0
[[1,3],[2],[4]]
=> [4,2,1,3] => [2,4,1,3] => [3,1,4,2] => 1
[[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] => 0
[[1,2,3,4,5]]
=> [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[[1,3,4,5],[2]]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => 0
[[1,2,4,5],[3]]
=> [3,1,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 1
[[1,2,3,5],[4]]
=> [4,1,2,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 2
[[1,2,3,4],[5]]
=> [5,1,2,3,4] => [1,2,3,5,4] => [1,2,3,5,4] => 3
[[1,3,5],[2,4]]
=> [2,4,1,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => 2
[[1,2,5],[3,4]]
=> [3,4,1,2,5] => [1,3,4,2,5] => [1,4,2,3,5] => 0
[[1,3,4],[2,5]]
=> [2,5,1,3,4] => [2,1,3,5,4] => [2,1,3,5,4] => 3
[[1,2,4],[3,5]]
=> [3,5,1,2,4] => [1,3,2,5,4] => [1,3,2,5,4] => 4
[[1,2,3],[4,5]]
=> [4,5,1,2,3] => [1,2,4,5,3] => [1,2,5,3,4] => 1
[[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => 0
[[1,3,5],[2],[4]]
=> [4,2,1,3,5] => [2,4,1,3,5] => [3,1,4,2,5] => 1
[[1,2,5],[3],[4]]
=> [4,3,1,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => 2
[[1,3,4],[2],[5]]
=> [5,2,1,3,4] => [2,1,5,3,4] => [2,1,4,5,3] => 2
[[1,2,4],[3],[5]]
=> [5,3,1,2,4] => [1,3,5,2,4] => [1,4,2,5,3] => 3
[[1,2,3],[4],[5]]
=> [5,4,1,2,3] => [1,2,5,4,3] => [1,2,5,4,3] => 4
[[1,4],[2,5],[3]]
=> [3,2,5,1,4] => [3,2,1,5,4] => [3,2,1,5,4] => 3
[[1,3],[2,5],[4]]
=> [4,2,5,1,3] => [2,4,1,5,3] => [3,1,5,2,4] => 0
[[1,2],[3,5],[4]]
=> [4,3,5,1,2] => [1,4,3,5,2] => [1,5,3,2,4] => 1
[[1,3],[2,4],[5]]
=> [5,2,4,1,3] => [2,1,5,4,3] => [2,1,5,4,3] => 4
[[1,2],[3,4],[5]]
=> [5,3,4,1,2] => [1,3,5,4,2] => [1,5,2,4,3] => 2
[[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => [4,3,2,1,5] => [4,3,2,1,5] => 0
[[1,4],[2],[3],[5]]
=> [5,3,2,1,4] => [3,5,2,1,4] => [4,3,1,5,2] => 1
[[1,3],[2],[4],[5]]
=> [5,4,2,1,3] => [2,5,4,1,3] => [4,1,5,3,2] => 2
[[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => [1,5,4,3,2] => [1,5,4,3,2] => 3
[[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [5,4,3,2,1] => [5,4,3,2,1] => 0
[[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[[1,3,4,5,6],[2]]
=> [2,1,3,4,5,6] => [2,1,3,4,5,6] => [2,1,3,4,5,6] => 0
[[1,2,4,5,6],[3]]
=> [3,1,2,4,5,6] => [1,3,2,4,5,6] => [1,3,2,4,5,6] => 1
[[1,2,3,5,6],[4]]
=> [4,1,2,3,5,6] => [1,2,4,3,5,6] => [1,2,4,3,5,6] => 2
[[1,2,3,4,6],[5]]
=> [5,1,2,3,4,6] => [1,2,3,5,4,6] => [1,2,3,5,4,6] => 3
[[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => [1,2,3,4,6,5] => [1,2,3,4,6,5] => 4
[[1,3,5,6],[2,4]]
=> [2,4,1,3,5,6] => [2,1,4,3,5,6] => [2,1,4,3,5,6] => 2
Description
The major index minus the number of inversions of a permutation.
This is, the difference between [[St000004]] and [[St000018]].
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