Your data matches 3 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001699
Mapping
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
Mapping
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%
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
Mapping
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St001377: Permutations ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 73%
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
[[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] => ? = 0
[[1,3,5,6,7],[2,4]]
 => [2,4,1,3,5,6,7] => [2,1,4,3,5,6,7] => [2,1,4,3,5,6,7] => ? = 2
[[1,3,4,6,7],[2,5]]
 => [2,5,1,3,4,6,7] => [2,1,3,5,4,6,7] => [2,1,3,5,4,6,7] => ? = 3
[[1,3,4,5,7],[2,6]]
 => [2,6,1,3,4,5,7] => [2,1,3,4,6,5,7] => [2,1,3,4,6,5,7] => ? = 4
[[1,3,4,5,6],[2,7]]
 => [2,7,1,3,4,5,6] => [2,1,3,4,5,7,6] => [2,1,3,4,5,7,6] => ? = 5
[[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] => ? = 0
[[1,3,5,6,7],[2],[4]]
 => [4,2,1,3,5,6,7] => [2,4,1,3,5,6,7] => [3,1,4,2,5,6,7] => ? = 1
[[1,3,4,6,7],[2],[5]]
 => [5,2,1,3,4,6,7] => [2,1,5,3,4,6,7] => [2,1,4,5,3,6,7] => ? = 2
[[1,3,4,5,7],[2],[6]]
 => [6,2,1,3,4,5,7] => [2,1,3,6,4,5,7] => [2,1,3,5,6,4,7] => ? = 3
[[1,3,4,5,6],[2],[7]]
 => [7,2,1,3,4,5,6] => [2,1,3,4,7,5,6] => [2,1,3,4,6,7,5] => ? = 4
[[1,3,5,7],[2,4,6]]
 => [2,4,6,1,3,5,7] => [2,1,4,3,6,5,7] => [2,1,4,3,6,5,7] => ? = 6
[[1,3,4,7],[2,5,6]]
 => [2,5,6,1,3,4,7] => [2,1,3,5,6,4,7] => [2,1,3,6,4,5,7] => ? = 2
[[1,3,5,6],[2,4,7]]
 => [2,4,7,1,3,5,6] => [2,1,4,3,5,7,6] => [2,1,4,3,5,7,6] => ? = 7
[[1,3,4,6],[2,5,7]]
 => [2,5,7,1,3,4,6] => [2,1,3,5,4,7,6] => [2,1,3,5,4,7,6] => ? = 8
[[1,3,4,5],[2,6,7]]
 => [2,6,7,1,3,4,5] => [2,1,3,4,6,7,5] => [2,1,3,4,7,5,6] => ? = 3
[[1,4,6,7],[2,5],[3]]
 => [3,2,5,1,4,6,7] => [3,2,1,5,4,6,7] => [3,2,1,5,4,6,7] => ? = 3
[[1,3,6,7],[2,5],[4]]
 => [4,2,5,1,3,6,7] => [2,4,1,5,3,6,7] => [3,1,5,2,4,6,7] => ? = 0
[[1,2,6,7],[3,5],[4]]
 => [4,3,5,1,2,6,7] => [1,4,3,5,2,6,7] => [1,5,3,2,4,6,7] => ? = 1
[[1,3,6,7],[2,4],[5]]
 => [5,2,4,1,3,6,7] => [2,1,5,4,3,6,7] => [2,1,5,4,3,6,7] => ? = 4
[[1,2,6,7],[3,4],[5]]
 => [5,3,4,1,2,6,7] => [1,3,5,4,2,6,7] => [1,5,2,4,3,6,7] => ? = 2
[[1,4,5,7],[2,6],[3]]
 => [3,2,6,1,4,5,7] => [3,2,1,4,6,5,7] => [3,2,1,4,6,5,7] => ? = 4
[[1,3,5,7],[2,6],[4]]
 => [4,2,6,1,3,5,7] => [2,4,1,3,6,5,7] => [3,1,4,2,6,5,7] => ? = 5
[[1,3,4,7],[2,6],[5]]
 => [5,2,6,1,3,4,7] => [2,1,5,3,6,4,7] => [2,1,4,6,3,5,7] => ? = 1
[[1,3,5,7],[2,4],[6]]
 => [6,2,4,1,3,5,7] => [2,1,4,6,3,5,7] => [2,1,5,3,6,4,7] => ? = 5
[[1,2,5,7],[3,4],[6]]
 => [6,3,4,1,2,5,7] => [1,3,4,6,2,5,7] => [1,5,2,3,6,4,7] => ? = 3
[[1,3,4,7],[2,5],[6]]
 => [6,2,5,1,3,4,7] => [2,1,3,6,5,4,7] => [2,1,3,6,5,4,7] => ? = 6
[[1,4,5,6],[2,7],[3]]
 => [3,2,7,1,4,5,6] => [3,2,1,4,5,7,6] => [3,2,1,4,5,7,6] => ? = 5
[[1,3,5,6],[2,7],[4]]
 => [4,2,7,1,3,5,6] => [2,4,1,3,5,7,6] => [3,1,4,2,5,7,6] => ? = 6
[[1,3,4,6],[2,7],[5]]
 => [5,2,7,1,3,4,6] => [2,1,5,3,4,7,6] => [2,1,4,5,3,7,6] => ? = 7
[[1,3,4,5],[2,7],[6]]
 => [6,2,7,1,3,4,5] => [2,1,3,6,4,7,5] => [2,1,3,5,7,4,6] => ? = 2
[[1,3,5,6],[2,4],[7]]
 => [7,2,4,1,3,5,6] => [2,1,4,3,7,5,6] => [2,1,4,3,6,7,5] => ? = 6
[[1,3,4,6],[2,5],[7]]
 => [7,2,5,1,3,4,6] => [2,1,3,5,7,4,6] => [2,1,3,6,4,7,5] => ? = 7
[[1,3,4,5],[2,6],[7]]
 => [7,2,6,1,3,4,5] => [2,1,3,4,7,6,5] => [2,1,3,4,7,6,5] => ? = 8
[[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] => ? = 0
[[1,4,6,7],[2],[3],[5]]
 => [5,3,2,1,4,6,7] => [3,5,2,1,4,6,7] => [4,3,1,5,2,6,7] => ? = 1
[[1,3,6,7],[2],[4],[5]]
 => [5,4,2,1,3,6,7] => [2,5,4,1,3,6,7] => [4,1,5,3,2,6,7] => ? = 2
[[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] => ? = 3
[[1,4,5,7],[2],[3],[6]]
 => [6,3,2,1,4,5,7] => [3,2,6,1,4,5,7] => [4,2,1,5,6,3,7] => ? = 2
[[1,3,5,7],[2],[4],[6]]
 => [6,4,2,1,3,5,7] => [2,4,6,1,3,5,7] => [4,1,5,2,6,3,7] => ? = 3
[[1,2,5,7],[3],[4],[6]]
 => [6,4,3,1,2,5,7] => [1,4,6,3,2,5,7] => [1,5,4,2,6,3,7] => ? = 4
[[1,3,4,7],[2],[5],[6]]
 => [6,5,2,1,3,4,7] => [2,1,6,5,3,4,7] => [2,1,5,6,4,3,7] => ? = 4
[[1,2,4,7],[3],[5],[6]]
 => [6,5,3,1,2,4,7] => [1,3,6,5,2,4,7] => [1,5,2,6,4,3,7] => ? = 5
[[1,4,5,6],[2],[3],[7]]
 => [7,3,2,1,4,5,6] => [3,2,1,7,4,5,6] => [3,2,1,5,6,7,4] => ? = 3
[[1,3,5,6],[2],[4],[7]]
 => [7,4,2,1,3,5,6] => [2,4,1,7,3,5,6] => [3,1,5,2,6,7,4] => ? = 4
[[1,2,5,6],[3],[4],[7]]
 => [7,4,3,1,2,5,6] => [1,4,3,7,2,5,6] => [1,5,3,2,6,7,4] => ? = 5
[[1,3,4,6],[2],[5],[7]]
 => [7,5,2,1,3,4,6] => [2,1,5,7,3,4,6] => [2,1,5,6,3,7,4] => ? = 5
[[1,2,4,6],[3],[5],[7]]
 => [7,5,3,1,2,4,6] => [1,3,5,7,2,4,6] => [1,5,2,6,3,7,4] => ? = 6
[[1,3,4,5],[2],[6],[7]]
 => [7,6,2,1,3,4,5] => [2,1,3,7,6,4,5] => [2,1,3,6,7,5,4] => ? = 6
[[1,4,6],[2,5,7],[3]]
 => [3,2,5,7,1,4,6] => [3,2,1,5,4,7,6] => [3,2,1,5,4,7,6] => ? = 8
[[1,3,6],[2,5,7],[4]]
 => [4,2,5,7,1,3,6] => [2,4,1,5,3,7,6] => [3,1,5,2,4,7,6] => ? = 5
Description
The major index minus the number of inversions of a permutation. This is, the difference between [[St000004]] and [[St000018]].