Your data matches 7 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001696
(load all 2 compositions to match this statistic)
Mapping
St001696: 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]]
 => 2
[[1,2],[3]]
 => 0
[[1],[2],[3]]
 => 0
[[1,2,3,4]]
 => 0
[[1,3,4],[2]]
 => 2
[[1,2,4],[3]]
 => 3
[[1,2,3],[4]]
 => 0
[[1,3],[2,4]]
 => 2
[[1,2],[3,4]]
 => 0
[[1,4],[2],[3]]
 => 3
[[1,3],[2],[4]]
 => 2
[[1,2],[3],[4]]
 => 0
[[1],[2],[3],[4]]
 => 0
[[1,2,3,4,5]]
 => 0
[[1,3,4,5],[2]]
 => 2
[[1,2,4,5],[3]]
 => 3
[[1,2,3,5],[4]]
 => 4
[[1,2,3,4],[5]]
 => 0
[[1,3,5],[2,4]]
 => 6
[[1,2,5],[3,4]]
 => 4
[[1,3,4],[2,5]]
 => 2
[[1,2,4],[3,5]]
 => 3
[[1,2,3],[4,5]]
 => 0
[[1,4,5],[2],[3]]
 => 3
[[1,3,5],[2],[4]]
 => 6
[[1,2,5],[3],[4]]
 => 4
[[1,3,4],[2],[5]]
 => 2
[[1,2,4],[3],[5]]
 => 3
[[1,2,3],[4],[5]]
 => 0
[[1,4],[2,5],[3]]
 => 3
[[1,3],[2,5],[4]]
 => 6
[[1,2],[3,5],[4]]
 => 4
[[1,3],[2,4],[5]]
 => 2
[[1,2],[3,4],[5]]
 => 0
[[1,5],[2],[3],[4]]
 => 4
[[1,4],[2],[3],[5]]
 => 3
[[1,3],[2],[4],[5]]
 => 2
[[1,2],[3],[4],[5]]
 => 0
[[1],[2],[3],[4],[5]]
 => 0
[[1,2,3,4,5,6]]
 => 0
[[1,3,4,5,6],[2]]
 => 2
[[1,2,4,5,6],[3]]
 => 3
[[1,2,3,5,6],[4]]
 => 4
[[1,2,3,4,6],[5]]
 => 5
[[1,2,3,4,5],[6]]
 => 0
[[1,3,5,6],[2,4]]
 => 6
Description
The natural major index of a standard Young tableau. A natural descent of a standard tableau $T$ is an entry $i$ such that $i+1$ appears in a higher row than $i$ in English notation. The natural major index of a tableau with natural descent set $D$ is then $\sum_{d\in D} d$.
Matching statistic: St000305
(load all 4 compositions to match this statistic)
Mapping
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
St000305: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
 => [1] => [1] => 0
[[1,2]]
 => [1,2] => [1,2] => 0
[[1],[2]]
 => [2,1] => [1,2] => 0
[[1,2,3]]
 => [1,2,3] => [1,2,3] => 0
[[1,3],[2]]
 => [2,1,3] => [1,3,2] => 2
[[1,2],[3]]
 => [3,1,2] => [1,2,3] => 0
[[1],[2],[3]]
 => [3,2,1] => [1,2,3] => 0
[[1,2,3,4]]
 => [1,2,3,4] => [1,2,3,4] => 0
[[1,3,4],[2]]
 => [2,1,3,4] => [1,3,4,2] => 2
[[1,2,4],[3]]
 => [3,1,2,4] => [1,2,4,3] => 3
[[1,2,3],[4]]
 => [4,1,2,3] => [1,2,3,4] => 0
[[1,3],[2,4]]
 => [2,4,1,3] => [1,3,2,4] => 2
[[1,2],[3,4]]
 => [3,4,1,2] => [1,2,3,4] => 0
[[1,4],[2],[3]]
 => [3,2,1,4] => [1,4,2,3] => 3
[[1,3],[2],[4]]
 => [4,2,1,3] => [1,3,2,4] => 2
[[1,2],[3],[4]]
 => [4,3,1,2] => [1,2,3,4] => 0
[[1],[2],[3],[4]]
 => [4,3,2,1] => [1,2,3,4] => 0
[[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] => [1,3,4,5,2] => 2
[[1,2,4,5],[3]]
 => [3,1,2,4,5] => [1,2,4,5,3] => 3
[[1,2,3,5],[4]]
 => [4,1,2,3,5] => [1,2,3,5,4] => 4
[[1,2,3,4],[5]]
 => [5,1,2,3,4] => [1,2,3,4,5] => 0
[[1,3,5],[2,4]]
 => [2,4,1,3,5] => [1,3,5,2,4] => 6
[[1,2,5],[3,4]]
 => [3,4,1,2,5] => [1,2,5,3,4] => 4
[[1,3,4],[2,5]]
 => [2,5,1,3,4] => [1,3,4,2,5] => 2
[[1,2,4],[3,5]]
 => [3,5,1,2,4] => [1,2,4,3,5] => 3
[[1,2,3],[4,5]]
 => [4,5,1,2,3] => [1,2,3,4,5] => 0
[[1,4,5],[2],[3]]
 => [3,2,1,4,5] => [1,4,5,2,3] => 3
[[1,3,5],[2],[4]]
 => [4,2,1,3,5] => [1,3,5,2,4] => 6
[[1,2,5],[3],[4]]
 => [4,3,1,2,5] => [1,2,5,3,4] => 4
[[1,3,4],[2],[5]]
 => [5,2,1,3,4] => [1,3,4,2,5] => 2
[[1,2,4],[3],[5]]
 => [5,3,1,2,4] => [1,2,4,3,5] => 3
[[1,2,3],[4],[5]]
 => [5,4,1,2,3] => [1,2,3,4,5] => 0
[[1,4],[2,5],[3]]
 => [3,2,5,1,4] => [1,4,2,5,3] => 3
[[1,3],[2,5],[4]]
 => [4,2,5,1,3] => [1,3,2,5,4] => 6
[[1,2],[3,5],[4]]
 => [4,3,5,1,2] => [1,2,3,5,4] => 4
[[1,3],[2,4],[5]]
 => [5,2,4,1,3] => [1,3,2,4,5] => 2
[[1,2],[3,4],[5]]
 => [5,3,4,1,2] => [1,2,3,4,5] => 0
[[1,5],[2],[3],[4]]
 => [4,3,2,1,5] => [1,5,2,3,4] => 4
[[1,4],[2],[3],[5]]
 => [5,3,2,1,4] => [1,4,2,3,5] => 3
[[1,3],[2],[4],[5]]
 => [5,4,2,1,3] => [1,3,2,4,5] => 2
[[1,2],[3],[4],[5]]
 => [5,4,3,1,2] => [1,2,3,4,5] => 0
[[1],[2],[3],[4],[5]]
 => [5,4,3,2,1] => [1,2,3,4,5] => 0
[[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] => [1,3,4,5,6,2] => 2
[[1,2,4,5,6],[3]]
 => [3,1,2,4,5,6] => [1,2,4,5,6,3] => 3
[[1,2,3,5,6],[4]]
 => [4,1,2,3,5,6] => [1,2,3,5,6,4] => 4
[[1,2,3,4,6],[5]]
 => [5,1,2,3,4,6] => [1,2,3,4,6,5] => 5
[[1,2,3,4,5],[6]]
 => [6,1,2,3,4,5] => [1,2,3,4,5,6] => 0
[[1,3,5,6],[2,4]]
 => [2,4,1,3,5,6] => [1,3,5,6,2,4] => 6
Description
The inverse major index of a permutation. This is the major index [[St000004]] of the inverse permutation [[Mp00066]].
Matching statistic: St000004
Mapping
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000004: 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] => [1,2] => [1,2] => 0
[[1,2,3]]
 => [1,2,3] => [1,2,3] => [1,2,3] => 0
[[1,3],[2]]
 => [2,1,3] => [1,3,2] => [1,3,2] => 2
[[1,2],[3]]
 => [3,1,2] => [1,2,3] => [1,2,3] => 0
[[1],[2],[3]]
 => [3,2,1] => [1,2,3] => [1,2,3] => 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] => [1,3,4,2] => [1,4,2,3] => 2
[[1,2,4],[3]]
 => [3,1,2,4] => [1,2,4,3] => [1,2,4,3] => 3
[[1,2,3],[4]]
 => [4,1,2,3] => [1,2,3,4] => [1,2,3,4] => 0
[[1,3],[2,4]]
 => [2,4,1,3] => [1,3,2,4] => [1,3,2,4] => 2
[[1,2],[3,4]]
 => [3,4,1,2] => [1,2,3,4] => [1,2,3,4] => 0
[[1,4],[2],[3]]
 => [3,2,1,4] => [1,4,2,3] => [1,3,4,2] => 3
[[1,3],[2],[4]]
 => [4,2,1,3] => [1,3,2,4] => [1,3,2,4] => 2
[[1,2],[3],[4]]
 => [4,3,1,2] => [1,2,3,4] => [1,2,3,4] => 0
[[1],[2],[3],[4]]
 => [4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 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] => [1,3,4,5,2] => [1,5,2,3,4] => 2
[[1,2,4,5],[3]]
 => [3,1,2,4,5] => [1,2,4,5,3] => [1,2,5,3,4] => 3
[[1,2,3,5],[4]]
 => [4,1,2,3,5] => [1,2,3,5,4] => [1,2,3,5,4] => 4
[[1,2,3,4],[5]]
 => [5,1,2,3,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[[1,3,5],[2,4]]
 => [2,4,1,3,5] => [1,3,5,2,4] => [1,4,2,5,3] => 6
[[1,2,5],[3,4]]
 => [3,4,1,2,5] => [1,2,5,3,4] => [1,2,4,5,3] => 4
[[1,3,4],[2,5]]
 => [2,5,1,3,4] => [1,3,4,2,5] => [1,4,2,3,5] => 2
[[1,2,4],[3,5]]
 => [3,5,1,2,4] => [1,2,4,3,5] => [1,2,4,3,5] => 3
[[1,2,3],[4,5]]
 => [4,5,1,2,3] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[[1,4,5],[2],[3]]
 => [3,2,1,4,5] => [1,4,5,2,3] => [1,4,5,2,3] => 3
[[1,3,5],[2],[4]]
 => [4,2,1,3,5] => [1,3,5,2,4] => [1,4,2,5,3] => 6
[[1,2,5],[3],[4]]
 => [4,3,1,2,5] => [1,2,5,3,4] => [1,2,4,5,3] => 4
[[1,3,4],[2],[5]]
 => [5,2,1,3,4] => [1,3,4,2,5] => [1,4,2,3,5] => 2
[[1,2,4],[3],[5]]
 => [5,3,1,2,4] => [1,2,4,3,5] => [1,2,4,3,5] => 3
[[1,2,3],[4],[5]]
 => [5,4,1,2,3] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[[1,4],[2,5],[3]]
 => [3,2,5,1,4] => [1,4,2,5,3] => [1,3,5,2,4] => 3
[[1,3],[2,5],[4]]
 => [4,2,5,1,3] => [1,3,2,5,4] => [1,3,2,5,4] => 6
[[1,2],[3,5],[4]]
 => [4,3,5,1,2] => [1,2,3,5,4] => [1,2,3,5,4] => 4
[[1,3],[2,4],[5]]
 => [5,2,4,1,3] => [1,3,2,4,5] => [1,3,2,4,5] => 2
[[1,2],[3,4],[5]]
 => [5,3,4,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[[1,5],[2],[3],[4]]
 => [4,3,2,1,5] => [1,5,2,3,4] => [1,3,4,5,2] => 4
[[1,4],[2],[3],[5]]
 => [5,3,2,1,4] => [1,4,2,3,5] => [1,3,4,2,5] => 3
[[1,3],[2],[4],[5]]
 => [5,4,2,1,3] => [1,3,2,4,5] => [1,3,2,4,5] => 2
[[1,2],[3],[4],[5]]
 => [5,4,3,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[[1],[2],[3],[4],[5]]
 => [5,4,3,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => 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] => [1,3,4,5,6,2] => [1,6,2,3,4,5] => 2
[[1,2,4,5,6],[3]]
 => [3,1,2,4,5,6] => [1,2,4,5,6,3] => [1,2,6,3,4,5] => 3
[[1,2,3,5,6],[4]]
 => [4,1,2,3,5,6] => [1,2,3,5,6,4] => [1,2,3,6,4,5] => 4
[[1,2,3,4,6],[5]]
 => [5,1,2,3,4,6] => [1,2,3,4,6,5] => [1,2,3,4,6,5] => 5
[[1,2,3,4,5],[6]]
 => [6,1,2,3,4,5] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[[1,3,5,6],[2,4]]
 => [2,4,1,3,5,6] => [1,3,5,6,2,4] => [1,5,2,6,3,4] => 6
Description
The major index of a permutation. This is the sum of the positions of its descents, $$\operatorname{maj}(\sigma) = \sum_{\sigma(i) > \sigma(i+1)} i.$$ Its generating function is $[n]_q! = [1]_q \cdot [2]_q \dots [n]_q$ for $[k]_q = 1 + q + q^2 + \dots q^{k-1}$. A statistic equidistributed with the major index is called '''Mahonian statistic'''.
Matching statistic: St000304
Mapping
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000304: 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] => [2,1] => 0
[[1],[2]]
 => [2,1] => [1,2] => [2,1] => 0
[[1,2,3]]
 => [1,2,3] => [1,2,3] => [3,2,1] => 0
[[1,3],[2]]
 => [2,1,3] => [1,3,2] => [3,1,2] => 2
[[1,2],[3]]
 => [3,1,2] => [1,2,3] => [3,2,1] => 0
[[1],[2],[3]]
 => [3,2,1] => [1,2,3] => [3,2,1] => 0
[[1,2,3,4]]
 => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 0
[[1,3,4],[2]]
 => [2,1,3,4] => [1,3,4,2] => [4,2,1,3] => 2
[[1,2,4],[3]]
 => [3,1,2,4] => [1,2,4,3] => [4,3,1,2] => 3
[[1,2,3],[4]]
 => [4,1,2,3] => [1,2,3,4] => [4,3,2,1] => 0
[[1,3],[2,4]]
 => [2,4,1,3] => [1,3,2,4] => [4,2,3,1] => 2
[[1,2],[3,4]]
 => [3,4,1,2] => [1,2,3,4] => [4,3,2,1] => 0
[[1,4],[2],[3]]
 => [3,2,1,4] => [1,4,2,3] => [4,1,3,2] => 3
[[1,3],[2],[4]]
 => [4,2,1,3] => [1,3,2,4] => [4,2,3,1] => 2
[[1,2],[3],[4]]
 => [4,3,1,2] => [1,2,3,4] => [4,3,2,1] => 0
[[1],[2],[3],[4]]
 => [4,3,2,1] => [1,2,3,4] => [4,3,2,1] => 0
[[1,2,3,4,5]]
 => [1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 0
[[1,3,4,5],[2]]
 => [2,1,3,4,5] => [1,3,4,5,2] => [5,3,2,1,4] => 2
[[1,2,4,5],[3]]
 => [3,1,2,4,5] => [1,2,4,5,3] => [5,4,2,1,3] => 3
[[1,2,3,5],[4]]
 => [4,1,2,3,5] => [1,2,3,5,4] => [5,4,3,1,2] => 4
[[1,2,3,4],[5]]
 => [5,1,2,3,4] => [1,2,3,4,5] => [5,4,3,2,1] => 0
[[1,3,5],[2,4]]
 => [2,4,1,3,5] => [1,3,5,2,4] => [5,3,1,4,2] => 6
[[1,2,5],[3,4]]
 => [3,4,1,2,5] => [1,2,5,3,4] => [5,4,1,3,2] => 4
[[1,3,4],[2,5]]
 => [2,5,1,3,4] => [1,3,4,2,5] => [5,3,2,4,1] => 2
[[1,2,4],[3,5]]
 => [3,5,1,2,4] => [1,2,4,3,5] => [5,4,2,3,1] => 3
[[1,2,3],[4,5]]
 => [4,5,1,2,3] => [1,2,3,4,5] => [5,4,3,2,1] => 0
[[1,4,5],[2],[3]]
 => [3,2,1,4,5] => [1,4,5,2,3] => [5,2,1,4,3] => 3
[[1,3,5],[2],[4]]
 => [4,2,1,3,5] => [1,3,5,2,4] => [5,3,1,4,2] => 6
[[1,2,5],[3],[4]]
 => [4,3,1,2,5] => [1,2,5,3,4] => [5,4,1,3,2] => 4
[[1,3,4],[2],[5]]
 => [5,2,1,3,4] => [1,3,4,2,5] => [5,3,2,4,1] => 2
[[1,2,4],[3],[5]]
 => [5,3,1,2,4] => [1,2,4,3,5] => [5,4,2,3,1] => 3
[[1,2,3],[4],[5]]
 => [5,4,1,2,3] => [1,2,3,4,5] => [5,4,3,2,1] => 0
[[1,4],[2,5],[3]]
 => [3,2,5,1,4] => [1,4,2,5,3] => [5,2,4,1,3] => 3
[[1,3],[2,5],[4]]
 => [4,2,5,1,3] => [1,3,2,5,4] => [5,3,4,1,2] => 6
[[1,2],[3,5],[4]]
 => [4,3,5,1,2] => [1,2,3,5,4] => [5,4,3,1,2] => 4
[[1,3],[2,4],[5]]
 => [5,2,4,1,3] => [1,3,2,4,5] => [5,3,4,2,1] => 2
[[1,2],[3,4],[5]]
 => [5,3,4,1,2] => [1,2,3,4,5] => [5,4,3,2,1] => 0
[[1,5],[2],[3],[4]]
 => [4,3,2,1,5] => [1,5,2,3,4] => [5,1,4,3,2] => 4
[[1,4],[2],[3],[5]]
 => [5,3,2,1,4] => [1,4,2,3,5] => [5,2,4,3,1] => 3
[[1,3],[2],[4],[5]]
 => [5,4,2,1,3] => [1,3,2,4,5] => [5,3,4,2,1] => 2
[[1,2],[3],[4],[5]]
 => [5,4,3,1,2] => [1,2,3,4,5] => [5,4,3,2,1] => 0
[[1],[2],[3],[4],[5]]
 => [5,4,3,2,1] => [1,2,3,4,5] => [5,4,3,2,1] => 0
[[1,2,3,4,5,6]]
 => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => 0
[[1,3,4,5,6],[2]]
 => [2,1,3,4,5,6] => [1,3,4,5,6,2] => [6,4,3,2,1,5] => 2
[[1,2,4,5,6],[3]]
 => [3,1,2,4,5,6] => [1,2,4,5,6,3] => [6,5,3,2,1,4] => 3
[[1,2,3,5,6],[4]]
 => [4,1,2,3,5,6] => [1,2,3,5,6,4] => [6,5,4,2,1,3] => 4
[[1,2,3,4,6],[5]]
 => [5,1,2,3,4,6] => [1,2,3,4,6,5] => [6,5,4,3,1,2] => 5
[[1,2,3,4,5],[6]]
 => [6,1,2,3,4,5] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => 0
[[1,3,5,6],[2,4]]
 => [2,4,1,3,5,6] => [1,3,5,6,2,4] => [6,4,2,1,5,3] => 6
Description
The load of a permutation. The definition of the load of a finite word in a totally ordered alphabet can be found in [1], for permutations, it is given by the major index [[St000004]] of the reverse [[Mp00064]] of the inverse [[Mp00066]] permutation.
Matching statistic: St000330
Mapping
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
St000330: Standard tableaux ⟶ 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] => [1,2] => [[1,2]]
 => 0
[[1,2,3]]
 => [1,2,3] => [1,2,3] => [[1,2,3]]
 => 0
[[1,3],[2]]
 => [2,1,3] => [1,3,2] => [[1,2],[3]]
 => 2
[[1,2],[3]]
 => [3,1,2] => [1,2,3] => [[1,2,3]]
 => 0
[[1],[2],[3]]
 => [3,2,1] => [1,2,3] => [[1,2,3]]
 => 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] => [1,3,4,2] => [[1,2,4],[3]]
 => 2
[[1,2,4],[3]]
 => [3,1,2,4] => [1,2,4,3] => [[1,2,3],[4]]
 => 3
[[1,2,3],[4]]
 => [4,1,2,3] => [1,2,3,4] => [[1,2,3,4]]
 => 0
[[1,3],[2,4]]
 => [2,4,1,3] => [1,3,2,4] => [[1,2,4],[3]]
 => 2
[[1,2],[3,4]]
 => [3,4,1,2] => [1,2,3,4] => [[1,2,3,4]]
 => 0
[[1,4],[2],[3]]
 => [3,2,1,4] => [1,4,2,3] => [[1,2,3],[4]]
 => 3
[[1,3],[2],[4]]
 => [4,2,1,3] => [1,3,2,4] => [[1,2,4],[3]]
 => 2
[[1,2],[3],[4]]
 => [4,3,1,2] => [1,2,3,4] => [[1,2,3,4]]
 => 0
[[1],[2],[3],[4]]
 => [4,3,2,1] => [1,2,3,4] => [[1,2,3,4]]
 => 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] => [1,3,4,5,2] => [[1,2,4,5],[3]]
 => 2
[[1,2,4,5],[3]]
 => [3,1,2,4,5] => [1,2,4,5,3] => [[1,2,3,5],[4]]
 => 3
[[1,2,3,5],[4]]
 => [4,1,2,3,5] => [1,2,3,5,4] => [[1,2,3,4],[5]]
 => 4
[[1,2,3,4],[5]]
 => [5,1,2,3,4] => [1,2,3,4,5] => [[1,2,3,4,5]]
 => 0
[[1,3,5],[2,4]]
 => [2,4,1,3,5] => [1,3,5,2,4] => [[1,2,4],[3,5]]
 => 6
[[1,2,5],[3,4]]
 => [3,4,1,2,5] => [1,2,5,3,4] => [[1,2,3,4],[5]]
 => 4
[[1,3,4],[2,5]]
 => [2,5,1,3,4] => [1,3,4,2,5] => [[1,2,4,5],[3]]
 => 2
[[1,2,4],[3,5]]
 => [3,5,1,2,4] => [1,2,4,3,5] => [[1,2,3,5],[4]]
 => 3
[[1,2,3],[4,5]]
 => [4,5,1,2,3] => [1,2,3,4,5] => [[1,2,3,4,5]]
 => 0
[[1,4,5],[2],[3]]
 => [3,2,1,4,5] => [1,4,5,2,3] => [[1,2,3],[4,5]]
 => 3
[[1,3,5],[2],[4]]
 => [4,2,1,3,5] => [1,3,5,2,4] => [[1,2,4],[3,5]]
 => 6
[[1,2,5],[3],[4]]
 => [4,3,1,2,5] => [1,2,5,3,4] => [[1,2,3,4],[5]]
 => 4
[[1,3,4],[2],[5]]
 => [5,2,1,3,4] => [1,3,4,2,5] => [[1,2,4,5],[3]]
 => 2
[[1,2,4],[3],[5]]
 => [5,3,1,2,4] => [1,2,4,3,5] => [[1,2,3,5],[4]]
 => 3
[[1,2,3],[4],[5]]
 => [5,4,1,2,3] => [1,2,3,4,5] => [[1,2,3,4,5]]
 => 0
[[1,4],[2,5],[3]]
 => [3,2,5,1,4] => [1,4,2,5,3] => [[1,2,3],[4,5]]
 => 3
[[1,3],[2,5],[4]]
 => [4,2,5,1,3] => [1,3,2,5,4] => [[1,2,4],[3,5]]
 => 6
[[1,2],[3,5],[4]]
 => [4,3,5,1,2] => [1,2,3,5,4] => [[1,2,3,4],[5]]
 => 4
[[1,3],[2,4],[5]]
 => [5,2,4,1,3] => [1,3,2,4,5] => [[1,2,4,5],[3]]
 => 2
[[1,2],[3,4],[5]]
 => [5,3,4,1,2] => [1,2,3,4,5] => [[1,2,3,4,5]]
 => 0
[[1,5],[2],[3],[4]]
 => [4,3,2,1,5] => [1,5,2,3,4] => [[1,2,3,4],[5]]
 => 4
[[1,4],[2],[3],[5]]
 => [5,3,2,1,4] => [1,4,2,3,5] => [[1,2,3,5],[4]]
 => 3
[[1,3],[2],[4],[5]]
 => [5,4,2,1,3] => [1,3,2,4,5] => [[1,2,4,5],[3]]
 => 2
[[1,2],[3],[4],[5]]
 => [5,4,3,1,2] => [1,2,3,4,5] => [[1,2,3,4,5]]
 => 0
[[1],[2],[3],[4],[5]]
 => [5,4,3,2,1] => [1,2,3,4,5] => [[1,2,3,4,5]]
 => 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] => [1,3,4,5,6,2] => [[1,2,4,5,6],[3]]
 => 2
[[1,2,4,5,6],[3]]
 => [3,1,2,4,5,6] => [1,2,4,5,6,3] => [[1,2,3,5,6],[4]]
 => 3
[[1,2,3,5,6],[4]]
 => [4,1,2,3,5,6] => [1,2,3,5,6,4] => [[1,2,3,4,6],[5]]
 => 4
[[1,2,3,4,6],[5]]
 => [5,1,2,3,4,6] => [1,2,3,4,6,5] => [[1,2,3,4,5],[6]]
 => 5
[[1,2,3,4,5],[6]]
 => [6,1,2,3,4,5] => [1,2,3,4,5,6] => [[1,2,3,4,5,6]]
 => 0
[[1,3,5,6],[2,4]]
 => [2,4,1,3,5,6] => [1,3,5,6,2,4] => [[1,2,4,6],[3,5]]
 => 6
Description
The (standard) major index of a standard tableau. A descent of a standard tableau $T$ is an index $i$ such that $i+1$ appears in a row strictly below the row of $i$. The (standard) major index is the the sum of the descents.
Matching statistic: St000961
Mapping
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000961: Permutations ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
[[1]]
 => [1] => [1] => [1] => ? = 0
[[1,2]]
 => [1,2] => [2,1] => [2,1] => 0
[[1],[2]]
 => [2,1] => [1,2] => [1,2] => 0
[[1,2,3]]
 => [1,2,3] => [3,2,1] => [3,2,1] => 0
[[1,3],[2]]
 => [2,1,3] => [3,1,2] => [2,3,1] => 2
[[1,2],[3]]
 => [3,1,2] => [2,1,3] => [2,1,3] => 0
[[1],[2],[3]]
 => [3,2,1] => [1,2,3] => [1,2,3] => 0
[[1,2,3,4]]
 => [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 0
[[1,3,4],[2]]
 => [2,1,3,4] => [4,3,1,2] => [3,4,2,1] => 2
[[1,2,4],[3]]
 => [3,1,2,4] => [4,2,1,3] => [3,2,4,1] => 3
[[1,2,3],[4]]
 => [4,1,2,3] => [3,2,1,4] => [3,2,1,4] => 0
[[1,3],[2,4]]
 => [2,4,1,3] => [3,1,4,2] => [2,4,1,3] => 2
[[1,2],[3,4]]
 => [3,4,1,2] => [2,1,4,3] => [2,1,4,3] => 0
[[1,4],[2],[3]]
 => [3,2,1,4] => [4,1,2,3] => [2,3,4,1] => 3
[[1,3],[2],[4]]
 => [4,2,1,3] => [3,1,2,4] => [2,3,1,4] => 2
[[1,2],[3],[4]]
 => [4,3,1,2] => [2,1,3,4] => [2,1,3,4] => 0
[[1],[2],[3],[4]]
 => [4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 0
[[1,2,3,4,5]]
 => [1,2,3,4,5] => [5,4,3,2,1] => [5,4,3,2,1] => 0
[[1,3,4,5],[2]]
 => [2,1,3,4,5] => [5,4,3,1,2] => [4,5,3,2,1] => 2
[[1,2,4,5],[3]]
 => [3,1,2,4,5] => [5,4,2,1,3] => [4,3,5,2,1] => 3
[[1,2,3,5],[4]]
 => [4,1,2,3,5] => [5,3,2,1,4] => [4,3,2,5,1] => 4
[[1,2,3,4],[5]]
 => [5,1,2,3,4] => [4,3,2,1,5] => [4,3,2,1,5] => 0
[[1,3,5],[2,4]]
 => [2,4,1,3,5] => [5,3,1,4,2] => [3,5,2,4,1] => 6
[[1,2,5],[3,4]]
 => [3,4,1,2,5] => [5,2,1,4,3] => [3,2,5,4,1] => 4
[[1,3,4],[2,5]]
 => [2,5,1,3,4] => [4,3,1,5,2] => [3,5,2,1,4] => 2
[[1,2,4],[3,5]]
 => [3,5,1,2,4] => [4,2,1,5,3] => [3,2,5,1,4] => 3
[[1,2,3],[4,5]]
 => [4,5,1,2,3] => [3,2,1,5,4] => [3,2,1,5,4] => 0
[[1,4,5],[2],[3]]
 => [3,2,1,4,5] => [5,4,1,2,3] => [3,4,5,2,1] => 3
[[1,3,5],[2],[4]]
 => [4,2,1,3,5] => [5,3,1,2,4] => [3,4,2,5,1] => 6
[[1,2,5],[3],[4]]
 => [4,3,1,2,5] => [5,2,1,3,4] => [3,2,4,5,1] => 4
[[1,3,4],[2],[5]]
 => [5,2,1,3,4] => [4,3,1,2,5] => [3,4,2,1,5] => 2
[[1,2,4],[3],[5]]
 => [5,3,1,2,4] => [4,2,1,3,5] => [3,2,4,1,5] => 3
[[1,2,3],[4],[5]]
 => [5,4,1,2,3] => [3,2,1,4,5] => [3,2,1,4,5] => 0
[[1,4],[2,5],[3]]
 => [3,2,5,1,4] => [4,1,5,2,3] => [2,4,5,1,3] => 3
[[1,3],[2,5],[4]]
 => [4,2,5,1,3] => [3,1,5,2,4] => [2,4,1,5,3] => 6
[[1,2],[3,5],[4]]
 => [4,3,5,1,2] => [2,1,5,3,4] => [2,1,4,5,3] => 4
[[1,3],[2,4],[5]]
 => [5,2,4,1,3] => [3,1,4,2,5] => [2,4,1,3,5] => 2
[[1,2],[3,4],[5]]
 => [5,3,4,1,2] => [2,1,4,3,5] => [2,1,4,3,5] => 0
[[1,5],[2],[3],[4]]
 => [4,3,2,1,5] => [5,1,2,3,4] => [2,3,4,5,1] => 4
[[1,4],[2],[3],[5]]
 => [5,3,2,1,4] => [4,1,2,3,5] => [2,3,4,1,5] => 3
[[1,3],[2],[4],[5]]
 => [5,4,2,1,3] => [3,1,2,4,5] => [2,3,1,4,5] => 2
[[1,2],[3],[4],[5]]
 => [5,4,3,1,2] => [2,1,3,4,5] => [2,1,3,4,5] => 0
[[1],[2],[3],[4],[5]]
 => [5,4,3,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[[1,2,3,4,5,6]]
 => [1,2,3,4,5,6] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => 0
[[1,3,4,5,6],[2]]
 => [2,1,3,4,5,6] => [6,5,4,3,1,2] => [5,6,4,3,2,1] => 2
[[1,2,4,5,6],[3]]
 => [3,1,2,4,5,6] => [6,5,4,2,1,3] => [5,4,6,3,2,1] => 3
[[1,2,3,5,6],[4]]
 => [4,1,2,3,5,6] => [6,5,3,2,1,4] => [5,4,3,6,2,1] => 4
[[1,2,3,4,6],[5]]
 => [5,1,2,3,4,6] => [6,4,3,2,1,5] => [5,4,3,2,6,1] => 5
[[1,2,3,4,5],[6]]
 => [6,1,2,3,4,5] => [5,4,3,2,1,6] => [5,4,3,2,1,6] => 0
[[1,3,5,6],[2,4]]
 => [2,4,1,3,5,6] => [6,5,3,1,4,2] => [4,6,3,5,2,1] => 6
[[1,2,5,6],[3,4]]
 => [3,4,1,2,5,6] => [6,5,2,1,4,3] => [4,3,6,5,2,1] => 4
Description
The shifted major index of a permutation. This is given by the sum of all indices $i$ such that $\pi(i)-\pi(i+1) > 1$. Summing with [[St000354]] yields Rawlings' Mahonian statistic, see [1, p. 50].
Matching statistic: St001207
Mapping
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
St001207: Permutations ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 30%
Values
[[1]]
 => [1] => [1] => [1] => ? = 0
[[1,2]]
 => [1,2] => [1,2] => [1,2] => 0
[[1],[2]]
 => [2,1] => [1,2] => [1,2] => 0
[[1,2,3]]
 => [1,2,3] => [1,2,3] => [1,2,3] => 0
[[1,3],[2]]
 => [2,1,3] => [1,3,2] => [3,1,2] => 2
[[1,2],[3]]
 => [3,1,2] => [1,2,3] => [1,2,3] => 0
[[1],[2],[3]]
 => [3,2,1] => [1,2,3] => [1,2,3] => 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] => [1,3,4,2] => [2,4,1,3] => 2
[[1,2,4],[3]]
 => [3,1,2,4] => [1,2,4,3] => [4,1,2,3] => 3
[[1,2,3],[4]]
 => [4,1,2,3] => [1,2,3,4] => [1,2,3,4] => 0
[[1,3],[2,4]]
 => [2,4,1,3] => [1,3,2,4] => [3,1,2,4] => 2
[[1,2],[3,4]]
 => [3,4,1,2] => [1,2,3,4] => [1,2,3,4] => 0
[[1,4],[2],[3]]
 => [3,2,1,4] => [1,4,2,3] => [3,4,1,2] => 3
[[1,3],[2],[4]]
 => [4,2,1,3] => [1,3,2,4] => [3,1,2,4] => 2
[[1,2],[3],[4]]
 => [4,3,1,2] => [1,2,3,4] => [1,2,3,4] => 0
[[1],[2],[3],[4]]
 => [4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 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] => [1,3,4,5,2] => [2,3,5,1,4] => ? = 2
[[1,2,4,5],[3]]
 => [3,1,2,4,5] => [1,2,4,5,3] => [3,5,1,2,4] => ? = 3
[[1,2,3,5],[4]]
 => [4,1,2,3,5] => [1,2,3,5,4] => [5,1,2,3,4] => ? = 4
[[1,2,3,4],[5]]
 => [5,1,2,3,4] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 0
[[1,3,5],[2,4]]
 => [2,4,1,3,5] => [1,3,5,2,4] => [2,4,5,1,3] => ? = 6
[[1,2,5],[3,4]]
 => [3,4,1,2,5] => [1,2,5,3,4] => [4,5,1,2,3] => ? = 4
[[1,3,4],[2,5]]
 => [2,5,1,3,4] => [1,3,4,2,5] => [2,4,1,3,5] => ? = 2
[[1,2,4],[3,5]]
 => [3,5,1,2,4] => [1,2,4,3,5] => [4,1,2,3,5] => ? = 3
[[1,2,3],[4,5]]
 => [4,5,1,2,3] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 0
[[1,4,5],[2],[3]]
 => [3,2,1,4,5] => [1,4,5,2,3] => [4,2,5,1,3] => ? = 3
[[1,3,5],[2],[4]]
 => [4,2,1,3,5] => [1,3,5,2,4] => [2,4,5,1,3] => ? = 6
[[1,2,5],[3],[4]]
 => [4,3,1,2,5] => [1,2,5,3,4] => [4,5,1,2,3] => ? = 4
[[1,3,4],[2],[5]]
 => [5,2,1,3,4] => [1,3,4,2,5] => [2,4,1,3,5] => ? = 2
[[1,2,4],[3],[5]]
 => [5,3,1,2,4] => [1,2,4,3,5] => [4,1,2,3,5] => ? = 3
[[1,2,3],[4],[5]]
 => [5,4,1,2,3] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 0
[[1,4],[2,5],[3]]
 => [3,2,5,1,4] => [1,4,2,5,3] => [5,3,1,2,4] => ? = 3
[[1,3],[2,5],[4]]
 => [4,2,5,1,3] => [1,3,2,5,4] => [2,5,1,3,4] => ? = 6
[[1,2],[3,5],[4]]
 => [4,3,5,1,2] => [1,2,3,5,4] => [5,1,2,3,4] => ? = 4
[[1,3],[2,4],[5]]
 => [5,2,4,1,3] => [1,3,2,4,5] => [3,1,2,4,5] => ? = 2
[[1,2],[3,4],[5]]
 => [5,3,4,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 0
[[1,5],[2],[3],[4]]
 => [4,3,2,1,5] => [1,5,2,3,4] => [3,4,5,1,2] => ? = 4
[[1,4],[2],[3],[5]]
 => [5,3,2,1,4] => [1,4,2,3,5] => [3,4,1,2,5] => ? = 3
[[1,3],[2],[4],[5]]
 => [5,4,2,1,3] => [1,3,2,4,5] => [3,1,2,4,5] => ? = 2
[[1,2],[3],[4],[5]]
 => [5,4,3,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 0
[[1],[2],[3],[4],[5]]
 => [5,4,3,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 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] => [1,3,4,5,6,2] => [2,3,4,6,1,5] => ? = 2
[[1,2,4,5,6],[3]]
 => [3,1,2,4,5,6] => [1,2,4,5,6,3] => [3,4,6,1,2,5] => ? = 3
[[1,2,3,5,6],[4]]
 => [4,1,2,3,5,6] => [1,2,3,5,6,4] => [4,6,1,2,3,5] => ? = 4
[[1,2,3,4,6],[5]]
 => [5,1,2,3,4,6] => [1,2,3,4,6,5] => [6,1,2,3,4,5] => ? = 5
[[1,2,3,4,5],[6]]
 => [6,1,2,3,4,5] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 0
[[1,3,5,6],[2,4]]
 => [2,4,1,3,5,6] => [1,3,5,6,2,4] => [5,2,3,6,1,4] => ? = 6
[[1,2,5,6],[3,4]]
 => [3,4,1,2,5,6] => [1,2,5,6,3,4] => [5,3,6,1,2,4] => ? = 4
[[1,3,4,6],[2,5]]
 => [2,5,1,3,4,6] => [1,3,4,6,2,5] => [2,3,5,6,1,4] => ? = 7
[[1,2,4,6],[3,5]]
 => [3,5,1,2,4,6] => [1,2,4,6,3,5] => [3,5,6,1,2,4] => ? = 8
[[1,2,3,6],[4,5]]
 => [4,5,1,2,3,6] => [1,2,3,6,4,5] => [5,6,1,2,3,4] => ? = 5
[[1,3,4,5],[2,6]]
 => [2,6,1,3,4,5] => [1,3,4,5,2,6] => [2,3,5,1,4,6] => ? = 2
[[1,2,4,5],[3,6]]
 => [3,6,1,2,4,5] => [1,2,4,5,3,6] => [3,5,1,2,4,6] => ? = 3
[[1,2,3,5],[4,6]]
 => [4,6,1,2,3,5] => [1,2,3,5,4,6] => [5,1,2,3,4,6] => ? = 4
[[1,2,3,4],[5,6]]
 => [5,6,1,2,3,4] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 0
[[1,4,5,6],[2],[3]]
 => [3,2,1,4,5,6] => [1,4,5,6,2,3] => [2,5,3,6,1,4] => ? = 3
[[1,3,5,6],[2],[4]]
 => [4,2,1,3,5,6] => [1,3,5,6,2,4] => [5,2,3,6,1,4] => ? = 6
[[1,2,5,6],[3],[4]]
 => [4,3,1,2,5,6] => [1,2,5,6,3,4] => [5,3,6,1,2,4] => ? = 4
[[1,3,4,6],[2],[5]]
 => [5,2,1,3,4,6] => [1,3,4,6,2,5] => [2,3,5,6,1,4] => ? = 7
[[1,2,4,6],[3],[5]]
 => [5,3,1,2,4,6] => [1,2,4,6,3,5] => [3,5,6,1,2,4] => ? = 8
[[1,2,3,6],[4],[5]]
 => [5,4,1,2,3,6] => [1,2,3,6,4,5] => [5,6,1,2,3,4] => ? = 5
[[1,3,4,5],[2],[6]]
 => [6,2,1,3,4,5] => [1,3,4,5,2,6] => [2,3,5,1,4,6] => ? = 2
[[1,2,4,5],[3],[6]]
 => [6,3,1,2,4,5] => [1,2,4,5,3,6] => [3,5,1,2,4,6] => ? = 3
Description
The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$.