Identifier
Values
[[1,2]] => [[1,2]] => [1,2] => [1,2] => 0
[[1],[2]] => [[1],[2]] => [2,1] => [1,2] => 0
[[1,2,3]] => [[1,2,3]] => [1,2,3] => [1,2,3] => 0
[[1,3],[2]] => [[1,2],[3]] => [3,1,2] => [1,3,2] => 1
[[1,2],[3]] => [[1,3],[2]] => [2,1,3] => [1,2,3] => 0
[[1],[2],[3]] => [[1],[2],[3]] => [3,2,1] => [1,3,2] => 1
[[1,2,3,4]] => [[1,2,3,4]] => [1,2,3,4] => [1,2,3,4] => 0
[[1,3,4],[2]] => [[1,2,4],[3]] => [3,1,2,4] => [1,3,2,4] => 1
[[1,2,4],[3]] => [[1,2,3],[4]] => [4,1,2,3] => [1,4,3,2] => 2
[[1,2,3],[4]] => [[1,3,4],[2]] => [2,1,3,4] => [1,2,3,4] => 0
[[1,3],[2,4]] => [[1,2],[3,4]] => [3,4,1,2] => [1,3,2,4] => 1
[[1,2],[3,4]] => [[1,3],[2,4]] => [2,4,1,3] => [1,2,4,3] => 0
[[1,4],[2],[3]] => [[1,2],[3],[4]] => [4,3,1,2] => [1,4,2,3] => 2
[[1,3],[2],[4]] => [[1,4],[2],[3]] => [3,2,1,4] => [1,3,2,4] => 1
[[1,2],[3],[4]] => [[1,3],[2],[4]] => [4,2,1,3] => [1,4,3,2] => 2
[[1],[2],[3],[4]] => [[1],[2],[3],[4]] => [4,3,2,1] => [1,4,2,3] => 2
[[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]] => [[1,2,4,5],[3]] => [3,1,2,4,5] => [1,3,2,4,5] => 1
[[1,2,4,5],[3]] => [[1,2,3,5],[4]] => [4,1,2,3,5] => [1,4,3,2,5] => 2
[[1,2,3,5],[4]] => [[1,2,3,4],[5]] => [5,1,2,3,4] => [1,5,4,3,2] => 3
[[1,2,3,4],[5]] => [[1,3,4,5],[2]] => [2,1,3,4,5] => [1,2,3,4,5] => 0
[[1,3,5],[2,4]] => [[1,2,4],[3,5]] => [3,5,1,2,4] => [1,3,2,5,4] => 1
[[1,2,5],[3,4]] => [[1,2,3],[4,5]] => [4,5,1,2,3] => [1,4,2,5,3] => 2
[[1,3,4],[2,5]] => [[1,2,5],[3,4]] => [3,4,1,2,5] => [1,3,2,4,5] => 1
[[1,2,4],[3,5]] => [[1,3,5],[2,4]] => [2,4,1,3,5] => [1,2,4,3,5] => 0
[[1,2,3],[4,5]] => [[1,3,4],[2,5]] => [2,5,1,3,4] => [1,2,5,4,3] => 0
[[1,4,5],[2],[3]] => [[1,2,5],[3],[4]] => [4,3,1,2,5] => [1,4,2,3,5] => 2
[[1,3,5],[2],[4]] => [[1,2,4],[3],[5]] => [5,3,1,2,4] => [1,5,4,2,3] => 3
[[1,2,5],[3],[4]] => [[1,2,3],[4],[5]] => [5,4,1,2,3] => [1,5,3,2,4] => 3
[[1,3,4],[2],[5]] => [[1,4,5],[2],[3]] => [3,2,1,4,5] => [1,3,2,4,5] => 1
[[1,2,4],[3],[5]] => [[1,3,5],[2],[4]] => [4,2,1,3,5] => [1,4,3,2,5] => 2
[[1,2,3],[4],[5]] => [[1,3,4],[2],[5]] => [5,2,1,3,4] => [1,5,4,3,2] => 3
[[1,4],[2,5],[3]] => [[1,2],[3,5],[4]] => [4,3,5,1,2] => [1,4,2,3,5] => 2
[[1,3],[2,5],[4]] => [[1,2],[3,4],[5]] => [5,3,4,1,2] => [1,5,2,3,4] => 3
[[1,2],[3,5],[4]] => [[1,3],[2,4],[5]] => [5,2,4,1,3] => [1,5,3,4,2] => 3
[[1,3],[2,4],[5]] => [[1,4],[2,5],[3]] => [3,2,5,1,4] => [1,3,5,4,2] => 1
[[1,2],[3,4],[5]] => [[1,3],[2,5],[4]] => [4,2,5,1,3] => [1,4,2,3,5] => 2
[[1,5],[2],[3],[4]] => [[1,2],[3],[4],[5]] => [5,4,3,1,2] => [1,5,2,4,3] => 3
[[1,4],[2],[3],[5]] => [[1,5],[2],[3],[4]] => [4,3,2,1,5] => [1,4,2,3,5] => 2
[[1,3],[2],[4],[5]] => [[1,4],[2],[3],[5]] => [5,3,2,1,4] => [1,5,4,2,3] => 3
[[1,2],[3],[4],[5]] => [[1,3],[2],[4],[5]] => [5,4,2,1,3] => [1,5,3,2,4] => 3
[[1],[2],[3],[4],[5]] => [[1],[2],[3],[4],[5]] => [5,4,3,2,1] => [1,5,2,4,3] => 3
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Description
The number of inversions of the second entry of a permutation.
This is, for a permutation $\pi$ of length $n$,
$$\# \{2 < k \leq n \mid \pi(2) > \pi(k)\}.$$
The number of inversions of the first entry is St000054The first entry of the permutation. and the number of inversions of the third entry is St001556The number of inversions of the third entry of a permutation.. The sequence of inversions of all the entries define the Lehmer code of a permutation.
Map
cycle-as-one-line notation
Description
Return the permutation obtained by concatenating the cycles of a permutation, each written with minimal element first, sorted by minimal element.
Map
reading word permutation
Description
Return the permutation obtained by reading the entries of the tableau row by row, starting with the bottom-most row in English notation.
Map
promotion
Description
The promotion of a standard Young tableau.
This map replaces the largest entry of the tableau with a zero, uses the jeu de taquin to move it to the origin, and finally increases all entries by one.