Identifier
Values
[1] => [[1]] => [[1]] => [1] => 0
[2] => [[1,2]] => [[1,2]] => [1,2] => 1
[1,1] => [[1],[2]] => [[1],[2]] => [2,1] => 0
[3] => [[1,2,3]] => [[1,2,3]] => [1,2,3] => 2
[2,1] => [[1,2],[3]] => [[1,3],[2]] => [2,1,3] => 1
[1,1,1] => [[1],[2],[3]] => [[1],[2],[3]] => [3,2,1] => 0
[4] => [[1,2,3,4]] => [[1,2,3,4]] => [1,2,3,4] => 3
[3,1] => [[1,2,3],[4]] => [[1,3,4],[2]] => [2,1,3,4] => 2
[2,2] => [[1,2],[3,4]] => [[1,3],[2,4]] => [2,4,1,3] => 1
[2,1,1] => [[1,2],[3],[4]] => [[1,3],[2],[4]] => [4,2,1,3] => 1
[1,1,1,1] => [[1],[2],[3],[4]] => [[1],[2],[3],[4]] => [4,3,2,1] => 0
[5] => [[1,2,3,4,5]] => [[1,2,3,4,5]] => [1,2,3,4,5] => 4
[4,1] => [[1,2,3,4],[5]] => [[1,3,4,5],[2]] => [2,1,3,4,5] => 3
[3,2] => [[1,2,3],[4,5]] => [[1,3,4],[2,5]] => [2,5,1,3,4] => 2
[3,1,1] => [[1,2,3],[4],[5]] => [[1,3,4],[2],[5]] => [5,2,1,3,4] => 2
[2,2,1] => [[1,2],[3,4],[5]] => [[1,3],[2,5],[4]] => [4,2,5,1,3] => 1
[2,1,1,1] => [[1,2],[3],[4],[5]] => [[1,3],[2],[4],[5]] => [5,4,2,1,3] => 1
[1,1,1,1,1] => [[1],[2],[3],[4],[5]] => [[1],[2],[3],[4],[5]] => [5,4,3,2,1] => 0
[6] => [[1,2,3,4,5,6]] => [[1,2,3,4,5,6]] => [1,2,3,4,5,6] => 5
[5,1] => [[1,2,3,4,5],[6]] => [[1,3,4,5,6],[2]] => [2,1,3,4,5,6] => 4
[4,2] => [[1,2,3,4],[5,6]] => [[1,3,4,5],[2,6]] => [2,6,1,3,4,5] => 3
[4,1,1] => [[1,2,3,4],[5],[6]] => [[1,3,4,5],[2],[6]] => [6,2,1,3,4,5] => 3
[3,3] => [[1,2,3],[4,5,6]] => [[1,3,4],[2,5,6]] => [2,5,6,1,3,4] => 3
[3,2,1] => [[1,2,3],[4,5],[6]] => [[1,3,4],[2,6],[5]] => [5,2,6,1,3,4] => 2
[3,1,1,1] => [[1,2,3],[4],[5],[6]] => [[1,3,4],[2],[5],[6]] => [6,5,2,1,3,4] => 2
[2,2,2] => [[1,2],[3,4],[5,6]] => [[1,3],[2,5],[4,6]] => [4,6,2,5,1,3] => 1
[2,2,1,1] => [[1,2],[3,4],[5],[6]] => [[1,3],[2,5],[4],[6]] => [6,4,2,5,1,3] => 1
[2,1,1,1,1] => [[1,2],[3],[4],[5],[6]] => [[1,3],[2],[4],[5],[6]] => [6,5,4,2,1,3] => 1
[1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6]] => [[1],[2],[3],[4],[5],[6]] => [6,5,4,3,2,1] => 0
[7] => [[1,2,3,4,5,6,7]] => [[1,2,3,4,5,6,7]] => [1,2,3,4,5,6,7] => 6
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Description
The number of repeated entries in the Lehmer code of a permutation.
The Lehmer code of a permutation $\pi$ is the sequence $(v_1,\dots,v_n)$, with $v_i=|\{j > i: \pi(j) < \pi(i)\}$. This statistic counts the number of distinct elements in this sequence.
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.
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
initial tableau
Description
Sends an integer partition to the standard tableau obtained by filling the numbers $1$ through $n$ row by row.