Identifier
Values
[1,2] => [1,1] => [[1],[2]] => [2,1] => 0
[2,1] => [2] => [[1,2]] => [1,2] => 0
[1,2,3] => [1,1,1] => [[1],[2],[3]] => [3,2,1] => 0
[1,3,2] => [2,1] => [[1,3],[2]] => [2,1,3] => 0
[2,1,3] => [2,1] => [[1,3],[2]] => [2,1,3] => 0
[2,3,1] => [3] => [[1,2,3]] => [1,2,3] => 0
[3,1,2] => [3] => [[1,2,3]] => [1,2,3] => 0
[3,2,1] => [2,1] => [[1,3],[2]] => [2,1,3] => 0
[1,2,3,4] => [1,1,1,1] => [[1],[2],[3],[4]] => [4,3,2,1] => 1
[1,2,4,3] => [2,1,1] => [[1,4],[2],[3]] => [3,2,1,4] => 0
[1,3,2,4] => [2,1,1] => [[1,4],[2],[3]] => [3,2,1,4] => 0
[1,3,4,2] => [3,1] => [[1,3,4],[2]] => [2,1,3,4] => 0
[1,4,2,3] => [3,1] => [[1,3,4],[2]] => [2,1,3,4] => 0
[1,4,3,2] => [2,1,1] => [[1,4],[2],[3]] => [3,2,1,4] => 0
[2,1,3,4] => [2,1,1] => [[1,4],[2],[3]] => [3,2,1,4] => 0
[2,1,4,3] => [2,2] => [[1,2],[3,4]] => [3,4,1,2] => 1
[2,3,1,4] => [3,1] => [[1,3,4],[2]] => [2,1,3,4] => 0
[2,3,4,1] => [4] => [[1,2,3,4]] => [1,2,3,4] => 0
[2,4,1,3] => [4] => [[1,2,3,4]] => [1,2,3,4] => 0
[2,4,3,1] => [3,1] => [[1,3,4],[2]] => [2,1,3,4] => 0
[3,1,2,4] => [3,1] => [[1,3,4],[2]] => [2,1,3,4] => 0
[3,1,4,2] => [4] => [[1,2,3,4]] => [1,2,3,4] => 0
[3,2,1,4] => [2,1,1] => [[1,4],[2],[3]] => [3,2,1,4] => 0
[3,2,4,1] => [3,1] => [[1,3,4],[2]] => [2,1,3,4] => 0
[3,4,1,2] => [2,2] => [[1,2],[3,4]] => [3,4,1,2] => 1
[3,4,2,1] => [4] => [[1,2,3,4]] => [1,2,3,4] => 0
[4,1,2,3] => [4] => [[1,2,3,4]] => [1,2,3,4] => 0
[4,1,3,2] => [3,1] => [[1,3,4],[2]] => [2,1,3,4] => 0
[4,2,1,3] => [3,1] => [[1,3,4],[2]] => [2,1,3,4] => 0
[4,2,3,1] => [2,1,1] => [[1,4],[2],[3]] => [3,2,1,4] => 0
[4,3,1,2] => [4] => [[1,2,3,4]] => [1,2,3,4] => 0
[4,3,2,1] => [2,2] => [[1,2],[3,4]] => [3,4,1,2] => 1
[1,2,3,4,5] => [1,1,1,1,1] => [[1],[2],[3],[4],[5]] => [5,4,3,2,1] => 2
[1,2,3,5,4] => [2,1,1,1] => [[1,5],[2],[3],[4]] => [4,3,2,1,5] => 1
[1,2,4,3,5] => [2,1,1,1] => [[1,5],[2],[3],[4]] => [4,3,2,1,5] => 1
[1,2,4,5,3] => [3,1,1] => [[1,4,5],[2],[3]] => [3,2,1,4,5] => 0
[1,2,5,3,4] => [3,1,1] => [[1,4,5],[2],[3]] => [3,2,1,4,5] => 0
[1,2,5,4,3] => [2,1,1,1] => [[1,5],[2],[3],[4]] => [4,3,2,1,5] => 1
[1,3,2,4,5] => [2,1,1,1] => [[1,5],[2],[3],[4]] => [4,3,2,1,5] => 1
[1,3,2,5,4] => [2,2,1] => [[1,3],[2,5],[4]] => [4,2,5,1,3] => 1
[1,3,4,2,5] => [3,1,1] => [[1,4,5],[2],[3]] => [3,2,1,4,5] => 0
[1,3,4,5,2] => [4,1] => [[1,3,4,5],[2]] => [2,1,3,4,5] => 0
[1,3,5,2,4] => [4,1] => [[1,3,4,5],[2]] => [2,1,3,4,5] => 0
[1,3,5,4,2] => [3,1,1] => [[1,4,5],[2],[3]] => [3,2,1,4,5] => 0
[1,4,2,3,5] => [3,1,1] => [[1,4,5],[2],[3]] => [3,2,1,4,5] => 0
[1,4,2,5,3] => [4,1] => [[1,3,4,5],[2]] => [2,1,3,4,5] => 0
[1,4,3,2,5] => [2,1,1,1] => [[1,5],[2],[3],[4]] => [4,3,2,1,5] => 1
[1,4,3,5,2] => [3,1,1] => [[1,4,5],[2],[3]] => [3,2,1,4,5] => 0
[1,4,5,2,3] => [2,2,1] => [[1,3],[2,5],[4]] => [4,2,5,1,3] => 1
[1,4,5,3,2] => [4,1] => [[1,3,4,5],[2]] => [2,1,3,4,5] => 0
[1,5,2,3,4] => [4,1] => [[1,3,4,5],[2]] => [2,1,3,4,5] => 0
[1,5,2,4,3] => [3,1,1] => [[1,4,5],[2],[3]] => [3,2,1,4,5] => 0
[1,5,3,2,4] => [3,1,1] => [[1,4,5],[2],[3]] => [3,2,1,4,5] => 0
[1,5,3,4,2] => [2,1,1,1] => [[1,5],[2],[3],[4]] => [4,3,2,1,5] => 1
[1,5,4,2,3] => [4,1] => [[1,3,4,5],[2]] => [2,1,3,4,5] => 0
[1,5,4,3,2] => [2,2,1] => [[1,3],[2,5],[4]] => [4,2,5,1,3] => 1
[2,1,3,4,5] => [2,1,1,1] => [[1,5],[2],[3],[4]] => [4,3,2,1,5] => 1
[2,1,3,5,4] => [2,2,1] => [[1,3],[2,5],[4]] => [4,2,5,1,3] => 1
[2,1,4,3,5] => [2,2,1] => [[1,3],[2,5],[4]] => [4,2,5,1,3] => 1
[2,1,4,5,3] => [3,2] => [[1,2,5],[3,4]] => [3,4,1,2,5] => 1
[2,1,5,3,4] => [3,2] => [[1,2,5],[3,4]] => [3,4,1,2,5] => 1
[2,1,5,4,3] => [2,2,1] => [[1,3],[2,5],[4]] => [4,2,5,1,3] => 1
[2,3,1,4,5] => [3,1,1] => [[1,4,5],[2],[3]] => [3,2,1,4,5] => 0
[2,3,1,5,4] => [3,2] => [[1,2,5],[3,4]] => [3,4,1,2,5] => 1
[2,3,4,1,5] => [4,1] => [[1,3,4,5],[2]] => [2,1,3,4,5] => 0
[2,3,4,5,1] => [5] => [[1,2,3,4,5]] => [1,2,3,4,5] => 0
[2,3,5,1,4] => [5] => [[1,2,3,4,5]] => [1,2,3,4,5] => 0
[2,3,5,4,1] => [4,1] => [[1,3,4,5],[2]] => [2,1,3,4,5] => 0
[2,4,1,3,5] => [4,1] => [[1,3,4,5],[2]] => [2,1,3,4,5] => 0
[2,4,1,5,3] => [5] => [[1,2,3,4,5]] => [1,2,3,4,5] => 0
[2,4,3,1,5] => [3,1,1] => [[1,4,5],[2],[3]] => [3,2,1,4,5] => 0
[2,4,3,5,1] => [4,1] => [[1,3,4,5],[2]] => [2,1,3,4,5] => 0
[2,4,5,1,3] => [3,2] => [[1,2,5],[3,4]] => [3,4,1,2,5] => 1
[2,4,5,3,1] => [5] => [[1,2,3,4,5]] => [1,2,3,4,5] => 0
[2,5,1,3,4] => [5] => [[1,2,3,4,5]] => [1,2,3,4,5] => 0
[2,5,1,4,3] => [4,1] => [[1,3,4,5],[2]] => [2,1,3,4,5] => 0
[2,5,3,1,4] => [4,1] => [[1,3,4,5],[2]] => [2,1,3,4,5] => 0
[2,5,3,4,1] => [3,1,1] => [[1,4,5],[2],[3]] => [3,2,1,4,5] => 0
[2,5,4,1,3] => [5] => [[1,2,3,4,5]] => [1,2,3,4,5] => 0
[2,5,4,3,1] => [3,2] => [[1,2,5],[3,4]] => [3,4,1,2,5] => 1
[3,1,2,4,5] => [3,1,1] => [[1,4,5],[2],[3]] => [3,2,1,4,5] => 0
[3,1,2,5,4] => [3,2] => [[1,2,5],[3,4]] => [3,4,1,2,5] => 1
[3,1,4,2,5] => [4,1] => [[1,3,4,5],[2]] => [2,1,3,4,5] => 0
[3,1,4,5,2] => [5] => [[1,2,3,4,5]] => [1,2,3,4,5] => 0
[3,1,5,2,4] => [5] => [[1,2,3,4,5]] => [1,2,3,4,5] => 0
[3,1,5,4,2] => [4,1] => [[1,3,4,5],[2]] => [2,1,3,4,5] => 0
[3,2,1,4,5] => [2,1,1,1] => [[1,5],[2],[3],[4]] => [4,3,2,1,5] => 1
[3,2,1,5,4] => [2,2,1] => [[1,3],[2,5],[4]] => [4,2,5,1,3] => 1
[3,2,4,1,5] => [3,1,1] => [[1,4,5],[2],[3]] => [3,2,1,4,5] => 0
[3,2,4,5,1] => [4,1] => [[1,3,4,5],[2]] => [2,1,3,4,5] => 0
[3,2,5,1,4] => [4,1] => [[1,3,4,5],[2]] => [2,1,3,4,5] => 0
[3,2,5,4,1] => [3,1,1] => [[1,4,5],[2],[3]] => [3,2,1,4,5] => 0
[3,4,1,2,5] => [2,2,1] => [[1,3],[2,5],[4]] => [4,2,5,1,3] => 1
[3,4,1,5,2] => [3,2] => [[1,2,5],[3,4]] => [3,4,1,2,5] => 1
[3,4,2,1,5] => [4,1] => [[1,3,4,5],[2]] => [2,1,3,4,5] => 0
[3,4,2,5,1] => [5] => [[1,2,3,4,5]] => [1,2,3,4,5] => 0
[3,4,5,1,2] => [5] => [[1,2,3,4,5]] => [1,2,3,4,5] => 0
[3,4,5,2,1] => [3,2] => [[1,2,5],[3,4]] => [3,4,1,2,5] => 1
[3,5,1,2,4] => [3,2] => [[1,2,5],[3,4]] => [3,4,1,2,5] => 1
[3,5,1,4,2] => [2,2,1] => [[1,3],[2,5],[4]] => [4,2,5,1,3] => 1
[3,5,2,1,4] => [5] => [[1,2,3,4,5]] => [1,2,3,4,5] => 0
>>> Load all 152 entries. <<<
[3,5,2,4,1] => [4,1] => [[1,3,4,5],[2]] => [2,1,3,4,5] => 0
[3,5,4,1,2] => [3,2] => [[1,2,5],[3,4]] => [3,4,1,2,5] => 1
[3,5,4,2,1] => [5] => [[1,2,3,4,5]] => [1,2,3,4,5] => 0
[4,1,2,3,5] => [4,1] => [[1,3,4,5],[2]] => [2,1,3,4,5] => 0
[4,1,2,5,3] => [5] => [[1,2,3,4,5]] => [1,2,3,4,5] => 0
[4,1,3,2,5] => [3,1,1] => [[1,4,5],[2],[3]] => [3,2,1,4,5] => 0
[4,1,3,5,2] => [4,1] => [[1,3,4,5],[2]] => [2,1,3,4,5] => 0
[4,1,5,2,3] => [3,2] => [[1,2,5],[3,4]] => [3,4,1,2,5] => 1
[4,1,5,3,2] => [5] => [[1,2,3,4,5]] => [1,2,3,4,5] => 0
[4,2,1,3,5] => [3,1,1] => [[1,4,5],[2],[3]] => [3,2,1,4,5] => 0
[4,2,1,5,3] => [4,1] => [[1,3,4,5],[2]] => [2,1,3,4,5] => 0
[4,2,3,1,5] => [2,1,1,1] => [[1,5],[2],[3],[4]] => [4,3,2,1,5] => 1
[4,2,3,5,1] => [3,1,1] => [[1,4,5],[2],[3]] => [3,2,1,4,5] => 0
[4,2,5,1,3] => [2,2,1] => [[1,3],[2,5],[4]] => [4,2,5,1,3] => 1
[4,2,5,3,1] => [4,1] => [[1,3,4,5],[2]] => [2,1,3,4,5] => 0
[4,3,1,2,5] => [4,1] => [[1,3,4,5],[2]] => [2,1,3,4,5] => 0
[4,3,1,5,2] => [5] => [[1,2,3,4,5]] => [1,2,3,4,5] => 0
[4,3,2,1,5] => [2,2,1] => [[1,3],[2,5],[4]] => [4,2,5,1,3] => 1
[4,3,2,5,1] => [3,2] => [[1,2,5],[3,4]] => [3,4,1,2,5] => 1
[4,3,5,1,2] => [3,2] => [[1,2,5],[3,4]] => [3,4,1,2,5] => 1
[4,3,5,2,1] => [5] => [[1,2,3,4,5]] => [1,2,3,4,5] => 0
[4,5,1,2,3] => [5] => [[1,2,3,4,5]] => [1,2,3,4,5] => 0
[4,5,1,3,2] => [3,2] => [[1,2,5],[3,4]] => [3,4,1,2,5] => 1
[4,5,2,1,3] => [3,2] => [[1,2,5],[3,4]] => [3,4,1,2,5] => 1
[4,5,2,3,1] => [5] => [[1,2,3,4,5]] => [1,2,3,4,5] => 0
[4,5,3,1,2] => [2,2,1] => [[1,3],[2,5],[4]] => [4,2,5,1,3] => 1
[4,5,3,2,1] => [4,1] => [[1,3,4,5],[2]] => [2,1,3,4,5] => 0
[5,1,2,3,4] => [5] => [[1,2,3,4,5]] => [1,2,3,4,5] => 0
[5,1,2,4,3] => [4,1] => [[1,3,4,5],[2]] => [2,1,3,4,5] => 0
[5,1,3,2,4] => [4,1] => [[1,3,4,5],[2]] => [2,1,3,4,5] => 0
[5,1,3,4,2] => [3,1,1] => [[1,4,5],[2],[3]] => [3,2,1,4,5] => 0
[5,1,4,2,3] => [5] => [[1,2,3,4,5]] => [1,2,3,4,5] => 0
[5,1,4,3,2] => [3,2] => [[1,2,5],[3,4]] => [3,4,1,2,5] => 1
[5,2,1,3,4] => [4,1] => [[1,3,4,5],[2]] => [2,1,3,4,5] => 0
[5,2,1,4,3] => [3,1,1] => [[1,4,5],[2],[3]] => [3,2,1,4,5] => 0
[5,2,3,1,4] => [3,1,1] => [[1,4,5],[2],[3]] => [3,2,1,4,5] => 0
[5,2,3,4,1] => [2,1,1,1] => [[1,5],[2],[3],[4]] => [4,3,2,1,5] => 1
[5,2,4,1,3] => [4,1] => [[1,3,4,5],[2]] => [2,1,3,4,5] => 0
[5,2,4,3,1] => [2,2,1] => [[1,3],[2,5],[4]] => [4,2,5,1,3] => 1
[5,3,1,2,4] => [5] => [[1,2,3,4,5]] => [1,2,3,4,5] => 0
[5,3,1,4,2] => [4,1] => [[1,3,4,5],[2]] => [2,1,3,4,5] => 0
[5,3,2,1,4] => [3,2] => [[1,2,5],[3,4]] => [3,4,1,2,5] => 1
[5,3,2,4,1] => [2,2,1] => [[1,3],[2,5],[4]] => [4,2,5,1,3] => 1
[5,3,4,1,2] => [5] => [[1,2,3,4,5]] => [1,2,3,4,5] => 0
[5,3,4,2,1] => [3,2] => [[1,2,5],[3,4]] => [3,4,1,2,5] => 1
[5,4,1,2,3] => [3,2] => [[1,2,5],[3,4]] => [3,4,1,2,5] => 1
[5,4,1,3,2] => [5] => [[1,2,3,4,5]] => [1,2,3,4,5] => 0
[5,4,2,1,3] => [5] => [[1,2,3,4,5]] => [1,2,3,4,5] => 0
[5,4,2,3,1] => [3,2] => [[1,2,5],[3,4]] => [3,4,1,2,5] => 1
[5,4,3,1,2] => [4,1] => [[1,3,4,5],[2]] => [2,1,3,4,5] => 0
[5,4,3,2,1] => [2,2,1] => [[1,3],[2,5],[4]] => [4,2,5,1,3] => 1
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Description
The number of strict 3-descents.
A strict 3-descent of a permutation $\pi$ of $\{1,2, \dots ,n \}$ is a pair $(i,i+3)$ with $ i+3 \leq n$ and $\pi(i) > \pi(i+3)$.
Map
cycle type
Description
The cycle type of a permutation as a partition.
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
reading tableau
Description
Return the RSK recording tableau of the reading word of the (standard) tableau $T$ labeled down (in English convention) each column to the shape of a partition.