- FindStat

Identifier

Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
St000157: Standard tableaux ⟶ ℤ

Values

[1,0] => [1] => [[1]] => 0

[1,0,1,0] => [1,2] => [[1,2]] => 0

[1,1,0,0] => [2,1] => [[1],[2]] => 1

[1,0,1,0,1,0] => [1,2,3] => [[1,2,3]] => 0

[1,0,1,1,0,0] => [1,3,2] => [[1,2],[3]] => 1

[1,1,0,0,1,0] => [2,1,3] => [[1,3],[2]] => 1

[1,1,0,1,0,0] => [2,3,1] => [[1,3],[2]] => 1

[1,1,1,0,0,0] => [3,1,2] => [[1,2],[3]] => 1

[1,0,1,0,1,0,1,0] => [1,2,3,4] => [[1,2,3,4]] => 0

[1,0,1,0,1,1,0,0] => [1,2,4,3] => [[1,2,3],[4]] => 1

[1,0,1,1,0,0,1,0] => [1,3,2,4] => [[1,2,4],[3]] => 1

[1,0,1,1,0,1,0,0] => [1,3,4,2] => [[1,2,4],[3]] => 1

[1,0,1,1,1,0,0,0] => [1,4,2,3] => [[1,2,3],[4]] => 1

[1,1,0,0,1,0,1,0] => [2,1,3,4] => [[1,3,4],[2]] => 1

[1,1,0,0,1,1,0,0] => [2,1,4,3] => [[1,3],[2,4]] => 2

[1,1,0,1,0,0,1,0] => [2,3,1,4] => [[1,3,4],[2]] => 1

[1,1,0,1,0,1,0,0] => [2,3,4,1] => [[1,3,4],[2]] => 1

[1,1,0,1,1,0,0,0] => [2,4,1,3] => [[1,3],[2,4]] => 2

[1,1,1,0,0,0,1,0] => [3,1,2,4] => [[1,2,4],[3]] => 1

[1,1,1,0,0,1,0,0] => [3,1,4,2] => [[1,2],[3,4]] => 1

[1,1,1,0,1,0,0,0] => [3,4,1,2] => [[1,2],[3,4]] => 1

[1,1,1,1,0,0,0,0] => [4,1,2,3] => [[1,2,3],[4]] => 1

[1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5] => [[1,2,3,4,5]] => 0

[1,0,1,0,1,0,1,1,0,0] => [1,2,3,5,4] => [[1,2,3,4],[5]] => 1

[1,0,1,0,1,1,0,0,1,0] => [1,2,4,3,5] => [[1,2,3,5],[4]] => 1

[1,0,1,0,1,1,0,1,0,0] => [1,2,4,5,3] => [[1,2,3,5],[4]] => 1

[1,0,1,0,1,1,1,0,0,0] => [1,2,5,3,4] => [[1,2,3,4],[5]] => 1

[1,0,1,1,0,0,1,0,1,0] => [1,3,2,4,5] => [[1,2,4,5],[3]] => 1

[1,0,1,1,0,0,1,1,0,0] => [1,3,2,5,4] => [[1,2,4],[3,5]] => 2

[1,0,1,1,0,1,0,0,1,0] => [1,3,4,2,5] => [[1,2,4,5],[3]] => 1

[1,0,1,1,0,1,0,1,0,0] => [1,3,4,5,2] => [[1,2,4,5],[3]] => 1

[1,0,1,1,0,1,1,0,0,0] => [1,3,5,2,4] => [[1,2,4],[3,5]] => 2

[1,0,1,1,1,0,0,0,1,0] => [1,4,2,3,5] => [[1,2,3,5],[4]] => 1

[1,0,1,1,1,0,0,1,0,0] => [1,4,2,5,3] => [[1,2,3],[4,5]] => 1

[1,0,1,1,1,0,1,0,0,0] => [1,4,5,2,3] => [[1,2,3],[4,5]] => 1

[1,0,1,1,1,1,0,0,0,0] => [1,5,2,3,4] => [[1,2,3,4],[5]] => 1

[1,1,0,0,1,0,1,0,1,0] => [2,1,3,4,5] => [[1,3,4,5],[2]] => 1

[1,1,0,0,1,0,1,1,0,0] => [2,1,3,5,4] => [[1,3,4],[2,5]] => 2

[1,1,0,0,1,1,0,0,1,0] => [2,1,4,3,5] => [[1,3,5],[2,4]] => 2

[1,1,0,0,1,1,0,1,0,0] => [2,1,4,5,3] => [[1,3,5],[2,4]] => 2

[1,1,0,0,1,1,1,0,0,0] => [2,1,5,3,4] => [[1,3,4],[2,5]] => 2

[1,1,0,1,0,0,1,0,1,0] => [2,3,1,4,5] => [[1,3,4,5],[2]] => 1

[1,1,0,1,0,0,1,1,0,0] => [2,3,1,5,4] => [[1,3,4],[2,5]] => 2

[1,1,0,1,0,1,0,0,1,0] => [2,3,4,1,5] => [[1,3,4,5],[2]] => 1

[1,1,0,1,0,1,0,1,0,0] => [2,3,4,5,1] => [[1,3,4,5],[2]] => 1

[1,1,0,1,0,1,1,0,0,0] => [2,3,5,1,4] => [[1,3,4],[2,5]] => 2

[1,1,0,1,1,0,0,0,1,0] => [2,4,1,3,5] => [[1,3,5],[2,4]] => 2

[1,1,0,1,1,0,0,1,0,0] => [2,4,1,5,3] => [[1,3,5],[2,4]] => 2

[1,1,0,1,1,0,1,0,0,0] => [2,4,5,1,3] => [[1,3,5],[2,4]] => 2

[1,1,0,1,1,1,0,0,0,0] => [2,5,1,3,4] => [[1,3,4],[2,5]] => 2

[1,1,1,0,0,0,1,0,1,0] => [3,1,2,4,5] => [[1,2,4,5],[3]] => 1

[1,1,1,0,0,0,1,1,0,0] => [3,1,2,5,4] => [[1,2,4],[3,5]] => 2

[1,1,1,0,0,1,0,0,1,0] => [3,1,4,2,5] => [[1,2,5],[3,4]] => 1

[1,1,1,0,0,1,0,1,0,0] => [3,1,4,5,2] => [[1,2,5],[3,4]] => 1

[1,1,1,0,0,1,1,0,0,0] => [3,1,5,2,4] => [[1,2,4],[3,5]] => 2

[1,1,1,0,1,0,0,0,1,0] => [3,4,1,2,5] => [[1,2,5],[3,4]] => 1

[1,1,1,0,1,0,0,1,0,0] => [3,4,1,5,2] => [[1,2,5],[3,4]] => 1

[1,1,1,0,1,0,1,0,0,0] => [3,4,5,1,2] => [[1,2,5],[3,4]] => 1

[1,1,1,0,1,1,0,0,0,0] => [3,5,1,2,4] => [[1,2,4],[3,5]] => 2

[1,1,1,1,0,0,0,0,1,0] => [4,1,2,3,5] => [[1,2,3,5],[4]] => 1

[1,1,1,1,0,0,0,1,0,0] => [4,1,2,5,3] => [[1,2,3],[4,5]] => 1

[1,1,1,1,0,0,1,0,0,0] => [4,1,5,2,3] => [[1,2,3],[4,5]] => 1

[1,1,1,1,0,1,0,0,0,0] => [4,5,1,2,3] => [[1,2,3],[4,5]] => 1

[1,1,1,1,1,0,0,0,0,0] => [5,1,2,3,4] => [[1,2,3,4],[5]] => 1

[1,0,1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5,6] => [[1,2,3,4,5,6]] => 0

[1,0,1,0,1,0,1,0,1,1,0,0] => [1,2,3,4,6,5] => [[1,2,3,4,5],[6]] => 1

[1,0,1,0,1,0,1,1,0,0,1,0] => [1,2,3,5,4,6] => [[1,2,3,4,6],[5]] => 1

[1,0,1,0,1,0,1,1,0,1,0,0] => [1,2,3,5,6,4] => [[1,2,3,4,6],[5]] => 1

[1,0,1,0,1,0,1,1,1,0,0,0] => [1,2,3,6,4,5] => [[1,2,3,4,5],[6]] => 1

[1,0,1,0,1,1,0,0,1,0,1,0] => [1,2,4,3,5,6] => [[1,2,3,5,6],[4]] => 1

[1,0,1,0,1,1,0,0,1,1,0,0] => [1,2,4,3,6,5] => [[1,2,3,5],[4,6]] => 2

[1,0,1,0,1,1,0,1,0,0,1,0] => [1,2,4,5,3,6] => [[1,2,3,5,6],[4]] => 1

[1,0,1,0,1,1,0,1,0,1,0,0] => [1,2,4,5,6,3] => [[1,2,3,5,6],[4]] => 1

[1,0,1,0,1,1,0,1,1,0,0,0] => [1,2,4,6,3,5] => [[1,2,3,5],[4,6]] => 2

[1,0,1,0,1,1,1,0,0,0,1,0] => [1,2,5,3,4,6] => [[1,2,3,4,6],[5]] => 1

[1,0,1,0,1,1,1,0,0,1,0,0] => [1,2,5,3,6,4] => [[1,2,3,4],[5,6]] => 1

[1,0,1,0,1,1,1,0,1,0,0,0] => [1,2,5,6,3,4] => [[1,2,3,4],[5,6]] => 1

[1,0,1,0,1,1,1,1,0,0,0,0] => [1,2,6,3,4,5] => [[1,2,3,4,5],[6]] => 1

[1,0,1,1,0,0,1,0,1,0,1,0] => [1,3,2,4,5,6] => [[1,2,4,5,6],[3]] => 1

[1,0,1,1,0,0,1,0,1,1,0,0] => [1,3,2,4,6,5] => [[1,2,4,5],[3,6]] => 2

[1,0,1,1,0,0,1,1,0,0,1,0] => [1,3,2,5,4,6] => [[1,2,4,6],[3,5]] => 2

[1,0,1,1,0,0,1,1,0,1,0,0] => [1,3,2,5,6,4] => [[1,2,4,6],[3,5]] => 2

[1,0,1,1,0,0,1,1,1,0,0,0] => [1,3,2,6,4,5] => [[1,2,4,5],[3,6]] => 2

[1,0,1,1,0,1,0,0,1,0,1,0] => [1,3,4,2,5,6] => [[1,2,4,5,6],[3]] => 1

[1,0,1,1,0,1,0,0,1,1,0,0] => [1,3,4,2,6,5] => [[1,2,4,5],[3,6]] => 2

[1,0,1,1,0,1,0,1,0,0,1,0] => [1,3,4,5,2,6] => [[1,2,4,5,6],[3]] => 1

[1,0,1,1,0,1,0,1,0,1,0,0] => [1,3,4,5,6,2] => [[1,2,4,5,6],[3]] => 1

[1,0,1,1,0,1,0,1,1,0,0,0] => [1,3,4,6,2,5] => [[1,2,4,5],[3,6]] => 2

[1,0,1,1,0,1,1,0,0,0,1,0] => [1,3,5,2,4,6] => [[1,2,4,6],[3,5]] => 2

[1,0,1,1,0,1,1,0,0,1,0,0] => [1,3,5,2,6,4] => [[1,2,4,6],[3,5]] => 2

[1,0,1,1,0,1,1,0,1,0,0,0] => [1,3,5,6,2,4] => [[1,2,4,6],[3,5]] => 2

[1,0,1,1,0,1,1,1,0,0,0,0] => [1,3,6,2,4,5] => [[1,2,4,5],[3,6]] => 2

[1,0,1,1,1,0,0,0,1,0,1,0] => [1,4,2,3,5,6] => [[1,2,3,5,6],[4]] => 1

[1,0,1,1,1,0,0,0,1,1,0,0] => [1,4,2,3,6,5] => [[1,2,3,5],[4,6]] => 2

[1,0,1,1,1,0,0,1,0,0,1,0] => [1,4,2,5,3,6] => [[1,2,3,6],[4,5]] => 1

[1,0,1,1,1,0,0,1,0,1,0,0] => [1,4,2,5,6,3] => [[1,2,3,6],[4,5]] => 1

[1,0,1,1,1,0,0,1,1,0,0,0] => [1,4,2,6,3,5] => [[1,2,3,5],[4,6]] => 2

[1,0,1,1,1,0,1,0,0,0,1,0] => [1,4,5,2,3,6] => [[1,2,3,6],[4,5]] => 1

[1,0,1,1,1,0,1,0,0,1,0,0] => [1,4,5,2,6,3] => [[1,2,3,6],[4,5]] => 1

[1,0,1,1,1,0,1,0,1,0,0,0] => [1,4,5,6,2,3] => [[1,2,3,6],[4,5]] => 1

[1,0,1,1,1,0,1,1,0,0,0,0] => [1,4,6,2,3,5] => [[1,2,3,5],[4,6]] => 2

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Description

The number of descents of a standard tableau.
Entry $i$ of a standard Young tableau is a descent if $i+1$ appears in a row below the row of $i$.

Map

to 321-avoiding permutation (Krattenthaler)

Description

Krattenthaler's bijection to 321-avoiding permutations.
Draw the path of semilength $n$ in an $n\times n$ square matrix, starting at the upper left corner, with right and down steps, and staying below the diagonal. Then the permutation matrix is obtained by placing ones into the cells corresponding to the peaks of the path and placing ones into the remaining columns from left to right, such that the row indices of the cells increase.

Map

Robinson-Schensted insertion tableau

Description

Sends a permutation to its Robinson-Schensted insertion tableau.
The Robinson-Schensted corrspondence is a bijection between permutations of length $n$ and pairs of standard Young tableaux of the same shape and of size $n$, see [1]. These two tableaux are the insertion tableau and the recording tableau.
This map sends a permutation to its corresponding insertion tableau.