- FindStat

Identifier

Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00294: Standard tableaux —peak composition⟶ Integer compositions
Mp00315: Integer compositions —inverse Foata bijection⟶ Integer compositions

Images

[1] => [[1]] => [1] => [1]

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[7] => [[1,2,3,4,5,6,7]] => [7] => [7]

[6,1] => [[1,3,4,5,6,7],[2]] => [7] => [7]

[5,2] => [[1,2,5,6,7],[3,4]] => [2,5] => [2,5]

[5,1,1] => [[1,4,5,6,7],[2],[3]] => [7] => [7]

[4,3] => [[1,2,3,7],[4,5,6]] => [3,4] => [3,4]

[4,2,1] => [[1,3,6,7],[2,5],[4]] => [3,4] => [3,4]

[4,1,1,1] => [[1,5,6,7],[2],[3],[4]] => [7] => [7]

[3,3,1] => [[1,3,4],[2,6,7],[5]] => [4,3] => [4,3]

[3,2,2] => [[1,2,7],[3,4],[5,6]] => [2,2,3] => [2,2,3]

[3,2,1,1] => [[1,4,7],[2,6],[3],[5]] => [4,3] => [4,3]

[3,1,1,1,1] => [[1,6,7],[2],[3],[4],[5]] => [7] => [7]

[2,2,2,1] => [[1,3],[2,5],[4,7],[6]] => [3,2,2] => [2,3,2]

[2,2,1,1,1] => [[1,5],[2,7],[3],[4],[6]] => [5,2] => [5,2]

[2,1,1,1,1,1] => [[1,7],[2],[3],[4],[5],[6]] => [7] => [7]

[1,1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6],[7]] => [7] => [7]

[8] => [[1,2,3,4,5,6,7,8]] => [8] => [8]

[7,1] => [[1,3,4,5,6,7,8],[2]] => [8] => [8]

[6,2] => [[1,2,5,6,7,8],[3,4]] => [2,6] => [2,6]

[6,1,1] => [[1,4,5,6,7,8],[2],[3]] => [8] => [8]

[5,3] => [[1,2,3,7,8],[4,5,6]] => [3,5] => [3,5]

[5,2,1] => [[1,3,6,7,8],[2,5],[4]] => [3,5] => [3,5]

[5,1,1,1] => [[1,5,6,7,8],[2],[3],[4]] => [8] => [8]

[4,4] => [[1,2,3,4],[5,6,7,8]] => [4,4] => [4,4]

[4,3,1] => [[1,3,4,8],[2,6,7],[5]] => [4,4] => [4,4]

[4,2,2] => [[1,2,7,8],[3,4],[5,6]] => [2,2,4] => [2,2,4]

[4,2,1,1] => [[1,4,7,8],[2,6],[3],[5]] => [4,4] => [4,4]

[4,1,1,1,1] => [[1,6,7,8],[2],[3],[4],[5]] => [8] => [8]

[3,3,2] => [[1,2,5],[3,4,8],[6,7]] => [2,3,3] => [2,3,3]

[3,3,1,1] => [[1,4,5],[2,7,8],[3],[6]] => [5,3] => [5,3]

[3,2,2,1] => [[1,3,8],[2,5],[4,7],[6]] => [3,2,3] => [3,2,3]

[3,2,1,1,1] => [[1,5,8],[2,7],[3],[4],[6]] => [5,3] => [5,3]

[3,1,1,1,1,1] => [[1,7,8],[2],[3],[4],[5],[6]] => [8] => [8]

[2,2,2,2] => [[1,2],[3,4],[5,6],[7,8]] => [2,2,2,2] => [2,2,2,2]

[2,2,2,1,1] => [[1,4],[2,6],[3,8],[5],[7]] => [4,2,2] => [2,4,2]

[2,2,1,1,1,1] => [[1,6],[2,8],[3],[4],[5],[7]] => [6,2] => [6,2]

[2,1,1,1,1,1,1] => [[1,8],[2],[3],[4],[5],[6],[7]] => [8] => [8]

[1,1,1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6],[7],[8]] => [8] => [8]

[9] => [[1,2,3,4,5,6,7,8,9]] => [9] => [9]

[8,1] => [[1,3,4,5,6,7,8,9],[2]] => [9] => [9]

[7,2] => [[1,2,5,6,7,8,9],[3,4]] => [2,7] => [2,7]

[7,1,1] => [[1,4,5,6,7,8,9],[2],[3]] => [9] => [9]

[6,3] => [[1,2,3,7,8,9],[4,5,6]] => [3,6] => [3,6]

[6,2,1] => [[1,3,6,7,8,9],[2,5],[4]] => [3,6] => [3,6]

[6,1,1,1] => [[1,5,6,7,8,9],[2],[3],[4]] => [9] => [9]

[5,4] => [[1,2,3,4,9],[5,6,7,8]] => [4,5] => [4,5]

[5,3,1] => [[1,3,4,8,9],[2,6,7],[5]] => [4,5] => [4,5]

[5,2,2] => [[1,2,7,8,9],[3,4],[5,6]] => [2,2,5] => [2,2,5]

[5,2,1,1] => [[1,4,7,8,9],[2,6],[3],[5]] => [4,5] => [4,5]

[5,1,1,1,1] => [[1,6,7,8,9],[2],[3],[4],[5]] => [9] => [9]

[4,4,1] => [[1,3,4,5],[2,7,8,9],[6]] => [5,4] => [5,4]

[4,3,2] => [[1,2,5,9],[3,4,8],[6,7]] => [2,3,4] => [2,3,4]

[4,3,1,1] => [[1,4,5,9],[2,7,8],[3],[6]] => [5,4] => [5,4]

[4,2,2,1] => [[1,3,8,9],[2,5],[4,7],[6]] => [3,2,4] => [3,2,4]

[4,2,1,1,1] => [[1,5,8,9],[2,7],[3],[4],[6]] => [5,4] => [5,4]

[4,1,1,1,1,1] => [[1,7,8,9],[2],[3],[4],[5],[6]] => [9] => [9]

[3,3,3] => [[1,2,3],[4,5,6],[7,8,9]] => [3,3,3] => [3,3,3]

[3,3,2,1] => [[1,3,6],[2,5,9],[4,8],[7]] => [3,3,3] => [3,3,3]

[3,3,1,1,1] => [[1,5,6],[2,8,9],[3],[4],[7]] => [6,3] => [6,3]

[3,2,2,2] => [[1,2,9],[3,4],[5,6],[7,8]] => [2,2,2,3] => [2,2,2,3]

[3,2,2,1,1] => [[1,4,9],[2,6],[3,8],[5],[7]] => [4,2,3] => [2,4,3]

[3,2,1,1,1,1] => [[1,6,9],[2,8],[3],[4],[5],[7]] => [6,3] => [6,3]

[3,1,1,1,1,1,1] => [[1,8,9],[2],[3],[4],[5],[6],[7]] => [9] => [9]

[2,2,2,2,1] => [[1,3],[2,5],[4,7],[6,9],[8]] => [3,2,2,2] => [2,2,3,2]

[2,2,2,1,1,1] => [[1,5],[2,7],[3,9],[4],[6],[8]] => [5,2,2] => [2,5,2]

[2,2,1,1,1,1,1] => [[1,7],[2,9],[3],[4],[5],[6],[8]] => [7,2] => [7,2]

[2,1,1,1,1,1,1,1] => [[1,9],[2],[3],[4],[5],[6],[7],[8]] => [9] => [9]

[1,1,1,1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6],[7],[8],[9]] => [9] => [9]

[10] => [[1,2,3,4,5,6,7,8,9,10]] => [10] => [10]

[9,1] => [[1,3,4,5,6,7,8,9,10],[2]] => [10] => [10]

[8,2] => [[1,2,5,6,7,8,9,10],[3,4]] => [2,8] => [2,8]

[8,1,1] => [[1,4,5,6,7,8,9,10],[2],[3]] => [10] => [10]

[7,3] => [[1,2,3,7,8,9,10],[4,5,6]] => [3,7] => [3,7]

>>> Load all 138 entries. <<<

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.

Map

peak composition

Description

The composition corresponding to the peak set of a standard tableau.
Let

$T$ be a standard tableau of size

$n$ .
An entry

$i$ of

$T$ is a descent if

$i+1$ is in a lower row (in English notation), otherwise

$i$ is an ascent.
An entry

$2 \leq i \leq n-1$ is a peak, if

$i-1$ is an ascent and

$i$ is a descent.
This map returns the composition

$c_1,\dots,c_k$ of

$n$ such that

$\{c_1, c_1+c_2,\dots, c_1+\dots+c_k\}$ is the peak set of

$T$ .

Map

inverse Foata bijection

Description

The inverse of Foata's bijection.
See Mp00314Foata bijection.