- FindStat

Identifier

Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00295: Standard tableaux —valley 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] => [[1,2],[3]] => [3] => [3]

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 138 entries. <<<

Map

initial tableau

Description

Sends an integer partition to the standard tableau obtained by filling the numbers

$1$ through

$n$ row by row.

Map

valley composition

Description

The composition corresponding to the valley 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 valley if

$i-1$ is a descent and

$i$ is an ascent.
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 valley set of

$T$ .

Map

inverse Foata bijection

Description

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