- FindStat

Identifier

Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00187: Skew partitions —conjugate⟶ Skew partitions

Images

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

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

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

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

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

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

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

[1,1,1,1] => [[1,1,1,1],[]] => [[4],[]]

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

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

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

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

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

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

[4] => [[4],[]] => [[1,1,1,1],[]]

[1,1,1,1,1] => [[1,1,1,1,1],[]] => [[5],[]]

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[5] => [[5],[]] => [[1,1,1,1,1],[]]

[1,1,1,1,1,1] => [[1,1,1,1,1,1],[]] => [[6],[]]

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[6] => [[6],[]] => [[1,1,1,1,1,1],[]]

[1,1,1,1,1,1,1] => [[1,1,1,1,1,1,1],[]] => [[7],[]]

[1,1,1,1,1,2] => [[2,1,1,1,1,1],[]] => [[6,1],[]]

[1,1,1,1,2,1] => [[2,2,1,1,1,1],[1]] => [[6,2],[1]]

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

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

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

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

[1,1,1,4] => [[4,1,1,1],[]] => [[4,1,1,1],[]]

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[1,6] => [[6,1],[]] => [[2,1,1,1,1,1],[]]

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

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

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

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

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

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

>>> Load all 216 entries. <<<

Map

to ribbon

Description

The ribbon shape corresponding to an integer composition.
For an integer composition

$(a_1, \dots, a_n)$ , this is the ribbon shape whose

$i$ th row from the bottom has

$a_i$ cells.

Map

conjugate

Description

The conjugate of the skew partition.
The conjugate of a skew partition

$\lambda$ is the skew partition

$\lambda^*$ whose Ferrers diagram is obtained from the Ferrers diagram of

$\lambda$ by interchanging rows with columns.
This is also called the associated partition or the transpose in the literature.