- FindStat

Identifier

Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths

Images

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

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

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

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

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

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

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

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

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

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

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

[5] => [1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,0,0,0]

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

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

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

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

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

[1,1,1,1,1] => [5] => [1,1,1,1,1,0,0,0,0,0,1,0]

[6] => [1,1,1,1,1,1] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]

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

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

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

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

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

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

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

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

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

[1,1,1,1,1,1] => [6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]

[7] => [1,1,1,1,1,1,1] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]

[6,1] => [2,1,1,1,1,1] => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]

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

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

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

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

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

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

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

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

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

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

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

[2,1,1,1,1,1] => [6,1] => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]

[1,1,1,1,1,1,1] => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]

[8] => [1,1,1,1,1,1,1,1] => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]

[7,1] => [2,1,1,1,1,1,1] => [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]

[6,2] => [2,2,1,1,1,1] => [1,0,1,1,1,1,0,1,1,0,0,0,0,0]

[6,1,1] => [3,1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,1,0,0,0,0]

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

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

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

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

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

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

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

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

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

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

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

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

[3,1,1,1,1,1] => [6,1,1] => [1,1,1,1,0,1,1,0,0,0,0,0,1,0]

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

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

[2,2,1,1,1,1] => [6,2] => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]

[2,1,1,1,1,1,1] => [7,1] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]

[1,1,1,1,1,1,1,1] => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]

[9] => [1,1,1,1,1,1,1,1,1] => [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]

[8,1] => [2,1,1,1,1,1,1,1] => [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]

[7,2] => [2,2,1,1,1,1,1] => [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]

[7,1,1] => [3,1,1,1,1,1,1] => [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]

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

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

[6,1,1,1] => [4,1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,0,1,0,0,0]

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

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

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

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

[5,1,1,1,1] => [5,1,1,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0]

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

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

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

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

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

[4,1,1,1,1,1] => [6,1,1,1] => [1,1,1,0,1,1,1,0,0,0,0,0,1,0]

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

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

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

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

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

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

[3,1,1,1,1,1,1] => [7,1,1] => [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]

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

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

[2,2,1,1,1,1,1] => [7,2] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]

[2,1,1,1,1,1,1,1] => [8,1] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]

[1,1,1,1,1,1,1,1,1] => [9] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]

[10] => [1,1,1,1,1,1,1,1,1,1] => [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]

[9,1] => [2,1,1,1,1,1,1,1,1] => [1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]

[8,2] => [2,2,1,1,1,1,1,1] => [1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]

[8,1,1] => [3,1,1,1,1,1,1,1] => [1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]

[7,3] => [2,2,2,1,1,1,1] => [1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]

>>> Load all 1404 entries. <<<

Map

conjugate

Description

Return the conjugate partition of the partition.
The conjugate partition of the partition $\lambda$ of $n$ is the partition $\lambda^*$ whose Ferrers diagram is obtained from the diagram of $\lambda$ by interchanging rows with columns.
This is also called the associated partition or the transpose in the literature.

Map

to Dyck path

Description

Sends a partition to the shortest Dyck path tracing the shape of its Ferrers diagram.