- FindStat

Identifier

Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions

Images

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 2480 entries. <<<

Map

to partition

Description

The cut-out partition of a Dyck path.
The partition $\lambda$ associated to a Dyck path is defined to be the complementary partition inside the staircase partition $(n-1,\ldots,2,1)$ when cutting out $D$ considered as a path from $(0,0)$ to $(n,n)$.
In other words, $\lambda_{i}$ is the number of down-steps before the $(n+1-i)$-th up-step of $D$.
This map is a bijection between Dyck paths of size $n$ and partitions inside the staircase partition $(n-1,\ldots,2,1)$.

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.