- FindStat

Identifier

Mp00027: Dyck paths —to partition⟶ Integer partitions
St000159: Integer partitions ⟶ ℤ (values match St000318The number of addable cells of the Ferrers diagram of an integer partition., St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition.)

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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Description

The number of distinct parts of the integer partition.
This statistic is also the number of removeable cells of the partition, and the number of valleys of the Dyck path tracing the shape of the partition.

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 $(n-1,\ldots,2,1)$.