- FindStat

Identifier

Mp00056: Parking functions —to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000228: Integer partitions ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 7958 entries. <<<

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Description

The size of a partition.
This statistic is the constant statistic of the level sets.

Map

to Dyck path

Description

Sends a parking function to its support Dyck word.

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