- FindStat

Identifier

Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
St001487: Skew partitions ⟶ ℤ

Values

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

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$F_{1} = q$

$F_{2} = q + q^{2}$

$F_{3} = 2\ q + q^{2}$

Description

The number of inner corners of a skew partition.

Map

to Dyck path

Description

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

Map

skew partition

Description

The parallelogram polyomino corresponding to a Dyck path, interpreted as a skew partition.
Let

$D$ be a Dyck path of semilength

$n$ . The parallelogram polyomino

$\gamma(D)$ is defined as follows: let

$\tilde D = d_0 d_1 \dots d_{2n+1}$ be the Dyck path obtained by prepending an up step and appending a down step to

$D$ . Then, the upper path of

$\gamma(D)$ corresponds to the sequence of steps of

$\tilde D$ with even indices, and the lower path of

$\gamma(D)$ corresponds to the sequence of steps of

$\tilde D$ with odd indices.
This map returns the skew partition definded by the diagram of

$\gamma(D)$ .