- FindStat

Identifier

Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00124: Dyck paths —Adin-Bagno-Roichman transformation⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
St001490: Skew partitions ⟶ ℤ

Values

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

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$F_{1} = q$

$F_{2} = 2\ q$

Description

The number of connected components of a skew partition.

Map

Adin-Bagno-Roichman transformation

Description

The Adin-Bagno-Roichman transformation of a Dyck path.
This is a bijection preserving the number of up steps before each peak and sending the number of returns to the number of up steps after the last double up step.

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

$\gamma(D)$ .

Map

to Dyck path

Description

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