- FindStat

Identifier

Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
St001597: Skew partitions ⟶ ℤ

Values

[1,0] => [1,0] => [1,1,0,0] => [[2],[]] => 1
[1,0,1,0] => [1,1,0,0] => [1,1,1,0,0,0] => [[2,2],[]] => 2
[1,1,0,0] => [1,0,1,0] => [1,1,0,1,0,0] => [[3],[]] => 1
[1,0,1,0,1,0] => [1,1,0,1,0,0] => [1,1,1,0,1,0,0,0] => [[2,2,2],[]] => 2
[1,0,1,1,0,0] => [1,1,0,0,1,0] => [1,1,1,0,0,1,0,0] => [[3,2],[]] => 2
[1,1,0,0,1,0] => [1,0,1,1,0,0] => [1,1,0,1,1,0,0,0] => [[3,3],[1]] => 2
[1,1,0,1,0,0] => [1,1,1,0,0,0] => [1,1,1,1,0,0,0,0] => [[3,3],[]] => 2
[1,1,1,0,0,0] => [1,0,1,0,1,0] => [1,1,0,1,0,1,0,0] => [[4],[]] => 1
[1,0,1,0,1,1,0,0] => [1,1,0,1,0,0,1,0] => [1,1,1,0,1,0,0,1,0,0] => [[3,2,2],[]] => 2
[1,0,1,1,0,0,1,0] => [1,1,0,0,1,1,0,0] => [1,1,1,0,0,1,1,0,0,0] => [[3,3,2],[1]] => 3
[1,0,1,1,1,0,0,0] => [1,1,0,0,1,0,1,0] => [1,1,1,0,0,1,0,1,0,0] => [[4,2],[]] => 2
[1,1,0,0,1,0,1,0] => [1,0,1,1,0,1,0,0] => [1,1,0,1,1,0,1,0,0,0] => [[3,3,3],[1,1]] => 2
[1,1,0,0,1,1,0,0] => [1,0,1,1,0,0,1,0] => [1,1,0,1,1,0,0,1,0,0] => [[4,3],[1]] => 2
[1,1,0,1,1,0,0,0] => [1,1,1,0,0,0,1,0] => [1,1,1,1,0,0,0,1,0,0] => [[4,3],[]] => 2
[1,1,1,0,0,0,1,0] => [1,0,1,0,1,1,0,0] => [1,1,0,1,0,1,1,0,0,0] => [[4,4],[2]] => 2
[1,1,1,0,0,1,0,0] => [1,0,1,1,1,0,0,0] => [1,1,0,1,1,1,0,0,0,0] => [[4,4],[1]] => 2
[1,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,0] => [[5],[]] => 1
[1,0,1,1,1,1,0,0,0,0] => [1,1,0,0,1,0,1,0,1,0] => [1,1,1,0,0,1,0,1,0,1,0,0] => [[5,2],[]] => 2
[1,1,0,0,1,1,1,0,0,0] => [1,0,1,1,0,0,1,0,1,0] => [1,1,0,1,1,0,0,1,0,1,0,0] => [[5,3],[1]] => 2
[1,1,1,0,0,0,1,1,0,0] => [1,0,1,0,1,1,0,0,1,0] => [1,1,0,1,0,1,1,0,0,1,0,0] => [[5,4],[2]] => 2
[1,1,1,1,0,0,0,0,1,0] => [1,0,1,0,1,0,1,1,0,0] => [1,1,0,1,0,1,0,1,1,0,0,0] => [[5,5],[3]] => 2
[1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,1,0,0] => [[6],[]] => 1
[1,1,1,1,1,1,0,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,1,0,1,0,0] => [[7],[]] => 1
[] => [] => [1,0] => [[1],[]] => 1

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$F_{1} = q$

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

$F_{3} = q + 4\ q^{2}$

Description

The Frobenius rank of a skew partition.
This is the minimal number of border strips in a border strip decomposition of the skew partition.

Map

Delest-Viennot

Description

Return the Dyck path corresponding to the parallelogram polyomino obtained by applying Delest-Viennot's bijection.
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.
The Delest-Viennot bijection

$\beta$ returns the parallelogram polyomino, whose column heights are the heights of the peaks of the Dyck path, and the intersection heights between columns are the heights of the valleys of the Dyck path.
This map returns the Dyck path

$(\gamma^{(-1)}\circ\beta)(D)$ .

Map

prime Dyck path

Description

Return the Dyck path obtained by adding an initial up and a final down 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

$\gamma(D)$ .