- FindStat

Identifier

Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00124: Dyck paths —Adin-Bagno-Roichman transformation⟶ Dyck paths
St001232: Dyck paths ⟶ ℤ

Values

[1] => [1,0] => [1,0] => [1,0] => 0
[2] => [1,0,1,0] => [1,1,0,0] => [1,1,0,0] => 0
[1,1] => [1,1,0,0] => [1,0,1,0] => [1,0,1,0] => 1
[3] => [1,0,1,0,1,0] => [1,1,1,0,0,0] => [1,1,1,0,0,0] => 0
[2,1] => [1,0,1,1,0,0] => [1,1,0,0,1,0] => [1,1,0,0,1,0] => 1
[4] => [1,0,1,0,1,0,1,0] => [1,1,1,1,0,0,0,0] => [1,1,1,1,0,0,0,0] => 0
[3,1] => [1,0,1,0,1,1,0,0] => [1,1,1,0,0,0,1,0] => [1,1,1,0,0,0,1,0] => 1
[2,2] => [1,1,1,0,0,0] => [1,1,0,1,0,0] => [1,0,1,1,0,0] => 2
[5] => [1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,0,0,0,0,0] => [1,1,1,1,1,0,0,0,0,0] => 0
[4,1] => [1,0,1,0,1,0,1,1,0,0] => [1,1,1,1,0,0,0,0,1,0] => [1,1,1,1,0,0,0,0,1,0] => 1
[3,2] => [1,0,1,1,1,0,0,0] => [1,1,1,0,0,1,0,0] => [1,1,0,0,1,1,0,0] => 2
[2,2,1] => [1,1,1,0,0,1,0,0] => [1,1,0,1,0,0,1,0] => [1,0,1,1,0,0,1,0] => 3
[6] => [1,0,1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,1,0,0,0,0,0,0] => [1,1,1,1,1,1,0,0,0,0,0,0] => 0
[5,1] => [1,0,1,0,1,0,1,0,1,1,0,0] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,1,1,1,1,0,0,0,0,0,1,0] => 1
[4,2] => [1,0,1,0,1,1,1,0,0,0] => [1,1,1,1,0,0,0,1,0,0] => [1,1,1,0,0,0,1,1,0,0] => 2
[3,3] => [1,1,1,0,1,0,0,0] => [1,1,1,0,1,0,0,0] => [1,1,0,1,1,0,0,0] => 4
[3,2,1] => [1,0,1,1,1,0,0,1,0,0] => [1,1,1,0,0,1,0,0,1,0] => [1,1,0,0,1,1,0,0,1,0] => 3
[7] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0] => 0
[6,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => 1
[5,2] => [1,0,1,0,1,0,1,1,1,0,0,0] => [1,1,1,1,1,0,0,0,0,1,0,0] => [1,1,1,1,0,0,0,0,1,1,0,0] => 2
[4,3] => [1,0,1,1,1,0,1,0,0,0] => [1,1,1,1,0,0,1,0,0,0] => [1,1,1,0,0,1,1,0,0,0] => 4
[4,2,1] => [1,0,1,0,1,1,1,0,0,1,0,0] => [1,1,1,1,0,0,0,1,0,0,1,0] => [1,1,1,0,0,0,1,1,0,0,1,0] => 3
[3,3,1] => [1,1,1,0,1,0,0,1,0,0] => [1,1,1,0,1,0,0,0,1,0] => [1,1,0,1,1,0,0,0,1,0] => 5
[6,2] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0] => [1,1,1,1,1,1,0,0,0,0,0,1,0,0] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0] => 2
[5,3] => [1,0,1,0,1,1,1,0,1,0,0,0] => [1,1,1,1,1,0,0,0,1,0,0,0] => [1,1,1,1,0,0,0,1,1,0,0,0] => 4
[5,2,1] => [1,0,1,0,1,0,1,1,1,0,0,1,0,0] => [1,1,1,1,1,0,0,0,0,1,0,0,1,0] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0] => 3
[4,4] => [1,1,1,0,1,0,1,0,0,0] => [1,1,1,1,0,1,0,0,0,0] => [1,1,1,0,1,1,0,0,0,0] => 6
[4,3,1] => [1,0,1,1,1,0,1,0,0,1,0,0] => [1,1,1,1,0,0,1,0,0,0,1,0] => [1,1,1,0,0,1,1,0,0,0,1,0] => 5
[6,3] => [1,0,1,0,1,0,1,1,1,0,1,0,0,0] => [1,1,1,1,1,1,0,0,0,0,1,0,0,0] => [1,1,1,1,1,0,0,0,0,1,1,0,0,0] => 4
[5,4] => [1,0,1,1,1,0,1,0,1,0,0,0] => [1,1,1,1,1,0,0,1,0,0,0,0] => [1,1,1,1,0,0,1,1,0,0,0,0] => 6
[5,3,1] => [1,0,1,0,1,1,1,0,1,0,0,1,0,0] => [1,1,1,1,1,0,0,0,1,0,0,0,1,0] => [1,1,1,1,0,0,0,1,1,0,0,0,1,0] => 5
[4,4,1] => [1,1,1,0,1,0,1,0,0,1,0,0] => [1,1,1,1,0,1,0,0,0,0,1,0] => [1,1,1,0,1,1,0,0,0,0,1,0] => 7
[6,4] => [1,0,1,0,1,1,1,0,1,0,1,0,0,0] => [1,1,1,1,1,1,0,0,0,1,0,0,0,0] => [1,1,1,1,1,0,0,0,1,1,0,0,0,0] => 6
[5,5] => [1,1,1,0,1,0,1,0,1,0,0,0] => [1,1,1,1,1,0,1,0,0,0,0,0] => [1,1,1,1,0,1,1,0,0,0,0,0] => 8
[5,4,1] => [1,0,1,1,1,0,1,0,1,0,0,1,0,0] => [1,1,1,1,1,0,0,1,0,0,0,0,1,0] => [1,1,1,1,0,0,1,1,0,0,0,0,1,0] => 7
[6,5] => [1,0,1,1,1,0,1,0,1,0,1,0,0,0] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0] => [1,1,1,1,1,0,0,1,1,0,0,0,0,0] => 8
[5,5,1] => [1,1,1,0,1,0,1,0,1,0,0,1,0,0] => [1,1,1,1,1,0,1,0,0,0,0,0,1,0] => [1,1,1,1,0,1,1,0,0,0,0,0,1,0] => 9
[6,6] => [1,1,1,0,1,0,1,0,1,0,1,0,0,0] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0] => [1,1,1,1,1,0,1,1,0,0,0,0,0,0] => 10

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$F_{1} = 1$

$F_{2} = 1 + q$

Description

The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.

Map

parallelogram polyomino

Description

Return the Dyck path corresponding to the partition interpreted as a parallogram polyomino.
The Ferrers diagram of an integer partition can be interpreted as a parallogram polyomino, such that each part corresponds to a column.
This map returns the corresponding Dyck path.

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

Delest-Viennot-inverse

Description

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

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