- FindStat

Identifier

Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00102: Dyck paths —rise composition⟶ Integer compositions
St001235: Integer compositions ⟶ ℤ

Values

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

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$F_{1} = q^{2}$

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

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

$F_{4} = 4\ q^{2} + 6\ q^{3} + 3\ q^{4} + q^{5}$

$F_{5} = 9\ q^{2} + 18\ q^{3} + 10\ q^{4} + 4\ q^{5} + q^{6}$

Description

The global dimension of the corresponding Comp-Nakayama algebra.
We identify the composition [n1-1,n2-1,...,nr-1] with the Nakayama algebra with Kupisch series [n1,n1-1,...,2,n2,n2-1,...,2,...,nr,nr-1,...,3,2,1]. We call such Nakayama algebras with Kupisch series corresponding to a integer composition "Comp-Nakayama algebra".

Map

prime Dyck path

Description

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

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

Map

rise composition

Description

Send a Dyck path to the composition of sizes of its rises.