- FindStat

Identifier

Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St000676: Dyck paths ⟶ ℤ

Values

[1,0] => [1,0] => 1

[1,0,1,0] => [1,1,0,0] => 1

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

[1,0,1,0,1,0] => [1,1,0,1,0,0] => 1

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

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

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

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

[1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,0] => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

[1,0,1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,0] => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0] => 5

[1,0,1,0,1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,1,0,0] => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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$F_{1} = q$

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

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

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

$F_{5} = q + 10\ q^{2} + 20\ q^{3} + 10\ q^{4} + q^{5}$

$F_{6} = q + 15\ q^{2} + 50\ q^{3} + 50\ q^{4} + 15\ q^{5} + q^{6}$

$F_{7} = q + 21\ q^{2} + 105\ q^{3} + 175\ q^{4} + 105\ q^{5} + 21\ q^{6} + q^{7}$

Description

The number of odd rises of a Dyck path.
This is the number of ones at an odd position, with the initial position equal to 1.
The number of Dyck paths of semilength

$n$ with

$k$ up steps in odd positions and

$k$ returns to the main diagonal are counted by the binomial coefficient

$\binom{n-1}{k-1}$ [3,4].

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

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