- FindStat

Identifier

Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00118: Dyck paths —swap returns and last descent⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000353: Permutations ⟶ ℤ (values match St000092The number of outer peaks of a permutation.)

Values

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

search for individual values

searching the database for the individual values of this statistic

/ search for generating function

searching the database for statistics with the same generating function

click to show known generating functions

$F_{1} = 1$

$F_{2} = 1 + q$

$F_{3} = 1 + 4\ q$

$F_{4} = 1 + 10\ q + 3\ q^{2}$

$F_{5} = 1 + 21\ q + 20\ q^{2}$

Description

The number of inner valleys of a permutation.
The number of valleys including the boundary is St000099The number of valleys of a permutation, including the boundary..

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

Ringel

Description

The Ringel permutation of the LNakayama algebra corresponding to a Dyck path.

Map

swap returns and last descent

Description

Return a Dyck path with number of returns and length of the last descent interchanged.
This is the specialisation of the map

$\Phi$ in [1] to Dyck paths. It is characterised by the fact that the number of up steps before a down step that is neither a return nor part of the last descent is preserved.