- FindStat

Identifier

Mp00122: Dyck paths —Elizalde-Deutsch bijection⟶ Dyck paths
St001032: Dyck paths ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 1200 entries. <<<

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

Description

The number of horizontal steps in the bicoloured Motzkin path associated with the Dyck path.
In other words, this is the number of valleys and peaks whose first step is in odd position, the initial position equal to 1.
The generating function is given in [1].

Map

Elizalde-Deutsch bijection

Description

The Elizalde-Deutsch bijection on Dyck paths.
.Let $n$ be the length of the Dyck path. Consider the steps $1,n,2,n-1,\dots$ of $D$. When considering the $i$-th step its corresponding matching step has not yet been read, let the $i$-th step of the image of $D$ be an up step, otherwise let it be a down step.