- FindStat

Identifier

Mp00122: Dyck paths —Elizalde-Deutsch bijection⟶ Dyck paths
St001139: Dyck paths ⟶ ℤ

Values

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

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

[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] => 0

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

[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] => 0

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

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

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

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

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

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

[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] => 0

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

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

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

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

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

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

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

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

[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] => 0

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

[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] => 0

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

[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] => 0

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

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

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

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

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

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

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

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

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

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

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

[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] => 2

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[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] => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[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] => 1

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

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

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

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

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

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

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

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

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

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

[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] => 1

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

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

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

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Description

The number of occurrences of hills of size 2 in a Dyck path.
A hill of size two is a subpath beginning at height zero, consisting of two up steps followed by two down steps.

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.