- FindStat

Identifier

Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St000980: Dyck paths ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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Description

The number of boxes weakly below the path and above the diagonal that lie below at least two peaks.
For example, the path $111011010000$ has three peaks in positions $03, 15, 26$. The boxes below $03$ are $01,02,\textbf{12}$, the boxes below $15$ are $\textbf{12},13,14,\textbf{23},\textbf{24},\textbf{34}$, and the boxes below $26$ are $\textbf{23},\textbf{24},25,\textbf{34},35,45$.
We thus obtain the four boxes in positions $12,23,24,34$ that are below at least two peaks.

Map

peaks-to-valleys

Description

Return the path that has a valley wherever the original path has a peak of height at least one.
More precisely, the height of a valley in the image is the height of the corresponding peak minus $2$.
This is also (the inverse of) rowmotion on Dyck paths regarded as order ideals in the triangular poset.