Identifier
-
Mp00231:
Integer compositions
—bounce path⟶
Dyck paths
St000980: Dyck paths ⟶ ℤ
Values
[1,1] => [1,0,1,0] => 0
[2] => [1,1,0,0] => 0
[1,1,1] => [1,0,1,0,1,0] => 0
[1,2] => [1,0,1,1,0,0] => 0
[2,1] => [1,1,0,0,1,0] => 0
[3] => [1,1,1,0,0,0] => 0
[1,1,1,1] => [1,0,1,0,1,0,1,0] => 0
[1,1,2] => [1,0,1,0,1,1,0,0] => 0
[1,2,1] => [1,0,1,1,0,0,1,0] => 0
[1,3] => [1,0,1,1,1,0,0,0] => 0
[2,1,1] => [1,1,0,0,1,0,1,0] => 0
[2,2] => [1,1,0,0,1,1,0,0] => 0
[3,1] => [1,1,1,0,0,0,1,0] => 0
[4] => [1,1,1,1,0,0,0,0] => 0
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => 0
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0] => 0
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => 0
[1,1,3] => [1,0,1,0,1,1,1,0,0,0] => 0
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => 0
[1,2,2] => [1,0,1,1,0,0,1,1,0,0] => 0
[1,3,1] => [1,0,1,1,1,0,0,0,1,0] => 0
[1,4] => [1,0,1,1,1,1,0,0,0,0] => 0
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => 0
[2,1,2] => [1,1,0,0,1,0,1,1,0,0] => 0
[2,2,1] => [1,1,0,0,1,1,0,0,1,0] => 0
[2,3] => [1,1,0,0,1,1,1,0,0,0] => 0
[3,1,1] => [1,1,1,0,0,0,1,0,1,0] => 0
[3,2] => [1,1,1,0,0,0,1,1,0,0] => 0
[4,1] => [1,1,1,1,0,0,0,0,1,0] => 0
[5] => [1,1,1,1,1,0,0,0,0,0] => 0
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => 0
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0] => 0
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0] => 0
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0] => 0
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0] => 0
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0] => 0
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0] => 0
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0] => 0
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0] => 0
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0] => 0
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0] => 0
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0] => 0
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0] => 0
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0] => 0
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0] => 0
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0] => 0
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0] => 0
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0] => 0
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0] => 0
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0] => 0
[2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0] => 0
[2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0] => 0
[2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0] => 0
[2,4] => [1,1,0,0,1,1,1,1,0,0,0,0] => 0
[3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0] => 0
[3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0] => 0
[3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0] => 0
[3,3] => [1,1,1,0,0,0,1,1,1,0,0,0] => 0
[4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0] => 0
[4,2] => [1,1,1,1,0,0,0,0,1,1,0,0] => 0
[5,1] => [1,1,1,1,1,0,0,0,0,0,1,0] => 0
[6] => [1,1,1,1,1,1,0,0,0,0,0,0] => 0
[1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0] => 0
[1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0] => 0
[1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0] => 0
[1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0] => 0
[1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0] => 0
[1,1,1,2,2] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0] => 0
[1,1,1,3,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0] => 0
[1,1,1,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0] => 0
[1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0] => 0
[1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0] => 0
[1,1,2,2,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0] => 0
[1,1,2,3] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0] => 0
[1,1,3,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0] => 0
[1,1,3,2] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0] => 0
[1,1,4,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0] => 0
[1,1,5] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0] => 0
[1,2,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0] => 0
[1,2,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0] => 0
[1,2,1,2,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0] => 0
[1,2,1,3] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0] => 0
[1,2,2,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0] => 0
[1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0] => 0
[1,2,3,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0] => 0
[1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0] => 0
[1,3,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0] => 0
[1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0] => 0
[1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0] => 0
[1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0] => 0
[1,4,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0] => 0
[1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0] => 0
[1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0] => 0
[1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0] => 0
[2,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0] => 0
[2,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0] => 0
[2,1,1,2,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0] => 0
[2,1,1,3] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0] => 0
[2,1,2,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0] => 0
[2,1,2,2] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0] => 0
[2,1,3,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0] => 0
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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.
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
bounce path
Description
The bounce path determined by an integer composition.
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