Identifier
-
Mp00231:
Integer compositions
—bounce path⟶
Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St001024: Dyck paths ⟶ ℤ
Values
[1] => [1,0] => [1,0] => [1,0] => 1
[1,1] => [1,0,1,0] => [1,1,0,0] => [1,0,1,0] => 2
[2] => [1,1,0,0] => [1,0,1,0] => [1,1,0,0] => 1
[1,1,1] => [1,0,1,0,1,0] => [1,1,1,0,0,0] => [1,0,1,0,1,0] => 3
[1,2] => [1,0,1,1,0,0] => [1,1,0,0,1,0] => [1,0,1,1,0,0] => 2
[2,1] => [1,1,0,0,1,0] => [1,0,1,1,0,0] => [1,1,0,0,1,0] => 1
[3] => [1,1,1,0,0,0] => [1,1,0,1,0,0] => [1,1,1,0,0,0] => 1
[1,1,1,1] => [1,0,1,0,1,0,1,0] => [1,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,0] => 4
[1,1,2] => [1,0,1,0,1,1,0,0] => [1,1,1,0,0,0,1,0] => [1,0,1,0,1,1,0,0] => 3
[1,2,1] => [1,0,1,1,0,0,1,0] => [1,1,0,0,1,1,0,0] => [1,0,1,1,0,0,1,0] => 2
[1,3] => [1,0,1,1,1,0,0,0] => [1,1,1,0,0,1,0,0] => [1,0,1,1,1,0,0,0] => 2
[2,1,1] => [1,1,0,0,1,0,1,0] => [1,0,1,1,1,0,0,0] => [1,1,0,0,1,0,1,0] => 2
[2,2] => [1,1,0,0,1,1,0,0] => [1,0,1,1,0,0,1,0] => [1,1,0,0,1,1,0,0] => 1
[3,1] => [1,1,1,0,0,0,1,0] => [1,1,0,1,1,0,0,0] => [1,1,1,0,0,0,1,0] => 1
[4] => [1,1,1,1,0,0,0,0] => [1,1,1,0,1,0,0,0] => [1,1,1,0,1,0,0,0] => 2
[1,1,1,1,1] => [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] => 5
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0] => [1,1,1,1,0,0,0,0,1,0] => [1,0,1,0,1,0,1,1,0,0] => 4
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => [1,1,1,0,0,0,1,1,0,0] => [1,0,1,0,1,1,0,0,1,0] => 3
[1,1,3] => [1,0,1,0,1,1,1,0,0,0] => [1,1,1,1,0,0,0,1,0,0] => [1,0,1,0,1,1,1,0,0,0] => 3
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => [1,1,0,0,1,1,1,0,0,0] => [1,0,1,1,0,0,1,0,1,0] => 2
[1,2,2] => [1,0,1,1,0,0,1,1,0,0] => [1,1,0,0,1,1,0,0,1,0] => [1,0,1,1,0,0,1,1,0,0] => 2
[1,3,1] => [1,0,1,1,1,0,0,0,1,0] => [1,1,1,0,0,1,1,0,0,0] => [1,0,1,1,1,0,0,0,1,0] => 2
[1,4] => [1,0,1,1,1,1,0,0,0,0] => [1,1,1,1,0,0,1,0,0,0] => [1,0,1,1,1,0,1,0,0,0] => 3
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => [1,0,1,1,1,1,0,0,0,0] => [1,1,0,0,1,0,1,0,1,0] => 3
[2,1,2] => [1,1,0,0,1,0,1,1,0,0] => [1,0,1,1,1,0,0,0,1,0] => [1,1,0,0,1,0,1,1,0,0] => 2
[2,2,1] => [1,1,0,0,1,1,0,0,1,0] => [1,0,1,1,0,0,1,1,0,0] => [1,1,0,0,1,1,0,0,1,0] => 1
[2,3] => [1,1,0,0,1,1,1,0,0,0] => [1,0,1,1,1,0,0,1,0,0] => [1,1,0,0,1,1,1,0,0,0] => 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0] => [1,1,0,1,1,1,0,0,0,0] => [1,1,1,0,0,0,1,0,1,0] => 2
[3,2] => [1,1,1,0,0,0,1,1,0,0] => [1,1,0,1,1,0,0,0,1,0] => [1,1,1,0,0,0,1,1,0,0] => 1
[4,1] => [1,1,1,1,0,0,0,0,1,0] => [1,1,1,0,1,1,0,0,0,0] => [1,1,1,0,1,0,0,0,1,0] => 3
[5] => [1,1,1,1,1,0,0,0,0,0] => [1,1,1,1,0,1,0,0,0,0] => [1,1,1,0,1,0,1,0,0,0] => 2
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,1,0,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0,1,0] => 6
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,0,1,0,1,0,1,0,1,1,0,0] => 5
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0] => [1,1,1,1,0,0,0,0,1,1,0,0] => [1,0,1,0,1,0,1,1,0,0,1,0] => 4
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0] => [1,1,1,1,1,0,0,0,0,1,0,0] => [1,0,1,0,1,0,1,1,1,0,0,0] => 4
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0] => [1,1,1,0,0,0,1,1,1,0,0,0] => [1,0,1,0,1,1,0,0,1,0,1,0] => 3
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0] => [1,1,1,0,0,0,1,1,0,0,1,0] => [1,0,1,0,1,1,0,0,1,1,0,0] => 3
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0] => [1,1,1,1,0,0,0,1,1,0,0,0] => [1,0,1,0,1,1,1,0,0,0,1,0] => 3
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0] => [1,1,1,1,1,0,0,0,1,0,0,0] => [1,0,1,0,1,1,1,0,1,0,0,0] => 4
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0] => [1,1,0,0,1,1,1,1,0,0,0,0] => [1,0,1,1,0,0,1,0,1,0,1,0] => 3
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0] => [1,1,0,0,1,1,1,0,0,0,1,0] => [1,0,1,1,0,0,1,0,1,1,0,0] => 2
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0] => [1,1,0,0,1,1,0,0,1,1,0,0] => [1,0,1,1,0,0,1,1,0,0,1,0] => 2
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0] => [1,1,0,0,1,1,1,0,0,1,0,0] => [1,0,1,1,0,0,1,1,1,0,0,0] => 2
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0] => [1,1,1,0,0,1,1,1,0,0,0,0] => [1,0,1,1,1,0,0,0,1,0,1,0] => 2
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0] => [1,1,1,0,0,1,1,0,0,0,1,0] => [1,0,1,1,1,0,0,0,1,1,0,0] => 2
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0] => [1,1,1,1,0,0,1,1,0,0,0,0] => [1,0,1,1,1,0,1,0,0,0,1,0] => 4
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0] => [1,1,1,1,1,0,0,1,0,0,0,0] => [1,0,1,1,1,0,1,0,1,0,0,0] => 3
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0] => [1,0,1,1,1,1,1,0,0,0,0,0] => [1,1,0,0,1,0,1,0,1,0,1,0] => 4
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0] => [1,0,1,1,1,1,0,0,0,0,1,0] => [1,1,0,0,1,0,1,0,1,1,0,0] => 3
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0] => [1,0,1,1,1,0,0,0,1,1,0,0] => [1,1,0,0,1,0,1,1,0,0,1,0] => 2
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0] => [1,0,1,1,1,1,0,0,0,1,0,0] => [1,1,0,0,1,0,1,1,1,0,0,0] => 2
[2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0] => [1,0,1,1,0,0,1,1,1,0,0,0] => [1,1,0,0,1,1,0,0,1,0,1,0] => 2
[2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0] => [1,0,1,1,0,0,1,1,0,0,1,0] => [1,1,0,0,1,1,0,0,1,1,0,0] => 1
[2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0] => [1,0,1,1,1,0,0,1,1,0,0,0] => [1,1,0,0,1,1,1,0,0,0,1,0] => 1
[2,4] => [1,1,0,0,1,1,1,1,0,0,0,0] => [1,0,1,1,1,1,0,0,1,0,0,0] => [1,1,0,0,1,1,1,0,1,0,0,0] => 2
[3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0] => [1,1,0,1,1,1,1,0,0,0,0,0] => [1,1,1,0,0,0,1,0,1,0,1,0] => 3
[3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0] => [1,1,0,1,1,1,0,0,0,0,1,0] => [1,1,1,0,0,0,1,0,1,1,0,0] => 2
[3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0] => [1,1,0,1,1,0,0,0,1,1,0,0] => [1,1,1,0,0,0,1,1,0,0,1,0] => 1
[3,3] => [1,1,1,0,0,0,1,1,1,0,0,0] => [1,1,0,1,1,1,0,0,0,1,0,0] => [1,1,1,0,0,0,1,1,1,0,0,0] => 1
[4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0] => [1,1,1,0,1,1,1,0,0,0,0,0] => [1,1,1,0,1,0,0,0,1,0,1,0] => 4
[4,2] => [1,1,1,1,0,0,0,0,1,1,0,0] => [1,1,1,0,1,1,0,0,0,0,1,0] => [1,1,1,0,1,0,0,0,1,1,0,0] => 3
[5,1] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,1,1,1,0,1,1,0,0,0,0,0] => [1,1,1,0,1,0,1,0,0,0,1,0] => 3
[6] => [1,1,1,1,1,1,0,0,0,0,0,0] => [1,1,1,1,1,0,1,0,0,0,0,0] => [1,1,1,0,1,0,1,0,1,0,0,0] => 2
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Description
Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path.
Map
Delest-Viennot
Description
Return the Dyck path corresponding to the parallelogram polyomino obtained by applying Delest-Viennot's bijection.
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 $(\gamma^{(-1)}\circ\beta)(D)$.
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 $(\gamma^{(-1)}\circ\beta)(D)$.
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.
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.
Map
bounce path
Description
The bounce path determined by an integer composition.
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