Identifier
-
Mp00099:
Dyck paths
—bounce path⟶
Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000619: Permutations ⟶ ℤ
Values
[1,0] => [1,0] => [2,1] => 1
[1,0,1,0] => [1,0,1,0] => [3,1,2] => 1
[1,1,0,0] => [1,1,0,0] => [2,3,1] => 1
[1,0,1,0,1,0] => [1,0,1,0,1,0] => [4,1,2,3] => 1
[1,0,1,1,0,0] => [1,0,1,1,0,0] => [3,1,4,2] => 2
[1,1,0,0,1,0] => [1,1,0,0,1,0] => [2,4,1,3] => 2
[1,1,0,1,0,0] => [1,0,1,1,0,0] => [3,1,4,2] => 2
[1,1,1,0,0,0] => [1,1,1,0,0,0] => [2,3,4,1] => 1
[1,0,1,0,1,0,1,0] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => 1
[1,0,1,0,1,1,0,0] => [1,0,1,0,1,1,0,0] => [4,1,2,5,3] => 2
[1,0,1,1,0,0,1,0] => [1,0,1,1,0,0,1,0] => [3,1,5,2,4] => 3
[1,0,1,1,0,1,0,0] => [1,0,1,0,1,1,0,0] => [4,1,2,5,3] => 2
[1,0,1,1,1,0,0,0] => [1,0,1,1,1,0,0,0] => [3,1,4,5,2] => 2
[1,1,0,0,1,0,1,0] => [1,1,0,0,1,0,1,0] => [2,5,1,3,4] => 2
[1,1,0,0,1,1,0,0] => [1,1,0,0,1,1,0,0] => [2,4,1,5,3] => 3
[1,1,0,1,0,0,1,0] => [1,0,1,1,0,0,1,0] => [3,1,5,2,4] => 3
[1,1,0,1,0,1,0,0] => [1,1,0,0,1,1,0,0] => [2,4,1,5,3] => 3
[1,1,0,1,1,0,0,0] => [1,0,1,1,1,0,0,0] => [3,1,4,5,2] => 2
[1,1,1,0,0,0,1,0] => [1,1,1,0,0,0,1,0] => [2,3,5,1,4] => 2
[1,1,1,0,0,1,0,0] => [1,1,0,0,1,1,0,0] => [2,4,1,5,3] => 3
[1,1,1,0,1,0,0,0] => [1,0,1,1,1,0,0,0] => [3,1,4,5,2] => 2
[1,1,1,1,0,0,0,0] => [1,1,1,1,0,0,0,0] => [2,3,4,5,1] => 1
[1,0,1,0,1,0,1,0,1,0] => [1,0,1,0,1,0,1,0,1,0] => [6,1,2,3,4,5] => 1
[1,0,1,0,1,0,1,1,0,0] => [1,0,1,0,1,0,1,1,0,0] => [5,1,2,3,6,4] => 2
[1,0,1,0,1,1,0,0,1,0] => [1,0,1,0,1,1,0,0,1,0] => [4,1,2,6,3,5] => 3
[1,0,1,0,1,1,0,1,0,0] => [1,0,1,0,1,0,1,1,0,0] => [5,1,2,3,6,4] => 2
[1,0,1,0,1,1,1,0,0,0] => [1,0,1,0,1,1,1,0,0,0] => [4,1,2,5,6,3] => 2
[1,0,1,1,0,0,1,0,1,0] => [1,0,1,1,0,0,1,0,1,0] => [3,1,6,2,4,5] => 3
[1,0,1,1,0,0,1,1,0,0] => [1,0,1,1,0,0,1,1,0,0] => [3,1,5,2,6,4] => 4
[1,0,1,1,0,1,0,0,1,0] => [1,0,1,0,1,1,0,0,1,0] => [4,1,2,6,3,5] => 3
[1,0,1,1,0,1,0,1,0,0] => [1,0,1,1,0,0,1,1,0,0] => [3,1,5,2,6,4] => 4
[1,0,1,1,0,1,1,0,0,0] => [1,0,1,0,1,1,1,0,0,0] => [4,1,2,5,6,3] => 2
[1,0,1,1,1,0,0,0,1,0] => [1,0,1,1,1,0,0,0,1,0] => [3,1,4,6,2,5] => 3
[1,0,1,1,1,0,0,1,0,0] => [1,0,1,1,0,0,1,1,0,0] => [3,1,5,2,6,4] => 4
[1,0,1,1,1,0,1,0,0,0] => [1,0,1,0,1,1,1,0,0,0] => [4,1,2,5,6,3] => 2
[1,0,1,1,1,1,0,0,0,0] => [1,0,1,1,1,1,0,0,0,0] => [3,1,4,5,6,2] => 2
[1,1,0,0,1,0,1,0,1,0] => [1,1,0,0,1,0,1,0,1,0] => [2,6,1,3,4,5] => 2
[1,1,0,0,1,0,1,1,0,0] => [1,1,0,0,1,0,1,1,0,0] => [2,5,1,3,6,4] => 3
[1,1,0,0,1,1,0,0,1,0] => [1,1,0,0,1,1,0,0,1,0] => [2,4,1,6,3,5] => 3
[1,1,0,0,1,1,0,1,0,0] => [1,1,0,0,1,0,1,1,0,0] => [2,5,1,3,6,4] => 3
[1,1,0,0,1,1,1,0,0,0] => [1,1,0,0,1,1,1,0,0,0] => [2,4,1,5,6,3] => 3
[1,1,0,1,0,0,1,0,1,0] => [1,0,1,1,0,0,1,0,1,0] => [3,1,6,2,4,5] => 3
[1,1,0,1,0,0,1,1,0,0] => [1,0,1,1,0,0,1,1,0,0] => [3,1,5,2,6,4] => 4
[1,1,0,1,0,1,0,0,1,0] => [1,1,0,0,1,1,0,0,1,0] => [2,4,1,6,3,5] => 3
[1,1,0,1,0,1,0,1,0,0] => [1,0,1,1,0,0,1,1,0,0] => [3,1,5,2,6,4] => 4
[1,1,0,1,0,1,1,0,0,0] => [1,1,0,0,1,1,1,0,0,0] => [2,4,1,5,6,3] => 3
[1,1,0,1,1,0,0,0,1,0] => [1,0,1,1,1,0,0,0,1,0] => [3,1,4,6,2,5] => 3
[1,1,0,1,1,0,0,1,0,0] => [1,0,1,1,0,0,1,1,0,0] => [3,1,5,2,6,4] => 4
[1,1,0,1,1,0,1,0,0,0] => [1,1,0,0,1,1,1,0,0,0] => [2,4,1,5,6,3] => 3
[1,1,0,1,1,1,0,0,0,0] => [1,0,1,1,1,1,0,0,0,0] => [3,1,4,5,6,2] => 2
[1,1,1,0,0,0,1,0,1,0] => [1,1,1,0,0,0,1,0,1,0] => [2,3,6,1,4,5] => 2
[1,1,1,0,0,0,1,1,0,0] => [1,1,1,0,0,0,1,1,0,0] => [2,3,5,1,6,4] => 3
[1,1,1,0,0,1,0,0,1,0] => [1,1,0,0,1,1,0,0,1,0] => [2,4,1,6,3,5] => 3
[1,1,1,0,0,1,0,1,0,0] => [1,1,1,0,0,0,1,1,0,0] => [2,3,5,1,6,4] => 3
[1,1,1,0,0,1,1,0,0,0] => [1,1,0,0,1,1,1,0,0,0] => [2,4,1,5,6,3] => 3
[1,1,1,0,1,0,0,0,1,0] => [1,0,1,1,1,0,0,0,1,0] => [3,1,4,6,2,5] => 3
[1,1,1,0,1,0,0,1,0,0] => [1,1,1,0,0,0,1,1,0,0] => [2,3,5,1,6,4] => 3
[1,1,1,0,1,0,1,0,0,0] => [1,1,0,0,1,1,1,0,0,0] => [2,4,1,5,6,3] => 3
[1,1,1,0,1,1,0,0,0,0] => [1,0,1,1,1,1,0,0,0,0] => [3,1,4,5,6,2] => 2
[1,1,1,1,0,0,0,0,1,0] => [1,1,1,1,0,0,0,0,1,0] => [2,3,4,6,1,5] => 2
[1,1,1,1,0,0,0,1,0,0] => [1,1,1,0,0,0,1,1,0,0] => [2,3,5,1,6,4] => 3
[1,1,1,1,0,0,1,0,0,0] => [1,1,0,0,1,1,1,0,0,0] => [2,4,1,5,6,3] => 3
[1,1,1,1,0,1,0,0,0,0] => [1,0,1,1,1,1,0,0,0,0] => [3,1,4,5,6,2] => 2
[1,1,1,1,1,0,0,0,0,0] => [1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => 1
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Description
The number of cyclic descents of a permutation.
For a permutation $\pi$ of $\{1,\ldots,n\}$, this is given by the number of indices $1 \leq i \leq n$ such that $\pi(i) > \pi(i+1)$ where we set $\pi(n+1) = \pi(1)$.
For a permutation $\pi$ of $\{1,\ldots,n\}$, this is given by the number of indices $1 \leq i \leq n$ such that $\pi(i) > \pi(i+1)$ where we set $\pi(n+1) = \pi(1)$.
Map
Ringel
Description
The Ringel permutation of the LNakayama algebra corresponding to a Dyck path.
Map
bounce path
Description
Sends a Dyck path $D$ of length $2n$ to its bounce path.
This path is formed by starting at the endpoint $(n,n)$ of $D$ and traveling west until encountering the first vertical step of $D$, then south until hitting the diagonal, then west again until hitting another vertical step of $D$, etc. until the point $(0,0)$ is reached.
This map is the first part of the zeta map Mp00030zeta map.
This path is formed by starting at the endpoint $(n,n)$ of $D$ and traveling west until encountering the first vertical step of $D$, then south until hitting the diagonal, then west again until hitting another vertical step of $D$, etc. until the point $(0,0)$ is reached.
This map is the first part of the zeta map Mp00030zeta map.
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