Identifier
Values
[1,0] => [1,0] => [2,1] => [1,2] => 0
[1,0,1,0] => [1,1,0,0] => [2,3,1] => [3,1,2] => 0
[1,1,0,0] => [1,0,1,0] => [3,1,2] => [1,3,2] => 0
[1,0,1,0,1,0] => [1,1,1,0,0,0] => [2,3,4,1] => [3,4,1,2] => 0
[1,0,1,1,0,0] => [1,1,0,0,1,0] => [2,4,1,3] => [3,1,4,2] => 0
[1,1,0,0,1,0] => [1,0,1,1,0,0] => [3,1,4,2] => [4,2,1,3] => 1
[1,1,0,1,0,0] => [1,0,1,0,1,0] => [4,1,2,3] => [1,3,4,2] => 0
[1,1,1,0,0,0] => [1,1,0,1,0,0] => [4,3,1,2] => [1,2,4,3] => 0
[1,0,1,0,1,0,1,0] => [1,1,1,1,0,0,0,0] => [2,3,4,5,1] => [3,4,5,1,2] => 0
[1,0,1,0,1,1,0,0] => [1,1,1,0,0,0,1,0] => [2,3,5,1,4] => [3,4,1,5,2] => 0
[1,0,1,1,0,0,1,0] => [1,1,0,0,1,1,0,0] => [2,4,1,5,3] => [3,5,2,1,4] => 1
[1,0,1,1,0,1,0,0] => [1,1,0,0,1,0,1,0] => [2,5,1,3,4] => [3,1,4,5,2] => 0
[1,0,1,1,1,0,0,0] => [1,1,1,0,0,1,0,0] => [2,5,4,1,3] => [3,1,2,5,4] => 0
[1,1,0,0,1,0,1,0] => [1,0,1,1,1,0,0,0] => [3,1,4,5,2] => [4,2,5,1,3] => 1
[1,1,0,0,1,1,0,0] => [1,0,1,1,0,0,1,0] => [3,1,5,2,4] => [4,2,1,5,3] => 1
[1,1,0,1,0,0,1,0] => [1,0,1,0,1,1,0,0] => [4,1,2,5,3] => [5,2,3,1,4] => 2
[1,1,0,1,0,1,0,0] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => [1,3,4,5,2] => 0
[1,1,0,1,1,0,0,0] => [1,0,1,1,0,1,0,0] => [5,1,4,2,3] => [1,3,2,5,4] => 0
[1,1,1,0,0,0,1,0] => [1,1,0,1,1,0,0,0] => [4,3,1,5,2] => [5,4,2,1,3] => 2
[1,1,1,0,0,1,0,0] => [1,1,0,1,0,0,1,0] => [5,3,1,2,4] => [1,5,3,4,2] => 2
[1,1,1,0,1,0,0,0] => [1,1,1,0,1,0,0,0] => [5,3,4,1,2] => [1,5,2,4,3] => 1
[1,1,1,1,0,0,0,0] => [1,1,0,1,0,1,0,0] => [5,4,1,2,3] => [1,2,4,5,3] => 0
[1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => [3,4,5,6,1,2] => 0
[1,0,1,0,1,0,1,1,0,0] => [1,1,1,1,0,0,0,0,1,0] => [2,3,4,6,1,5] => [3,4,5,1,6,2] => 0
[1,0,1,0,1,1,0,0,1,0] => [1,1,1,0,0,0,1,1,0,0] => [2,3,5,1,6,4] => [3,4,6,2,1,5] => 1
[1,0,1,0,1,1,0,1,0,0] => [1,1,1,0,0,0,1,0,1,0] => [2,3,6,1,4,5] => [3,4,1,5,6,2] => 0
[1,0,1,0,1,1,1,0,0,0] => [1,1,1,1,0,0,0,1,0,0] => [2,3,6,5,1,4] => [3,4,1,2,6,5] => 0
[1,0,1,1,0,0,1,0,1,0] => [1,1,0,0,1,1,1,0,0,0] => [2,4,1,5,6,3] => [3,5,2,6,1,4] => 1
[1,0,1,1,0,0,1,1,0,0] => [1,1,0,0,1,1,0,0,1,0] => [2,4,1,6,3,5] => [3,5,2,1,6,4] => 1
[1,0,1,1,0,1,0,0,1,0] => [1,1,0,0,1,0,1,1,0,0] => [2,5,1,3,6,4] => [3,6,2,4,1,5] => 2
[1,0,1,1,0,1,0,1,0,0] => [1,1,0,0,1,0,1,0,1,0] => [2,6,1,3,4,5] => [3,1,4,5,6,2] => 0
[1,0,1,1,0,1,1,0,0,0] => [1,1,0,0,1,1,0,1,0,0] => [2,6,1,5,3,4] => [3,1,4,2,6,5] => 0
[1,0,1,1,1,0,0,0,1,0] => [1,1,1,0,0,1,1,0,0,0] => [2,5,4,1,6,3] => [3,6,5,2,1,4] => 2
[1,0,1,1,1,0,0,1,0,0] => [1,1,1,0,0,1,0,0,1,0] => [2,6,4,1,3,5] => [3,1,6,4,5,2] => 2
[1,0,1,1,1,0,1,0,0,0] => [1,1,1,1,0,0,1,0,0,0] => [2,6,4,5,1,3] => [3,1,6,2,5,4] => 1
[1,0,1,1,1,1,0,0,0,0] => [1,1,1,0,0,1,0,1,0,0] => [2,6,5,1,3,4] => [3,1,2,5,6,4] => 0
[1,1,0,0,1,0,1,0,1,0] => [1,0,1,1,1,1,0,0,0,0] => [3,1,4,5,6,2] => [4,2,5,6,1,3] => 1
[1,1,0,0,1,0,1,1,0,0] => [1,0,1,1,1,0,0,0,1,0] => [3,1,4,6,2,5] => [4,2,5,1,6,3] => 1
[1,1,0,0,1,1,0,0,1,0] => [1,0,1,1,0,0,1,1,0,0] => [3,1,5,2,6,4] => [4,2,6,3,1,5] => 2
[1,1,0,0,1,1,0,1,0,0] => [1,0,1,1,0,0,1,0,1,0] => [3,1,6,2,4,5] => [4,2,1,5,6,3] => 1
[1,1,0,0,1,1,1,0,0,0] => [1,0,1,1,1,0,0,1,0,0] => [3,1,6,5,2,4] => [4,2,1,3,6,5] => 1
[1,1,0,1,0,0,1,0,1,0] => [1,0,1,0,1,1,1,0,0,0] => [4,1,2,5,6,3] => [5,2,3,6,1,4] => 2
[1,1,0,1,0,0,1,1,0,0] => [1,0,1,0,1,1,0,0,1,0] => [4,1,2,6,3,5] => [5,2,3,1,6,4] => 2
[1,1,0,1,0,1,0,0,1,0] => [1,0,1,0,1,0,1,1,0,0] => [5,1,2,3,6,4] => [6,2,3,4,1,5] => 3
[1,1,0,1,0,1,0,1,0,0] => [1,0,1,0,1,0,1,0,1,0] => [6,1,2,3,4,5] => [1,3,4,5,6,2] => 0
[1,1,0,1,0,1,1,0,0,0] => [1,0,1,0,1,1,0,1,0,0] => [6,1,2,5,3,4] => [1,3,4,2,6,5] => 0
[1,1,0,1,1,0,0,0,1,0] => [1,0,1,1,0,1,1,0,0,0] => [5,1,4,2,6,3] => [6,2,5,3,1,4] => 3
[1,1,0,1,1,0,0,1,0,0] => [1,0,1,1,0,1,0,0,1,0] => [6,1,4,2,3,5] => [1,3,6,4,5,2] => 2
[1,1,0,1,1,0,1,0,0,0] => [1,0,1,1,1,0,1,0,0,0] => [6,1,4,5,2,3] => [1,3,6,2,5,4] => 1
[1,1,0,1,1,1,0,0,0,0] => [1,0,1,1,0,1,0,1,0,0] => [6,1,5,2,3,4] => [1,3,2,5,6,4] => 0
[1,1,1,0,0,0,1,0,1,0] => [1,1,0,1,1,1,0,0,0,0] => [4,3,1,5,6,2] => [5,4,2,6,1,3] => 2
[1,1,1,0,0,0,1,1,0,0] => [1,1,0,1,1,0,0,0,1,0] => [4,3,1,6,2,5] => [5,4,2,1,6,3] => 2
[1,1,1,0,0,1,0,0,1,0] => [1,1,0,1,0,0,1,1,0,0] => [5,3,1,2,6,4] => [6,4,2,3,1,5] => 3
[1,1,1,0,0,1,0,1,0,0] => [1,1,0,1,0,0,1,0,1,0] => [6,3,1,2,4,5] => [1,5,3,4,6,2] => 2
[1,1,1,0,0,1,1,0,0,0] => [1,1,0,1,1,0,0,1,0,0] => [6,3,1,5,2,4] => [1,5,3,2,6,4] => 1
[1,1,1,0,1,0,0,0,1,0] => [1,1,1,0,1,1,0,0,0,0] => [5,3,4,1,6,2] => [6,4,5,2,1,3] => 3
[1,1,1,0,1,0,0,1,0,0] => [1,1,1,0,1,0,0,0,1,0] => [6,3,4,1,2,5] => [1,5,6,3,4,2] => 2
[1,1,1,0,1,0,1,0,0,0] => [1,1,1,1,0,1,0,0,0,0] => [6,3,4,5,1,2] => [1,5,6,2,4,3] => 1
[1,1,1,0,1,1,0,0,0,0] => [1,1,1,0,1,0,0,1,0,0] => [6,3,5,1,2,4] => [1,5,2,4,6,3] => 1
[1,1,1,1,0,0,0,0,1,0] => [1,1,0,1,0,1,1,0,0,0] => [5,4,1,2,6,3] => [6,5,2,3,1,4] => 3
[1,1,1,1,0,0,0,1,0,0] => [1,1,0,1,0,1,0,0,1,0] => [6,4,1,2,3,5] => [1,6,3,4,5,2] => 3
[1,1,1,1,0,0,1,0,0,0] => [1,1,0,1,1,0,1,0,0,0] => [6,4,1,5,2,3] => [1,6,3,2,5,4] => 2
[1,1,1,1,0,1,0,0,0,0] => [1,1,0,1,0,1,0,1,0,0] => [5,6,1,2,3,4] => [6,1,3,4,5,2] => 3
[1,1,1,1,1,0,0,0,0,0] => [1,1,1,0,1,0,1,0,0,0] => [6,5,4,1,2,3] => [1,2,3,5,6,4] => 0
[1,0,1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,1,0,0,0,0,0,0] => [2,3,4,5,6,7,1] => [3,4,5,6,7,1,2] => 0
[1,1,0,1,0,1,0,1,0,0,1,0] => [1,0,1,0,1,0,1,0,1,1,0,0] => [6,1,2,3,4,7,5] => [7,2,3,4,5,1,6] => 4
[1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0] => [2,3,4,5,6,7,8,1] => [3,4,5,6,7,8,1,2] => 0
[1,0,1,1,0,1,0,0,1,1,1,0,0,0] => [1,1,0,0,1,0,1,1,1,0,0,1,0,0] => [2,5,1,3,8,7,4,6] => [3,6,2,4,1,5,8,7] => 2
[1,0,1,1,0,1,1,0,0,0,1,0,1,0] => [1,1,0,0,1,1,0,1,1,1,0,0,0,0] => [2,6,1,5,3,7,8,4] => [3,7,2,6,4,8,1,5] => 3
[1,0,1,1,1,0,0,1,0,0,1,1,0,0] => [1,1,1,0,0,1,0,0,1,1,0,0,1,0] => [2,6,4,1,3,8,5,7] => [3,7,5,2,4,1,8,6] => 3
[1,1,0,0,1,0,1,1,0,1,1,0,0,0] => [1,0,1,1,1,0,0,0,1,1,0,1,0,0] => [3,1,4,8,2,7,5,6] => [4,2,5,1,6,3,8,7] => 1
[1,1,0,0,1,1,0,0,1,0,1,1,0,0] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0] => [3,1,5,2,6,8,4,7] => [4,2,6,3,7,1,8,5] => 2
[1,1,0,1,1,1,0,0,0,0,1,1,0,0] => [1,0,1,1,0,1,0,1,1,0,0,0,1,0] => [6,1,5,2,3,8,4,7] => [7,2,6,3,4,1,8,5] => 4
[1,1,1,0,1,0,0,1,0,0,1,0,1,0] => [1,1,1,0,1,0,0,0,1,1,1,0,0,0] => [6,3,4,1,2,7,8,5] => [7,4,5,2,3,8,1,6] => 4
[1,1,1,0,1,1,0,0,1,0,0,1,0,0] => [1,1,1,0,1,1,0,0,1,0,0,0,1,0] => [8,3,5,1,6,2,4,7] => [1,5,7,3,8,4,6,2] => 3
[1,1,1,1,0,0,1,0,0,0,1,0,1,0] => [1,1,0,1,1,0,1,1,1,0,0,0,0,0] => [6,4,1,5,2,7,8,3] => [7,5,2,6,3,8,1,4] => 4
[] => [] => [1] => [1] => 0
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Description
The number of nestings in the permutation.
Map
Delest-Viennot-inverse
Description
Return the Dyck path obtained by applying the inverse of Delest-Viennot's bijection to the corresponding parallelogram polyomino.
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 $(\beta^{(-1)}\circ\gamma)(D)$.
Map
Ringel
Description
The Ringel permutation of the LNakayama algebra corresponding to a Dyck path.
Map
Lehmer code rotation
Description
Sends a permutation $\pi$ to the unique permutation $\tau$ (of the same length) such that every entry in the Lehmer code of $\tau$ is cyclically one larger than the Lehmer code of $\pi$.