Identifier
Values
[1,0] => [1,1,0,0] => [2,3,1] => [1,2,3] => 1
[1,0,1,0] => [1,1,0,1,0,0] => [4,3,1,2] => [4,1,3,2] => 1
[1,1,0,0] => [1,1,1,0,0,0] => [2,3,4,1] => [1,2,3,4] => 2
[1,0,1,0,1,0] => [1,1,0,1,0,1,0,0] => [5,4,1,2,3] => [4,5,1,3,2] => 1
[1,0,1,1,0,0] => [1,1,0,1,1,0,0,0] => [4,3,1,5,2] => [4,1,3,2,5] => 2
[1,1,0,0,1,0] => [1,1,1,0,0,1,0,0] => [2,5,4,1,3] => [5,2,1,4,3] => 2
[1,1,0,1,0,0] => [1,1,1,0,1,0,0,0] => [5,3,4,1,2] => [5,1,3,4,2] => 1
[1,1,1,0,0,0] => [1,1,1,1,0,0,0,0] => [2,3,4,5,1] => [1,2,3,4,5] => 3
[1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,0] => [5,6,1,2,3,4] => [4,5,6,1,2,3] => 2
[1,0,1,0,1,1,0,0] => [1,1,0,1,0,1,1,0,0,0] => [5,4,1,2,6,3] => [4,5,1,3,2,6] => 2
[1,0,1,1,0,0,1,0] => [1,1,0,1,1,0,0,1,0,0] => [6,3,1,5,2,4] => [4,6,3,1,5,2] => 2
[1,0,1,1,0,1,0,0] => [1,1,0,1,1,0,1,0,0,0] => [6,4,1,5,2,3] => [4,6,1,3,5,2] => 1
[1,0,1,1,1,0,0,0] => [1,1,0,1,1,1,0,0,0,0] => [4,3,1,5,6,2] => [4,1,3,2,5,6] => 3
[1,1,0,0,1,0,1,0] => [1,1,1,0,0,1,0,1,0,0] => [2,6,5,1,3,4] => [5,2,6,1,4,3] => 2
[1,1,0,0,1,1,0,0] => [1,1,1,0,0,1,1,0,0,0] => [2,5,4,1,6,3] => [5,2,1,4,3,6] => 3
[1,1,0,1,0,0,1,0] => [1,1,1,0,1,0,0,1,0,0] => [6,3,5,1,2,4] => [5,6,3,1,4,2] => 2
[1,1,0,1,0,1,0,0] => [1,1,1,0,1,0,1,0,0,0] => [6,5,4,1,2,3] => [5,6,1,4,3,2] => 1
[1,1,0,1,1,0,0,0] => [1,1,1,0,1,1,0,0,0,0] => [5,3,4,1,6,2] => [5,1,3,4,2,6] => 2
[1,1,1,0,0,0,1,0] => [1,1,1,1,0,0,0,1,0,0] => [2,3,6,5,1,4] => [6,2,3,1,5,4] => 2
[1,1,1,0,0,1,0,0] => [1,1,1,1,0,0,1,0,0,0] => [2,6,4,5,1,3] => [6,2,1,4,5,3] => 2
[1,1,1,0,1,0,0,0] => [1,1,1,1,0,1,0,0,0,0] => [6,3,4,5,1,2] => [6,1,3,4,5,2] => 2
[1,1,1,1,0,0,0,0] => [1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => [1,2,3,4,5,6] => 4
[1,1,1,1,1,0,0,0,0,0] => [1,1,1,1,1,1,0,0,0,0,0,0] => [2,3,4,5,6,7,1] => [1,2,3,4,5,6,7] => 5
[] => [1,0] => [2,1] => [1,2] => 0
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Description
The number of ascents of distance 2 of a permutation.
This is, $\operatorname{asc}_2(\pi) = | \{ i : \pi(i) < \pi(i+2) \} |$.
Map
prime Dyck path
Description
Return the Dyck path obtained by adding an initial up and a final down step.
Map
Kreweras complement
Description
Sends the permutation $\pi \in \mathfrak{S}_n$ to the permutation $\pi^{-1}c$ where $c = (1,\ldots,n)$ is the long cycle.
Map
Ringel
Description
The Ringel permutation of the LNakayama algebra corresponding to a Dyck path.