Identifier
Values
[1,0] => [1,0] => [2,1] => [2,1] => 1
[1,0,1,0] => [1,1,0,0] => [2,3,1] => [3,1,2] => 3
[1,1,0,0] => [1,0,1,0] => [3,1,2] => [2,3,1] => 1
[1,0,1,0,1,0] => [1,1,0,0,1,0] => [2,4,1,3] => [3,1,4,2] => 3
[1,0,1,1,0,0] => [1,1,0,1,0,0] => [4,3,1,2] => [3,4,2,1] => 4
[1,1,0,0,1,0] => [1,1,1,0,0,0] => [2,3,4,1] => [4,1,2,3] => 4
[1,1,0,1,0,0] => [1,0,1,1,0,0] => [3,1,4,2] => [2,4,1,3] => 3
[1,1,1,0,0,0] => [1,0,1,0,1,0] => [4,1,2,3] => [2,3,4,1] => 1
[1,0,1,0,1,0,1,0] => [1,1,0,0,1,1,0,0] => [2,4,1,5,3] => [3,1,5,2,4] => 5
[1,0,1,0,1,1,0,0] => [1,1,0,1,1,0,0,0] => [4,3,1,5,2] => [3,5,2,1,4] => 5
[1,0,1,1,0,0,1,0] => [1,1,0,0,1,0,1,0] => [2,5,1,3,4] => [3,1,4,5,2] => 3
[1,0,1,1,0,1,0,0] => [1,1,0,1,0,0,1,0] => [5,3,1,2,4] => [3,4,2,5,1] => 4
[1,0,1,1,1,0,0,0] => [1,1,0,1,0,1,0,0] => [5,4,1,2,3] => [3,4,5,2,1] => 5
[1,1,0,0,1,0,1,0] => [1,1,1,0,0,1,0,0] => [2,5,4,1,3] => [4,1,5,3,2] => 5
[1,1,0,0,1,1,0,0] => [1,1,1,1,0,0,0,0] => [2,3,4,5,1] => [5,1,2,3,4] => 5
[1,1,0,1,0,0,1,0] => [1,1,1,0,0,0,1,0] => [2,3,5,1,4] => [4,1,2,5,3] => 4
[1,1,0,1,0,1,0,0] => [1,0,1,1,0,0,1,0] => [3,1,5,2,4] => [2,4,1,5,3] => 3
[1,1,0,1,1,0,0,0] => [1,0,1,1,0,1,0,0] => [5,1,4,2,3] => [2,4,5,3,1] => 4
[1,1,1,0,0,0,1,0] => [1,1,1,0,1,0,0,0] => [5,3,4,1,2] => [4,5,2,3,1] => 5
[1,1,1,0,0,1,0,0] => [1,0,1,1,1,0,0,0] => [3,1,4,5,2] => [2,5,1,3,4] => 4
[1,1,1,0,1,0,0,0] => [1,0,1,0,1,1,0,0] => [4,1,2,5,3] => [2,3,5,1,4] => 3
[1,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => [2,3,4,5,1] => 1
[1,0,1,0,1,0,1,0,1,0] => [1,1,0,0,1,1,0,0,1,0] => [2,4,1,6,3,5] => [3,1,5,2,6,4] => 5
[1,0,1,0,1,0,1,1,0,0] => [1,1,0,1,1,0,0,0,1,0] => [4,3,1,6,2,5] => [3,5,2,1,6,4] => 5
[1,0,1,0,1,1,0,0,1,0] => [1,1,0,0,1,1,0,1,0,0] => [2,6,1,5,3,4] => [3,1,5,6,4,2] => 6
[1,0,1,0,1,1,0,1,0,0] => [1,1,0,1,1,0,0,1,0,0] => [6,3,1,5,2,4] => [3,5,2,6,4,1] => 6
[1,0,1,0,1,1,1,0,0,0] => [1,1,0,1,1,1,0,0,0,0] => [4,3,1,5,6,2] => [3,6,2,1,4,5] => 6
[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] => [3,1,6,2,4,5] => 6
[1,0,1,1,0,0,1,1,0,0] => [1,1,0,1,1,0,1,0,0,0] => [6,4,1,5,2,3] => [3,5,6,2,4,1] => 6
[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] => [3,1,4,6,2,5] => 5
[1,0,1,1,0,1,0,1,0,0] => [1,1,0,1,0,0,1,1,0,0] => [5,3,1,2,6,4] => [3,4,2,6,1,5] => 6
[1,0,1,1,0,1,1,0,0,0] => [1,1,0,1,0,1,1,0,0,0] => [5,4,1,2,6,3] => [3,4,6,2,1,5] => 6
[1,0,1,1,1,0,0,0,1,0] => [1,1,0,0,1,0,1,0,1,0] => [2,6,1,3,4,5] => [3,1,4,5,6,2] => 3
[1,0,1,1,1,0,0,1,0,0] => [1,1,0,1,0,0,1,0,1,0] => [6,3,1,2,4,5] => [3,4,2,5,6,1] => 4
[1,0,1,1,1,0,1,0,0,0] => [1,1,0,1,0,1,0,0,1,0] => [6,4,1,2,3,5] => [3,4,5,2,6,1] => 5
[1,0,1,1,1,1,0,0,0,0] => [1,1,0,1,0,1,0,1,0,0] => [5,6,1,2,3,4] => [3,4,5,6,1,2] => 6
[1,1,0,0,1,0,1,0,1,0] => [1,1,1,0,0,1,0,0,1,0] => [2,6,4,1,3,5] => [4,1,5,3,6,2] => 5
[1,1,0,0,1,0,1,1,0,0] => [1,1,1,1,0,0,0,0,1,0] => [2,3,4,6,1,5] => [5,1,2,3,6,4] => 5
[1,1,0,0,1,1,0,0,1,0] => [1,1,1,0,0,1,0,1,0,0] => [2,6,5,1,3,4] => [4,1,5,6,3,2] => 6
[1,1,0,0,1,1,0,1,0,0] => [1,1,1,1,0,0,0,1,0,0] => [2,3,6,5,1,4] => [5,1,2,6,4,3] => 6
[1,1,0,0,1,1,1,0,0,0] => [1,1,1,1,0,1,0,0,0,0] => [6,3,4,5,1,2] => [5,6,2,3,4,1] => 6
[1,1,0,1,0,0,1,0,1,0] => [1,1,1,0,0,1,1,0,0,0] => [2,5,4,1,6,3] => [4,1,6,3,2,5] => 6
[1,1,0,1,0,0,1,1,0,0] => [1,1,1,1,0,0,1,0,0,0] => [2,6,4,5,1,3] => [5,1,6,3,4,2] => 6
[1,1,0,1,0,1,0,0,1,0] => [1,1,1,0,0,0,1,1,0,0] => [2,3,5,1,6,4] => [4,1,2,6,3,5] => 6
[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] => [2,4,1,6,3,5] => 5
[1,1,0,1,0,1,1,0,0,0] => [1,0,1,1,0,1,1,0,0,0] => [5,1,4,2,6,3] => [2,4,6,3,1,5] => 5
[1,1,0,1,1,0,0,0,1,0] => [1,1,1,0,0,0,1,0,1,0] => [2,3,6,1,4,5] => [4,1,2,5,6,3] => 4
[1,1,0,1,1,0,0,1,0,0] => [1,0,1,1,0,0,1,0,1,0] => [3,1,6,2,4,5] => [2,4,1,5,6,3] => 3
[1,1,0,1,1,0,1,0,0,0] => [1,0,1,1,0,1,0,0,1,0] => [6,1,4,2,3,5] => [2,4,5,3,6,1] => 4
[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] => [2,4,5,6,3,1] => 5
[1,1,1,0,0,0,1,0,1,0] => [1,1,1,0,1,1,0,0,0,0] => [5,3,4,1,6,2] => [4,6,2,3,1,5] => 6
[1,1,1,0,0,0,1,1,0,0] => [1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => [6,1,2,3,4,5] => 6
[1,1,1,0,0,1,0,0,1,0] => [1,1,1,0,1,0,0,1,0,0] => [6,3,5,1,2,4] => [4,5,2,6,3,1] => 6
[1,1,1,0,0,1,0,1,0,0] => [1,0,1,1,1,0,0,1,0,0] => [3,1,6,5,2,4] => [2,5,1,6,4,3] => 5
[1,1,1,0,0,1,1,0,0,0] => [1,0,1,1,1,1,0,0,0,0] => [3,1,4,5,6,2] => [2,6,1,3,4,5] => 5
[1,1,1,0,1,0,0,0,1,0] => [1,1,1,0,1,0,0,0,1,0] => [6,3,4,1,2,5] => [4,5,2,3,6,1] => 5
[1,1,1,0,1,0,0,1,0,0] => [1,0,1,1,1,0,0,0,1,0] => [3,1,4,6,2,5] => [2,5,1,3,6,4] => 4
[1,1,1,0,1,0,1,0,0,0] => [1,0,1,0,1,1,0,0,1,0] => [4,1,2,6,3,5] => [2,3,5,1,6,4] => 3
[1,1,1,0,1,1,0,0,0,0] => [1,0,1,0,1,1,0,1,0,0] => [6,1,2,5,3,4] => [2,3,5,6,4,1] => 4
[1,1,1,1,0,0,0,0,1,0] => [1,1,1,0,1,0,1,0,0,0] => [6,5,4,1,2,3] => [4,5,6,3,2,1] => 6
[1,1,1,1,0,0,0,1,0,0] => [1,0,1,1,1,0,1,0,0,0] => [6,1,4,5,2,3] => [2,5,6,3,4,1] => 5
[1,1,1,1,0,0,1,0,0,0] => [1,0,1,0,1,1,1,0,0,0] => [4,1,2,5,6,3] => [2,3,6,1,4,5] => 4
[1,1,1,1,0,1,0,0,0,0] => [1,0,1,0,1,0,1,1,0,0] => [5,1,2,3,6,4] => [2,3,4,6,1,5] => 3
[1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0] => [6,1,2,3,4,5] => [2,3,4,5,6,1] => 1
[] => [] => [1] => [1] => 1
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Description
The number of indices that are not small excedances.
A small excedance is an index $i$ for which $\pi_i = i+1$.
Map
inverse
Description
Sends a permutation to its inverse.
Map
Elizalde-Deutsch bijection
Description
The Elizalde-Deutsch bijection on Dyck paths.
.Let $n$ be the length of the Dyck path. Consider the steps $1,n,2,n-1,\dots$ of $D$. When considering the $i$-th step its corresponding matching step has not yet been read, let the $i$-th step of the image of $D$ be an up step, otherwise let it be a down step.
Map
Ringel
Description
The Ringel permutation of the LNakayama algebra corresponding to a Dyck path.