Identifier
Values
[1,0] => [1,0] => [2,1] => [2,1] => 0
[1,0,1,0] => [1,0,1,0] => [3,1,2] => [3,1,2] => 1
[1,1,0,0] => [1,1,0,0] => [2,3,1] => [3,2,1] => 0
[1,0,1,0,1,0] => [1,0,1,0,1,0] => [4,1,2,3] => [4,1,2,3] => 1
[1,0,1,1,0,0] => [1,1,0,1,0,0] => [4,3,1,2] => [3,1,4,2] => 3
[1,1,0,0,1,0] => [1,1,0,0,1,0] => [2,4,1,3] => [4,2,1,3] => 2
[1,1,0,1,0,0] => [1,0,1,1,0,0] => [3,1,4,2] => [4,3,1,2] => 1
[1,1,1,0,0,0] => [1,1,1,0,0,0] => [2,3,4,1] => [4,3,2,1] => 0
[1,0,1,0,1,0,1,0] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => [5,1,2,3,4] => 1
[1,0,1,0,1,1,0,0] => [1,1,0,1,0,1,0,0] => [5,4,1,2,3] => [4,1,5,2,3] => 3
[1,0,1,1,0,0,1,0] => [1,1,0,1,0,0,1,0] => [5,3,1,2,4] => [3,1,5,2,4] => 4
[1,0,1,1,0,1,0,0] => [1,0,1,1,0,1,0,0] => [5,1,4,2,3] => [4,1,2,5,3] => 3
[1,0,1,1,1,0,0,0] => [1,1,1,0,1,0,0,0] => [5,3,4,1,2] => [4,3,1,5,2] => 3
[1,1,0,0,1,0,1,0] => [1,1,0,0,1,0,1,0] => [2,5,1,3,4] => [5,2,1,3,4] => 2
[1,1,0,0,1,1,0,0] => [1,1,1,0,0,1,0,0] => [2,5,4,1,3] => [4,2,1,5,3] => 3
[1,1,0,1,0,0,1,0] => [1,0,1,1,0,0,1,0] => [3,1,5,2,4] => [5,3,1,2,4] => 3
[1,1,0,1,0,1,0,0] => [1,0,1,0,1,1,0,0] => [4,1,2,5,3] => [5,4,1,2,3] => 1
[1,1,0,1,1,0,0,0] => [1,1,0,1,1,0,0,0] => [4,3,1,5,2] => [3,1,5,4,2] => 3
[1,1,1,0,0,0,1,0] => [1,1,1,0,0,0,1,0] => [2,3,5,1,4] => [5,3,2,1,4] => 2
[1,1,1,0,0,1,0,0] => [1,1,0,0,1,1,0,0] => [2,4,1,5,3] => [5,4,2,1,3] => 2
[1,1,1,0,1,0,0,0] => [1,0,1,1,1,0,0,0] => [3,1,4,5,2] => [5,4,3,1,2] => 1
[1,1,1,1,0,0,0,0] => [1,1,1,1,0,0,0,0] => [2,3,4,5,1] => [5,4,3,2,1] => 0
[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] => [6,1,2,3,4,5] => 1
[1,0,1,0,1,0,1,1,0,0] => [1,1,0,1,0,1,0,1,0,0] => [5,6,1,2,3,4] => [6,1,5,2,3,4] => 3
[1,0,1,0,1,1,0,0,1,0] => [1,1,0,1,0,1,0,0,1,0] => [6,4,1,2,3,5] => [4,1,6,2,3,5] => 4
[1,0,1,0,1,1,0,1,0,0] => [1,0,1,1,0,1,0,1,0,0] => [6,1,5,2,3,4] => [5,1,2,6,3,4] => 3
[1,0,1,0,1,1,1,0,0,0] => [1,1,1,0,1,0,1,0,0,0] => [6,5,4,1,2,3] => [4,1,5,2,6,3] => 5
[1,0,1,1,0,0,1,0,1,0] => [1,1,0,1,0,0,1,0,1,0] => [6,3,1,2,4,5] => [3,1,6,2,4,5] => 4
[1,0,1,1,0,0,1,1,0,0] => [1,1,1,0,1,0,0,1,0,0] => [6,3,5,1,2,4] => [5,3,1,6,2,4] => 5
[1,0,1,1,0,1,0,0,1,0] => [1,0,1,1,0,1,0,0,1,0] => [6,1,4,2,3,5] => [4,1,2,6,3,5] => 4
[1,0,1,1,0,1,0,1,0,0] => [1,0,1,0,1,1,0,1,0,0] => [6,1,2,5,3,4] => [5,1,2,3,6,4] => 3
[1,0,1,1,0,1,1,0,0,0] => [1,1,0,1,1,0,1,0,0,0] => [6,4,1,5,2,3] => [5,4,1,6,2,3] => 3
[1,0,1,1,1,0,0,0,1,0] => [1,1,1,0,1,0,0,0,1,0] => [6,3,4,1,2,5] => [4,3,1,6,2,5] => 4
[1,0,1,1,1,0,0,1,0,0] => [1,0,1,1,1,0,0,1,0,0] => [3,1,6,5,2,4] => [5,3,1,2,6,4] => 4
[1,0,1,1,1,0,1,0,0,0] => [1,0,1,1,1,0,1,0,0,0] => [6,1,4,5,2,3] => [5,4,1,2,6,3] => 3
[1,0,1,1,1,1,0,0,0,0] => [1,1,1,1,0,1,0,0,0,0] => [6,3,4,5,1,2] => [5,4,3,1,6,2] => 3
[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] => [6,2,1,3,4,5] => 2
[1,1,0,0,1,0,1,1,0,0] => [1,1,1,0,0,1,0,1,0,0] => [2,6,5,1,3,4] => [5,2,1,6,3,4] => 3
[1,1,0,0,1,1,0,0,1,0] => [1,1,1,0,0,1,0,0,1,0] => [2,6,4,1,3,5] => [4,2,1,6,3,5] => 4
[1,1,0,0,1,1,0,1,0,0] => [1,1,0,0,1,1,0,1,0,0] => [2,6,1,5,3,4] => [5,2,1,3,6,4] => 4
[1,1,0,0,1,1,1,0,0,0] => [1,1,1,1,0,0,1,0,0,0] => [2,6,4,5,1,3] => [5,4,2,1,6,3] => 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] => [6,3,1,2,4,5] => 3
[1,1,0,1,0,0,1,1,0,0] => [1,1,0,1,1,0,0,1,0,0] => [6,3,1,5,2,4] => [3,1,5,2,6,4] => 5
[1,1,0,1,0,1,0,0,1,0] => [1,0,1,0,1,1,0,0,1,0] => [4,1,2,6,3,5] => [6,4,1,2,3,5] => 3
[1,1,0,1,0,1,0,1,0,0] => [1,0,1,0,1,0,1,1,0,0] => [5,1,2,3,6,4] => [6,5,1,2,3,4] => 1
[1,1,0,1,0,1,1,0,0,0] => [1,1,0,1,0,1,1,0,0,0] => [5,4,1,2,6,3] => [4,1,6,5,2,3] => 3
[1,1,0,1,1,0,0,0,1,0] => [1,1,0,1,1,0,0,0,1,0] => [4,3,1,6,2,5] => [3,1,6,4,2,5] => 5
[1,1,0,1,1,0,0,1,0,0] => [1,1,0,1,0,0,1,1,0,0] => [5,3,1,2,6,4] => [3,1,6,5,2,4] => 4
[1,1,0,1,1,0,1,0,0,0] => [1,0,1,1,0,1,1,0,0,0] => [5,1,4,2,6,3] => [4,1,2,6,5,3] => 3
[1,1,0,1,1,1,0,0,0,0] => [1,1,0,1,1,1,0,0,0,0] => [4,3,1,5,6,2] => [3,1,6,5,4,2] => 3
[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] => [6,3,2,1,4,5] => 2
[1,1,1,0,0,0,1,1,0,0] => [1,1,1,1,0,0,0,1,0,0] => [2,3,6,5,1,4] => [5,3,2,1,6,4] => 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] => [6,4,2,1,3,5] => 4
[1,1,1,0,0,1,0,1,0,0] => [1,1,0,0,1,0,1,1,0,0] => [2,5,1,3,6,4] => [6,5,2,1,3,4] => 2
[1,1,1,0,0,1,1,0,0,0] => [1,1,1,0,0,1,1,0,0,0] => [2,5,4,1,6,3] => [4,2,1,6,5,3] => 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] => [6,4,3,1,2,5] => 3
[1,1,1,0,1,0,0,1,0,0] => [1,0,1,1,0,0,1,1,0,0] => [3,1,5,2,6,4] => [6,5,3,1,2,4] => 3
[1,1,1,0,1,0,1,0,0,0] => [1,0,1,0,1,1,1,0,0,0] => [4,1,2,5,6,3] => [6,5,4,1,2,3] => 1
[1,1,1,0,1,1,0,0,0,0] => [1,1,1,0,1,1,0,0,0,0] => [5,3,4,1,6,2] => [4,3,1,6,5,2] => 3
[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] => [6,4,3,2,1,5] => 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] => [6,5,3,2,1,4] => 2
[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] => [6,5,4,2,1,3] => 2
[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] => [6,5,4,3,1,2] => 1
[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] => [6,5,4,3,2,1] => 0
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Description
The number of non-attacking neighbors of a permutation.
For a permutation $\sigma$, the indices $i$ and $i+1$ are attacking if $|\sigma(i)-\sigma(i+1)| = 1$.
Visually, this is, for $\sigma$ considered as a placement of kings on a chessboard, if the kings placed in columns $i$ and $i+1$ are non-attacking.
Map
Clarke-Steingrimsson-Zeng inverse
Description
The inverse of the Clarke-Steingrimsson-Zeng map, sending excedances to descents.
This is the inverse of the map $\Phi$ in [1, sec.3].
Map
switch returns and last double rise
Description
An alternative to the Adin-Bagno-Roichman transformation of a Dyck path.
This is a bijection preserving the number of up steps before each peak and exchanging the number of components with the position of the last double rise.
Map
Ringel
Description
The Ringel permutation of the LNakayama algebra corresponding to a Dyck path.