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
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
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
Ringel
Description
The Ringel permutation of the LNakayama algebra corresponding to a Dyck path.