Identifier
Values
[1,0] => [1,1,0,0] => [2,3,1] => [2,1] => 0
[1,0,1,0] => [1,1,0,1,0,0] => [4,3,1,2] => [3,1,2] => 0
[1,1,0,0] => [1,1,1,0,0,0] => [2,3,4,1] => [2,3,1] => 0
[1,0,1,0,1,0] => [1,1,0,1,0,1,0,0] => [5,4,1,2,3] => [4,1,2,3] => 0
[1,0,1,1,0,0] => [1,1,0,1,1,0,0,0] => [4,3,1,5,2] => [4,3,1,2] => 1
[1,1,0,0,1,0] => [1,1,1,0,0,1,0,0] => [2,5,4,1,3] => [2,4,1,3] => 1
[1,1,0,1,0,0] => [1,1,1,0,1,0,0,0] => [5,3,4,1,2] => [3,4,1,2] => 0
[1,1,1,0,0,0] => [1,1,1,1,0,0,0,0] => [2,3,4,5,1] => [2,3,4,1] => 0
[1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,0] => [5,6,1,2,3,4] => [5,1,2,3,4] => 0
[1,0,1,0,1,1,0,0] => [1,1,0,1,0,1,1,0,0,0] => [5,4,1,2,6,3] => [5,4,1,2,3] => 1
[1,0,1,1,0,0,1,0] => [1,1,0,1,1,0,0,1,0,0] => [6,3,1,5,2,4] => [3,1,5,2,4] => 1
[1,0,1,1,0,1,0,0] => [1,1,0,1,1,0,1,0,0,0] => [6,4,1,5,2,3] => [4,1,5,2,3] => 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] => [4,3,1,5,2] => 1
[1,1,0,0,1,0,1,0] => [1,1,1,0,0,1,0,1,0,0] => [2,6,5,1,3,4] => [2,5,1,3,4] => 1
[1,1,0,0,1,1,0,0] => [1,1,1,0,0,1,1,0,0,0] => [2,5,4,1,6,3] => [2,5,4,1,3] => 1
[1,1,0,1,0,0,1,0] => [1,1,1,0,1,0,0,1,0,0] => [6,3,5,1,2,4] => [3,5,1,2,4] => 1
[1,1,0,1,0,1,0,0] => [1,1,1,0,1,0,1,0,0,0] => [6,5,4,1,2,3] => [5,4,1,2,3] => 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] => [5,3,4,1,2] => 1
[1,1,1,0,0,0,1,0] => [1,1,1,1,0,0,0,1,0,0] => [2,3,6,5,1,4] => [2,3,5,1,4] => 1
[1,1,1,0,0,1,0,0] => [1,1,1,1,0,0,1,0,0,0] => [2,6,4,5,1,3] => [2,4,5,1,3] => 1
[1,1,1,0,1,0,0,0] => [1,1,1,1,0,1,0,0,0,0] => [6,3,4,5,1,2] => [3,4,5,1,2] => 0
[1,1,1,1,0,0,0,0] => [1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => [2,3,4,5,1] => 0
[1,0,1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,1,0,0] => [7,6,1,2,3,4,5] => [6,1,2,3,4,5] => 0
[1,0,1,0,1,0,1,1,0,0] => [1,1,0,1,0,1,0,1,1,0,0,0] => [5,6,1,2,3,7,4] => [5,6,1,2,3,4] => 0
[1,0,1,0,1,1,0,0,1,0] => [1,1,0,1,0,1,1,0,0,1,0,0] => [7,4,1,2,6,3,5] => [4,1,2,6,3,5] => 1
[1,0,1,0,1,1,0,1,0,0] => [1,1,0,1,0,1,1,0,1,0,0,0] => [5,7,1,2,6,3,4] => [5,1,2,6,3,4] => 1
[1,0,1,0,1,1,1,0,0,0] => [1,1,0,1,0,1,1,1,0,0,0,0] => [5,4,1,2,6,7,3] => [5,4,1,2,6,3] => 1
[1,0,1,1,0,0,1,0,1,0] => [1,1,0,1,1,0,0,1,0,1,0,0] => [7,3,1,6,2,4,5] => [3,1,6,2,4,5] => 2
[1,0,1,1,0,0,1,1,0,0] => [1,1,0,1,1,0,0,1,1,0,0,0] => [6,3,1,5,2,7,4] => [6,3,1,5,2,4] => 2
[1,0,1,1,0,1,0,0,1,0] => [1,1,0,1,1,0,1,0,0,1,0,0] => [7,4,1,6,2,3,5] => [4,1,6,2,3,5] => 1
[1,0,1,1,0,1,0,1,0,0] => [1,1,0,1,1,0,1,0,1,0,0,0] => [6,7,1,5,2,3,4] => [6,1,5,2,3,4] => 1
[1,0,1,1,0,1,1,0,0,0] => [1,1,0,1,1,0,1,1,0,0,0,0] => [6,4,1,5,2,7,3] => [6,4,1,5,2,3] => 1
[1,0,1,1,1,0,0,0,1,0] => [1,1,0,1,1,1,0,0,0,1,0,0] => [4,3,1,7,6,2,5] => [4,3,1,6,2,5] => 1
[1,0,1,1,1,0,0,1,0,0] => [1,1,0,1,1,1,0,0,1,0,0,0] => [7,3,1,5,6,2,4] => [3,1,5,6,2,4] => 2
[1,0,1,1,1,0,1,0,0,0] => [1,1,0,1,1,1,0,1,0,0,0,0] => [7,4,1,5,6,2,3] => [4,1,5,6,2,3] => 1
[1,0,1,1,1,1,0,0,0,0] => [1,1,0,1,1,1,1,0,0,0,0,0] => [4,3,1,5,6,7,2] => [4,3,1,5,6,2] => 2
[1,1,0,0,1,0,1,0,1,0] => [1,1,1,0,0,1,0,1,0,1,0,0] => [2,6,7,1,3,4,5] => [2,6,1,3,4,5] => 1
[1,1,0,0,1,0,1,1,0,0] => [1,1,1,0,0,1,0,1,1,0,0,0] => [2,6,5,1,3,7,4] => [2,6,5,1,3,4] => 2
[1,1,0,0,1,1,0,0,1,0] => [1,1,1,0,0,1,1,0,0,1,0,0] => [2,7,4,1,6,3,5] => [2,4,1,6,3,5] => 1
[1,1,0,0,1,1,0,1,0,0] => [1,1,1,0,0,1,1,0,1,0,0,0] => [2,7,5,1,6,3,4] => [2,5,1,6,3,4] => 1
[1,1,0,0,1,1,1,0,0,0] => [1,1,1,0,0,1,1,1,0,0,0,0] => [2,5,4,1,6,7,3] => [2,5,4,1,6,3] => 1
[1,1,0,1,0,0,1,0,1,0] => [1,1,1,0,1,0,0,1,0,1,0,0] => [6,3,7,1,2,4,5] => [6,3,1,2,4,5] => 1
[1,1,0,1,0,0,1,1,0,0] => [1,1,1,0,1,0,0,1,1,0,0,0] => [6,3,5,1,2,7,4] => [6,3,5,1,2,4] => 2
[1,1,0,1,0,1,0,0,1,0] => [1,1,1,0,1,0,1,0,0,1,0,0] => [6,7,4,1,2,3,5] => [6,4,1,2,3,5] => 1
[1,1,0,1,0,1,0,1,0,0] => [1,1,1,0,1,0,1,0,1,0,0,0] => [6,7,5,1,2,3,4] => [6,5,1,2,3,4] => 1
[1,1,0,1,0,1,1,0,0,0] => [1,1,1,0,1,0,1,1,0,0,0,0] => [6,5,4,1,2,7,3] => [6,5,4,1,2,3] => 1
[1,1,0,1,1,0,0,0,1,0] => [1,1,1,0,1,1,0,0,0,1,0,0] => [7,3,4,1,6,2,5] => [3,4,1,6,2,5] => 1
[1,1,0,1,1,0,0,1,0,0] => [1,1,1,0,1,1,0,0,1,0,0,0] => [7,3,5,1,6,2,4] => [3,5,1,6,2,4] => 2
[1,1,0,1,1,0,1,0,0,0] => [1,1,1,0,1,1,0,1,0,0,0,0] => [7,5,4,1,6,2,3] => [5,4,1,6,2,3] => 1
[1,1,0,1,1,1,0,0,0,0] => [1,1,1,0,1,1,1,0,0,0,0,0] => [5,3,4,1,6,7,2] => [5,3,4,1,6,2] => 1
[1,1,1,0,0,0,1,0,1,0] => [1,1,1,1,0,0,0,1,0,1,0,0] => [2,3,7,6,1,4,5] => [2,3,6,1,4,5] => 1
[1,1,1,0,0,0,1,1,0,0] => [1,1,1,1,0,0,0,1,1,0,0,0] => [2,3,6,5,1,7,4] => [2,3,6,5,1,4] => 1
[1,1,1,0,0,1,0,0,1,0] => [1,1,1,1,0,0,1,0,0,1,0,0] => [2,7,4,6,1,3,5] => [2,4,6,1,3,5] => 2
[1,1,1,0,0,1,0,1,0,0] => [1,1,1,1,0,0,1,0,1,0,0,0] => [2,7,6,5,1,3,4] => [2,6,5,1,3,4] => 2
[1,1,1,0,0,1,1,0,0,0] => [1,1,1,1,0,0,1,1,0,0,0,0] => [2,6,4,5,1,7,3] => [2,6,4,5,1,3] => 2
[1,1,1,0,1,0,0,0,1,0] => [1,1,1,1,0,1,0,0,0,1,0,0] => [7,3,4,6,1,2,5] => [3,4,6,1,2,5] => 1
[1,1,1,0,1,0,0,1,0,0] => [1,1,1,1,0,1,0,0,1,0,0,0] => [7,3,6,5,1,2,4] => [3,6,5,1,2,4] => 1
[1,1,1,0,1,0,1,0,0,0] => [1,1,1,1,0,1,0,1,0,0,0,0] => [7,6,4,5,1,2,3] => [6,4,5,1,2,3] => 1
[1,1,1,0,1,1,0,0,0,0] => [1,1,1,1,0,1,1,0,0,0,0,0] => [6,3,4,5,1,7,2] => [6,3,4,5,1,2] => 1
[1,1,1,1,0,0,0,0,1,0] => [1,1,1,1,1,0,0,0,0,1,0,0] => [2,3,4,7,6,1,5] => [2,3,4,6,1,5] => 1
[1,1,1,1,0,0,0,1,0,0] => [1,1,1,1,1,0,0,0,1,0,0,0] => [2,3,7,5,6,1,4] => [2,3,5,6,1,4] => 1
[1,1,1,1,0,0,1,0,0,0] => [1,1,1,1,1,0,0,1,0,0,0,0] => [2,7,4,5,6,1,3] => [2,4,5,6,1,3] => 1
[1,1,1,1,0,1,0,0,0,0] => [1,1,1,1,1,0,1,0,0,0,0,0] => [7,3,4,5,6,1,2] => [3,4,5,6,1,2] => 0
[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] => [2,3,4,5,6,1] => 0
[] => [1,0] => [2,1] => [1] => 0
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Description
The cyclic holeyness of a permutation.
For $S\subset [n]:=\{1,2,\dots,n\}$ let $\delta(S)$ be the number of elements $m\in S$ such that $(m\bmod n)+1\notin S$.
For a permutation $\pi$ of $[n]$ the cyclic holeyness of $\pi$ is $$\max_{S\subset [n]} (\delta(\pi(S))-\delta(S)).$$
Map
prime Dyck path
Description
Return the Dyck path obtained by adding an initial up and a final down step.
Map
Ringel
Description
The Ringel permutation of the LNakayama algebra corresponding to a Dyck path.
Map
restriction
Description
The permutation obtained by removing the largest letter.
This map is undefined for the empty permutation.