Identifier
Values
[1,2] => [1,2] => [2,1] => [2,1] => 0
[2,1] => [1,2] => [2,1] => [2,1] => 0
[1,2,3] => [1,2,3] => [2,3,1] => [3,2,1] => 0
[1,3,2] => [1,3,2] => [2,1,3] => [2,1,3] => 0
[2,1,3] => [1,3,2] => [2,1,3] => [2,1,3] => 0
[2,3,1] => [1,2,3] => [2,3,1] => [3,2,1] => 0
[3,1,2] => [1,2,3] => [2,3,1] => [3,2,1] => 0
[3,2,1] => [1,2,3] => [2,3,1] => [3,2,1] => 0
[1,2,3,4] => [1,2,3,4] => [2,3,4,1] => [4,3,2,1] => 0
[1,2,4,3] => [1,2,4,3] => [2,3,1,4] => [3,2,1,4] => 0
[1,3,2,4] => [1,3,2,4] => [2,4,3,1] => [3,4,2,1] => 1
[1,3,4,2] => [1,3,4,2] => [2,4,1,3] => [4,2,1,3] => 0
[1,4,2,3] => [1,4,2,3] => [2,1,4,3] => [2,1,4,3] => 0
[1,4,3,2] => [1,4,2,3] => [2,1,4,3] => [2,1,4,3] => 0
[2,1,3,4] => [1,3,4,2] => [2,4,1,3] => [4,2,1,3] => 0
[2,1,4,3] => [1,4,2,3] => [2,1,4,3] => [2,1,4,3] => 0
[2,3,1,4] => [1,4,2,3] => [2,1,4,3] => [2,1,4,3] => 0
[2,3,4,1] => [1,2,3,4] => [2,3,4,1] => [4,3,2,1] => 0
[2,4,1,3] => [1,3,2,4] => [2,4,3,1] => [3,4,2,1] => 1
[2,4,3,1] => [1,2,4,3] => [2,3,1,4] => [3,2,1,4] => 0
[3,1,2,4] => [1,2,4,3] => [2,3,1,4] => [3,2,1,4] => 0
[3,1,4,2] => [1,4,2,3] => [2,1,4,3] => [2,1,4,3] => 0
[3,2,1,4] => [1,4,2,3] => [2,1,4,3] => [2,1,4,3] => 0
[3,2,4,1] => [1,2,4,3] => [2,3,1,4] => [3,2,1,4] => 0
[3,4,1,2] => [1,2,3,4] => [2,3,4,1] => [4,3,2,1] => 0
[3,4,2,1] => [1,2,3,4] => [2,3,4,1] => [4,3,2,1] => 0
[4,1,2,3] => [1,2,3,4] => [2,3,4,1] => [4,3,2,1] => 0
[4,1,3,2] => [1,3,2,4] => [2,4,3,1] => [3,4,2,1] => 1
[4,2,1,3] => [1,3,2,4] => [2,4,3,1] => [3,4,2,1] => 1
[4,2,3,1] => [1,2,3,4] => [2,3,4,1] => [4,3,2,1] => 0
[4,3,1,2] => [1,2,3,4] => [2,3,4,1] => [4,3,2,1] => 0
[4,3,2,1] => [1,2,3,4] => [2,3,4,1] => [4,3,2,1] => 0
[1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => [5,4,3,2,1] => 0
[1,2,3,5,4] => [1,2,3,5,4] => [2,3,4,1,5] => [4,3,2,1,5] => 0
[1,2,4,3,5] => [1,2,4,3,5] => [2,3,5,4,1] => [4,5,3,2,1] => 1
[1,2,4,5,3] => [1,2,4,5,3] => [2,3,5,1,4] => [5,3,2,1,4] => 0
[1,2,5,3,4] => [1,2,5,3,4] => [2,3,1,5,4] => [3,2,1,5,4] => 0
[1,2,5,4,3] => [1,2,5,3,4] => [2,3,1,5,4] => [3,2,1,5,4] => 0
[1,3,2,4,5] => [1,3,2,4,5] => [2,4,3,5,1] => [3,5,4,2,1] => 1
[1,3,2,5,4] => [1,3,2,5,4] => [2,4,3,1,5] => [3,4,2,1,5] => 1
[1,3,4,2,5] => [1,3,4,2,5] => [2,4,5,3,1] => [5,3,4,2,1] => 0
[1,3,4,5,2] => [1,3,4,5,2] => [2,4,5,1,3] => [5,2,1,4,3] => 0
[1,3,5,2,4] => [1,3,5,2,4] => [2,4,1,5,3] => [5,4,2,1,3] => 0
[1,3,5,4,2] => [1,3,5,2,4] => [2,4,1,5,3] => [5,4,2,1,3] => 0
[1,4,2,3,5] => [1,4,2,3,5] => [2,5,3,4,1] => [3,4,5,2,1] => 2
[1,4,2,5,3] => [1,4,2,5,3] => [2,5,3,1,4] => [3,5,2,1,4] => 1
[1,4,3,2,5] => [1,4,2,5,3] => [2,5,3,1,4] => [3,5,2,1,4] => 1
[1,4,3,5,2] => [1,4,2,3,5] => [2,5,3,4,1] => [3,4,5,2,1] => 2
[1,4,5,2,3] => [1,4,5,2,3] => [2,5,1,4,3] => [4,5,2,1,3] => 1
[1,4,5,3,2] => [1,4,5,2,3] => [2,5,1,4,3] => [4,5,2,1,3] => 1
[1,5,2,3,4] => [1,5,2,3,4] => [2,1,4,5,3] => [2,1,5,4,3] => 0
[1,5,2,4,3] => [1,5,2,4,3] => [2,1,4,3,5] => [2,1,4,3,5] => 0
[1,5,3,2,4] => [1,5,2,4,3] => [2,1,4,3,5] => [2,1,4,3,5] => 0
[1,5,3,4,2] => [1,5,2,3,4] => [2,1,4,5,3] => [2,1,5,4,3] => 0
[1,5,4,2,3] => [1,5,2,3,4] => [2,1,4,5,3] => [2,1,5,4,3] => 0
[1,5,4,3,2] => [1,5,2,3,4] => [2,1,4,5,3] => [2,1,5,4,3] => 0
[2,1,3,4,5] => [1,3,4,5,2] => [2,4,5,1,3] => [5,2,1,4,3] => 0
[2,1,3,5,4] => [1,3,5,2,4] => [2,4,1,5,3] => [5,4,2,1,3] => 0
[2,1,4,3,5] => [1,4,2,3,5] => [2,5,3,4,1] => [3,4,5,2,1] => 2
[2,1,4,5,3] => [1,4,5,2,3] => [2,5,1,4,3] => [4,5,2,1,3] => 1
[2,1,5,3,4] => [1,5,2,3,4] => [2,1,4,5,3] => [2,1,5,4,3] => 0
[2,1,5,4,3] => [1,5,2,3,4] => [2,1,4,5,3] => [2,1,5,4,3] => 0
[2,3,1,4,5] => [1,4,5,2,3] => [2,5,1,4,3] => [4,5,2,1,3] => 1
[2,3,1,5,4] => [1,5,2,3,4] => [2,1,4,5,3] => [2,1,5,4,3] => 0
[2,3,4,1,5] => [1,5,2,3,4] => [2,1,4,5,3] => [2,1,5,4,3] => 0
[2,3,4,5,1] => [1,2,3,4,5] => [2,3,4,5,1] => [5,4,3,2,1] => 0
[2,3,5,1,4] => [1,4,2,3,5] => [2,5,3,4,1] => [3,4,5,2,1] => 2
[2,3,5,4,1] => [1,2,3,5,4] => [2,3,4,1,5] => [4,3,2,1,5] => 0
[2,4,1,3,5] => [1,3,5,2,4] => [2,4,1,5,3] => [5,4,2,1,3] => 0
[2,4,1,5,3] => [1,5,2,4,3] => [2,1,4,3,5] => [2,1,4,3,5] => 0
[2,4,3,1,5] => [1,5,2,4,3] => [2,1,4,3,5] => [2,1,4,3,5] => 0
[2,4,3,5,1] => [1,2,4,3,5] => [2,3,5,4,1] => [4,5,3,2,1] => 1
[2,4,5,1,3] => [1,3,2,4,5] => [2,4,3,5,1] => [3,5,4,2,1] => 1
[2,4,5,3,1] => [1,2,4,5,3] => [2,3,5,1,4] => [5,3,2,1,4] => 0
[2,5,1,3,4] => [1,3,4,2,5] => [2,4,5,3,1] => [5,3,4,2,1] => 0
[2,5,1,4,3] => [1,4,2,5,3] => [2,5,3,1,4] => [3,5,2,1,4] => 1
[2,5,3,1,4] => [1,4,2,5,3] => [2,5,3,1,4] => [3,5,2,1,4] => 1
[2,5,3,4,1] => [1,2,5,3,4] => [2,3,1,5,4] => [3,2,1,5,4] => 0
[2,5,4,1,3] => [1,3,2,5,4] => [2,4,3,1,5] => [3,4,2,1,5] => 1
[2,5,4,3,1] => [1,2,5,3,4] => [2,3,1,5,4] => [3,2,1,5,4] => 0
[3,1,2,4,5] => [1,2,4,5,3] => [2,3,5,1,4] => [5,3,2,1,4] => 0
[3,1,2,5,4] => [1,2,5,3,4] => [2,3,1,5,4] => [3,2,1,5,4] => 0
[3,1,4,2,5] => [1,4,2,5,3] => [2,5,3,1,4] => [3,5,2,1,4] => 1
[3,1,4,5,2] => [1,4,5,2,3] => [2,5,1,4,3] => [4,5,2,1,3] => 1
[3,1,5,2,4] => [1,5,2,4,3] => [2,1,4,3,5] => [2,1,4,3,5] => 0
[3,1,5,4,2] => [1,5,2,3,4] => [2,1,4,5,3] => [2,1,5,4,3] => 0
[3,2,1,4,5] => [1,4,5,2,3] => [2,5,1,4,3] => [4,5,2,1,3] => 1
[3,2,1,5,4] => [1,5,2,3,4] => [2,1,4,5,3] => [2,1,5,4,3] => 0
[3,2,4,1,5] => [1,5,2,4,3] => [2,1,4,3,5] => [2,1,4,3,5] => 0
[3,2,4,5,1] => [1,2,4,5,3] => [2,3,5,1,4] => [5,3,2,1,4] => 0
[3,2,5,1,4] => [1,4,2,5,3] => [2,5,3,1,4] => [3,5,2,1,4] => 1
[3,2,5,4,1] => [1,2,5,3,4] => [2,3,1,5,4] => [3,2,1,5,4] => 0
[3,4,1,2,5] => [1,2,5,3,4] => [2,3,1,5,4] => [3,2,1,5,4] => 0
[3,4,1,5,2] => [1,5,2,3,4] => [2,1,4,5,3] => [2,1,5,4,3] => 0
[3,4,2,1,5] => [1,5,2,3,4] => [2,1,4,5,3] => [2,1,5,4,3] => 0
[3,4,2,5,1] => [1,2,5,3,4] => [2,3,1,5,4] => [3,2,1,5,4] => 0
[3,4,5,1,2] => [1,2,3,4,5] => [2,3,4,5,1] => [5,4,3,2,1] => 0
[3,4,5,2,1] => [1,2,3,4,5] => [2,3,4,5,1] => [5,4,3,2,1] => 0
[3,5,1,2,4] => [1,2,4,3,5] => [2,3,5,4,1] => [4,5,3,2,1] => 1
[3,5,1,4,2] => [1,4,2,3,5] => [2,5,3,4,1] => [3,4,5,2,1] => 2
[3,5,2,1,4] => [1,4,2,3,5] => [2,5,3,4,1] => [3,4,5,2,1] => 2
>>> Load all 152 entries. <<<
[3,5,2,4,1] => [1,2,4,3,5] => [2,3,5,4,1] => [4,5,3,2,1] => 1
[3,5,4,1,2] => [1,2,3,5,4] => [2,3,4,1,5] => [4,3,2,1,5] => 0
[3,5,4,2,1] => [1,2,3,5,4] => [2,3,4,1,5] => [4,3,2,1,5] => 0
[4,1,2,3,5] => [1,2,3,5,4] => [2,3,4,1,5] => [4,3,2,1,5] => 0
[4,1,2,5,3] => [1,2,5,3,4] => [2,3,1,5,4] => [3,2,1,5,4] => 0
[4,1,3,2,5] => [1,3,2,5,4] => [2,4,3,1,5] => [3,4,2,1,5] => 1
[4,1,3,5,2] => [1,3,5,2,4] => [2,4,1,5,3] => [5,4,2,1,3] => 0
[4,1,5,2,3] => [1,5,2,3,4] => [2,1,4,5,3] => [2,1,5,4,3] => 0
[4,1,5,3,2] => [1,5,2,3,4] => [2,1,4,5,3] => [2,1,5,4,3] => 0
[4,2,1,3,5] => [1,3,5,2,4] => [2,4,1,5,3] => [5,4,2,1,3] => 0
[4,2,1,5,3] => [1,5,2,3,4] => [2,1,4,5,3] => [2,1,5,4,3] => 0
[4,2,3,1,5] => [1,5,2,3,4] => [2,1,4,5,3] => [2,1,5,4,3] => 0
[4,2,3,5,1] => [1,2,3,5,4] => [2,3,4,1,5] => [4,3,2,1,5] => 0
[4,2,5,1,3] => [1,3,2,5,4] => [2,4,3,1,5] => [3,4,2,1,5] => 1
[4,2,5,3,1] => [1,2,5,3,4] => [2,3,1,5,4] => [3,2,1,5,4] => 0
[4,3,1,2,5] => [1,2,5,3,4] => [2,3,1,5,4] => [3,2,1,5,4] => 0
[4,3,1,5,2] => [1,5,2,3,4] => [2,1,4,5,3] => [2,1,5,4,3] => 0
[4,3,2,1,5] => [1,5,2,3,4] => [2,1,4,5,3] => [2,1,5,4,3] => 0
[4,3,2,5,1] => [1,2,5,3,4] => [2,3,1,5,4] => [3,2,1,5,4] => 0
[4,3,5,1,2] => [1,2,3,5,4] => [2,3,4,1,5] => [4,3,2,1,5] => 0
[4,3,5,2,1] => [1,2,3,5,4] => [2,3,4,1,5] => [4,3,2,1,5] => 0
[4,5,1,2,3] => [1,2,3,4,5] => [2,3,4,5,1] => [5,4,3,2,1] => 0
[4,5,1,3,2] => [1,3,2,4,5] => [2,4,3,5,1] => [3,5,4,2,1] => 1
[4,5,2,1,3] => [1,3,2,4,5] => [2,4,3,5,1] => [3,5,4,2,1] => 1
[4,5,2,3,1] => [1,2,3,4,5] => [2,3,4,5,1] => [5,4,3,2,1] => 0
[4,5,3,1,2] => [1,2,3,4,5] => [2,3,4,5,1] => [5,4,3,2,1] => 0
[4,5,3,2,1] => [1,2,3,4,5] => [2,3,4,5,1] => [5,4,3,2,1] => 0
[5,1,2,3,4] => [1,2,3,4,5] => [2,3,4,5,1] => [5,4,3,2,1] => 0
[5,1,2,4,3] => [1,2,4,3,5] => [2,3,5,4,1] => [4,5,3,2,1] => 1
[5,1,3,2,4] => [1,3,2,4,5] => [2,4,3,5,1] => [3,5,4,2,1] => 1
[5,1,3,4,2] => [1,3,4,2,5] => [2,4,5,3,1] => [5,3,4,2,1] => 0
[5,1,4,2,3] => [1,4,2,3,5] => [2,5,3,4,1] => [3,4,5,2,1] => 2
[5,1,4,3,2] => [1,4,2,3,5] => [2,5,3,4,1] => [3,4,5,2,1] => 2
[5,2,1,3,4] => [1,3,4,2,5] => [2,4,5,3,1] => [5,3,4,2,1] => 0
[5,2,1,4,3] => [1,4,2,3,5] => [2,5,3,4,1] => [3,4,5,2,1] => 2
[5,2,3,1,4] => [1,4,2,3,5] => [2,5,3,4,1] => [3,4,5,2,1] => 2
[5,2,3,4,1] => [1,2,3,4,5] => [2,3,4,5,1] => [5,4,3,2,1] => 0
[5,2,4,1,3] => [1,3,2,4,5] => [2,4,3,5,1] => [3,5,4,2,1] => 1
[5,2,4,3,1] => [1,2,4,3,5] => [2,3,5,4,1] => [4,5,3,2,1] => 1
[5,3,1,2,4] => [1,2,4,3,5] => [2,3,5,4,1] => [4,5,3,2,1] => 1
[5,3,1,4,2] => [1,4,2,3,5] => [2,5,3,4,1] => [3,4,5,2,1] => 2
[5,3,2,1,4] => [1,4,2,3,5] => [2,5,3,4,1] => [3,4,5,2,1] => 2
[5,3,2,4,1] => [1,2,4,3,5] => [2,3,5,4,1] => [4,5,3,2,1] => 1
[5,3,4,1,2] => [1,2,3,4,5] => [2,3,4,5,1] => [5,4,3,2,1] => 0
[5,3,4,2,1] => [1,2,3,4,5] => [2,3,4,5,1] => [5,4,3,2,1] => 0
[5,4,1,2,3] => [1,2,3,4,5] => [2,3,4,5,1] => [5,4,3,2,1] => 0
[5,4,1,3,2] => [1,3,2,4,5] => [2,4,3,5,1] => [3,5,4,2,1] => 1
[5,4,2,1,3] => [1,3,2,4,5] => [2,4,3,5,1] => [3,5,4,2,1] => 1
[5,4,2,3,1] => [1,2,3,4,5] => [2,3,4,5,1] => [5,4,3,2,1] => 0
[5,4,3,1,2] => [1,2,3,4,5] => [2,3,4,5,1] => [5,4,3,2,1] => 0
[5,4,3,2,1] => [1,2,3,4,5] => [2,3,4,5,1] => [5,4,3,2,1] => 0
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Description
The number of augmented double ascents of a permutation.
An augmented double ascent of a permutation $\pi$ is a double ascent of the augmented permutation $\tilde\pi$ obtained from $\pi$ by adding an initial $0$.
A double ascent of $\tilde\pi$ then is a position $i$ such that $\tilde\pi(i) < \tilde\pi(i+1) < \tilde\pi(i+2)$.
Map
Lehmer code rotation
Description
Sends a permutation $\pi$ to the unique permutation $\tau$ (of the same length) such that every entry in the Lehmer code of $\tau$ is cyclically one larger than the Lehmer code of $\pi$.
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
runsort
Description
The permutation obtained by sorting the increasing runs lexicographically.