Identifier
Values
[1,2] => [1,2] => [2,1] => [2,1] => 1
[2,1] => [1,2] => [2,1] => [2,1] => 1
[1,2,3] => [1,2,3] => [2,3,1] => [2,3,1] => 1
[1,3,2] => [1,2,3] => [2,3,1] => [2,3,1] => 1
[2,1,3] => [1,2,3] => [2,3,1] => [2,3,1] => 1
[2,3,1] => [1,2,3] => [2,3,1] => [2,3,1] => 1
[3,1,2] => [1,3,2] => [2,1,3] => [2,1,3] => 1
[3,2,1] => [1,3,2] => [2,1,3] => [2,1,3] => 1
[1,2,3,4] => [1,2,3,4] => [2,3,4,1] => [2,4,3,1] => 2
[1,2,4,3] => [1,2,3,4] => [2,3,4,1] => [2,4,3,1] => 2
[1,3,2,4] => [1,2,3,4] => [2,3,4,1] => [2,4,3,1] => 2
[1,3,4,2] => [1,2,3,4] => [2,3,4,1] => [2,4,3,1] => 2
[1,4,2,3] => [1,2,4,3] => [2,3,1,4] => [2,4,1,3] => 2
[1,4,3,2] => [1,2,4,3] => [2,3,1,4] => [2,4,1,3] => 2
[2,1,3,4] => [1,2,3,4] => [2,3,4,1] => [2,4,3,1] => 2
[2,1,4,3] => [1,2,3,4] => [2,3,4,1] => [2,4,3,1] => 2
[2,3,1,4] => [1,2,3,4] => [2,3,4,1] => [2,4,3,1] => 2
[2,3,4,1] => [1,2,3,4] => [2,3,4,1] => [2,4,3,1] => 2
[2,4,1,3] => [1,2,4,3] => [2,3,1,4] => [2,4,1,3] => 2
[2,4,3,1] => [1,2,4,3] => [2,3,1,4] => [2,4,1,3] => 2
[3,1,2,4] => [1,3,2,4] => [2,4,3,1] => [2,4,3,1] => 2
[3,1,4,2] => [1,3,4,2] => [2,1,3,4] => [2,1,4,3] => 1
[3,2,1,4] => [1,3,2,4] => [2,4,3,1] => [2,4,3,1] => 2
[3,2,4,1] => [1,3,4,2] => [2,1,3,4] => [2,1,4,3] => 1
[3,4,1,2] => [1,3,2,4] => [2,4,3,1] => [2,4,3,1] => 2
[3,4,2,1] => [1,3,2,4] => [2,4,3,1] => [2,4,3,1] => 2
[4,1,2,3] => [1,4,3,2] => [2,1,4,3] => [2,1,4,3] => 1
[4,1,3,2] => [1,4,2,3] => [2,4,1,3] => [2,4,1,3] => 2
[4,2,1,3] => [1,4,3,2] => [2,1,4,3] => [2,1,4,3] => 1
[4,2,3,1] => [1,4,2,3] => [2,4,1,3] => [2,4,1,3] => 2
[4,3,1,2] => [1,4,2,3] => [2,4,1,3] => [2,4,1,3] => 2
[4,3,2,1] => [1,4,2,3] => [2,4,1,3] => [2,4,1,3] => 2
[1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => [2,5,4,3,1] => 2
[1,2,3,5,4] => [1,2,3,4,5] => [2,3,4,5,1] => [2,5,4,3,1] => 2
[1,2,4,3,5] => [1,2,3,4,5] => [2,3,4,5,1] => [2,5,4,3,1] => 2
[1,2,4,5,3] => [1,2,3,4,5] => [2,3,4,5,1] => [2,5,4,3,1] => 2
[1,2,5,3,4] => [1,2,3,5,4] => [2,3,4,1,5] => [2,5,4,1,3] => 2
[1,2,5,4,3] => [1,2,3,5,4] => [2,3,4,1,5] => [2,5,4,1,3] => 2
[1,3,2,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => [2,5,4,3,1] => 2
[1,3,2,5,4] => [1,2,3,4,5] => [2,3,4,5,1] => [2,5,4,3,1] => 2
[1,3,4,2,5] => [1,2,3,4,5] => [2,3,4,5,1] => [2,5,4,3,1] => 2
[1,3,4,5,2] => [1,2,3,4,5] => [2,3,4,5,1] => [2,5,4,3,1] => 2
[1,3,5,2,4] => [1,2,3,5,4] => [2,3,4,1,5] => [2,5,4,1,3] => 2
[1,3,5,4,2] => [1,2,3,5,4] => [2,3,4,1,5] => [2,5,4,1,3] => 2
[1,4,2,3,5] => [1,2,4,3,5] => [2,3,5,4,1] => [2,5,4,3,1] => 2
[1,4,2,5,3] => [1,2,4,5,3] => [2,3,1,4,5] => [2,5,1,4,3] => 2
[1,4,3,2,5] => [1,2,4,3,5] => [2,3,5,4,1] => [2,5,4,3,1] => 2
[1,4,3,5,2] => [1,2,4,5,3] => [2,3,1,4,5] => [2,5,1,4,3] => 2
[1,4,5,2,3] => [1,2,4,3,5] => [2,3,5,4,1] => [2,5,4,3,1] => 2
[1,4,5,3,2] => [1,2,4,3,5] => [2,3,5,4,1] => [2,5,4,3,1] => 2
[1,5,2,3,4] => [1,2,5,4,3] => [2,3,1,5,4] => [2,5,1,4,3] => 2
[1,5,2,4,3] => [1,2,5,3,4] => [2,3,5,1,4] => [2,5,4,1,3] => 2
[1,5,3,2,4] => [1,2,5,4,3] => [2,3,1,5,4] => [2,5,1,4,3] => 2
[1,5,3,4,2] => [1,2,5,3,4] => [2,3,5,1,4] => [2,5,4,1,3] => 2
[1,5,4,2,3] => [1,2,5,3,4] => [2,3,5,1,4] => [2,5,4,1,3] => 2
[1,5,4,3,2] => [1,2,5,3,4] => [2,3,5,1,4] => [2,5,4,1,3] => 2
[2,1,3,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => [2,5,4,3,1] => 2
[2,1,3,5,4] => [1,2,3,4,5] => [2,3,4,5,1] => [2,5,4,3,1] => 2
[2,1,4,3,5] => [1,2,3,4,5] => [2,3,4,5,1] => [2,5,4,3,1] => 2
[2,1,4,5,3] => [1,2,3,4,5] => [2,3,4,5,1] => [2,5,4,3,1] => 2
[2,1,5,3,4] => [1,2,3,5,4] => [2,3,4,1,5] => [2,5,4,1,3] => 2
[2,1,5,4,3] => [1,2,3,5,4] => [2,3,4,1,5] => [2,5,4,1,3] => 2
[2,3,1,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => [2,5,4,3,1] => 2
[2,3,1,5,4] => [1,2,3,4,5] => [2,3,4,5,1] => [2,5,4,3,1] => 2
[2,3,4,1,5] => [1,2,3,4,5] => [2,3,4,5,1] => [2,5,4,3,1] => 2
[2,3,4,5,1] => [1,2,3,4,5] => [2,3,4,5,1] => [2,5,4,3,1] => 2
[2,3,5,1,4] => [1,2,3,5,4] => [2,3,4,1,5] => [2,5,4,1,3] => 2
[2,3,5,4,1] => [1,2,3,5,4] => [2,3,4,1,5] => [2,5,4,1,3] => 2
[2,4,1,3,5] => [1,2,4,3,5] => [2,3,5,4,1] => [2,5,4,3,1] => 2
[2,4,1,5,3] => [1,2,4,5,3] => [2,3,1,4,5] => [2,5,1,4,3] => 2
[2,4,3,1,5] => [1,2,4,3,5] => [2,3,5,4,1] => [2,5,4,3,1] => 2
[2,4,3,5,1] => [1,2,4,5,3] => [2,3,1,4,5] => [2,5,1,4,3] => 2
[2,4,5,1,3] => [1,2,4,3,5] => [2,3,5,4,1] => [2,5,4,3,1] => 2
[2,4,5,3,1] => [1,2,4,3,5] => [2,3,5,4,1] => [2,5,4,3,1] => 2
[2,5,1,3,4] => [1,2,5,4,3] => [2,3,1,5,4] => [2,5,1,4,3] => 2
[2,5,1,4,3] => [1,2,5,3,4] => [2,3,5,1,4] => [2,5,4,1,3] => 2
[2,5,3,1,4] => [1,2,5,4,3] => [2,3,1,5,4] => [2,5,1,4,3] => 2
[2,5,3,4,1] => [1,2,5,3,4] => [2,3,5,1,4] => [2,5,4,1,3] => 2
[2,5,4,1,3] => [1,2,5,3,4] => [2,3,5,1,4] => [2,5,4,1,3] => 2
[2,5,4,3,1] => [1,2,5,3,4] => [2,3,5,1,4] => [2,5,4,1,3] => 2
[3,1,2,4,5] => [1,3,2,4,5] => [2,4,3,5,1] => [2,5,4,3,1] => 2
[3,1,2,5,4] => [1,3,2,4,5] => [2,4,3,5,1] => [2,5,4,3,1] => 2
[3,1,4,2,5] => [1,3,4,2,5] => [2,5,3,4,1] => [2,5,4,3,1] => 2
[3,1,4,5,2] => [1,3,4,5,2] => [2,1,3,4,5] => [2,1,5,4,3] => 2
[3,1,5,2,4] => [1,3,5,4,2] => [2,1,3,5,4] => [2,1,5,4,3] => 2
[3,1,5,4,2] => [1,3,5,2,4] => [2,5,3,1,4] => [2,5,4,1,3] => 2
[3,2,1,4,5] => [1,3,2,4,5] => [2,4,3,5,1] => [2,5,4,3,1] => 2
[3,2,1,5,4] => [1,3,2,4,5] => [2,4,3,5,1] => [2,5,4,3,1] => 2
[3,2,4,1,5] => [1,3,4,2,5] => [2,5,3,4,1] => [2,5,4,3,1] => 2
[3,2,4,5,1] => [1,3,4,5,2] => [2,1,3,4,5] => [2,1,5,4,3] => 2
[3,2,5,1,4] => [1,3,5,4,2] => [2,1,3,5,4] => [2,1,5,4,3] => 2
[3,2,5,4,1] => [1,3,5,2,4] => [2,5,3,1,4] => [2,5,4,1,3] => 2
[3,4,1,2,5] => [1,3,2,4,5] => [2,4,3,5,1] => [2,5,4,3,1] => 2
[3,4,1,5,2] => [1,3,2,4,5] => [2,4,3,5,1] => [2,5,4,3,1] => 2
[3,4,2,1,5] => [1,3,2,4,5] => [2,4,3,5,1] => [2,5,4,3,1] => 2
[3,4,2,5,1] => [1,3,2,4,5] => [2,4,3,5,1] => [2,5,4,3,1] => 2
[3,4,5,1,2] => [1,3,5,2,4] => [2,5,3,1,4] => [2,5,4,1,3] => 2
[3,4,5,2,1] => [1,3,5,2,4] => [2,5,3,1,4] => [2,5,4,1,3] => 2
[3,5,1,2,4] => [1,3,2,5,4] => [2,4,3,1,5] => [2,5,4,1,3] => 2
[3,5,1,4,2] => [1,3,2,5,4] => [2,4,3,1,5] => [2,5,4,1,3] => 2
[3,5,2,1,4] => [1,3,2,5,4] => [2,4,3,1,5] => [2,5,4,1,3] => 2
>>> Load all 152 entries. <<<
[3,5,2,4,1] => [1,3,2,5,4] => [2,4,3,1,5] => [2,5,4,1,3] => 2
[3,5,4,1,2] => [1,3,4,2,5] => [2,5,3,4,1] => [2,5,4,3,1] => 2
[3,5,4,2,1] => [1,3,4,2,5] => [2,5,3,4,1] => [2,5,4,3,1] => 2
[4,1,2,3,5] => [1,4,3,2,5] => [2,5,4,3,1] => [2,5,4,3,1] => 2
[4,1,2,5,3] => [1,4,5,3,2] => [2,1,5,3,4] => [2,1,5,4,3] => 2
[4,1,3,2,5] => [1,4,2,3,5] => [2,4,5,3,1] => [2,5,4,3,1] => 2
[4,1,3,5,2] => [1,4,5,2,3] => [2,5,1,3,4] => [2,5,1,4,3] => 2
[4,1,5,2,3] => [1,4,2,3,5] => [2,4,5,3,1] => [2,5,4,3,1] => 2
[4,1,5,3,2] => [1,4,3,5,2] => [2,1,4,3,5] => [2,1,5,4,3] => 2
[4,2,1,3,5] => [1,4,3,2,5] => [2,5,4,3,1] => [2,5,4,3,1] => 2
[4,2,1,5,3] => [1,4,5,3,2] => [2,1,5,3,4] => [2,1,5,4,3] => 2
[4,2,3,1,5] => [1,4,2,3,5] => [2,4,5,3,1] => [2,5,4,3,1] => 2
[4,2,3,5,1] => [1,4,5,2,3] => [2,5,1,3,4] => [2,5,1,4,3] => 2
[4,2,5,1,3] => [1,4,2,3,5] => [2,4,5,3,1] => [2,5,4,3,1] => 2
[4,2,5,3,1] => [1,4,3,5,2] => [2,1,4,3,5] => [2,1,5,4,3] => 2
[4,3,1,2,5] => [1,4,2,3,5] => [2,4,5,3,1] => [2,5,4,3,1] => 2
[4,3,1,5,2] => [1,4,5,2,3] => [2,5,1,3,4] => [2,5,1,4,3] => 2
[4,3,2,1,5] => [1,4,2,3,5] => [2,4,5,3,1] => [2,5,4,3,1] => 2
[4,3,2,5,1] => [1,4,5,2,3] => [2,5,1,3,4] => [2,5,1,4,3] => 2
[4,3,5,1,2] => [1,4,2,3,5] => [2,4,5,3,1] => [2,5,4,3,1] => 2
[4,3,5,2,1] => [1,4,2,3,5] => [2,4,5,3,1] => [2,5,4,3,1] => 2
[4,5,1,2,3] => [1,4,2,5,3] => [2,4,1,3,5] => [2,5,1,4,3] => 2
[4,5,1,3,2] => [1,4,3,2,5] => [2,5,4,3,1] => [2,5,4,3,1] => 2
[4,5,2,1,3] => [1,4,2,5,3] => [2,4,1,3,5] => [2,5,1,4,3] => 2
[4,5,2,3,1] => [1,4,3,2,5] => [2,5,4,3,1] => [2,5,4,3,1] => 2
[4,5,3,1,2] => [1,4,2,5,3] => [2,4,1,3,5] => [2,5,1,4,3] => 2
[4,5,3,2,1] => [1,4,2,5,3] => [2,4,1,3,5] => [2,5,1,4,3] => 2
[5,1,2,3,4] => [1,5,4,3,2] => [2,1,5,4,3] => [2,1,5,4,3] => 2
[5,1,2,4,3] => [1,5,3,2,4] => [2,5,4,1,3] => [2,5,4,1,3] => 2
[5,1,3,2,4] => [1,5,4,2,3] => [2,5,1,4,3] => [2,5,1,4,3] => 2
[5,1,3,4,2] => [1,5,2,3,4] => [2,4,5,1,3] => [2,5,4,1,3] => 2
[5,1,4,2,3] => [1,5,3,4,2] => [2,1,4,5,3] => [2,1,5,4,3] => 2
[5,1,4,3,2] => [1,5,2,3,4] => [2,4,5,1,3] => [2,5,4,1,3] => 2
[5,2,1,3,4] => [1,5,4,3,2] => [2,1,5,4,3] => [2,1,5,4,3] => 2
[5,2,1,4,3] => [1,5,3,2,4] => [2,5,4,1,3] => [2,5,4,1,3] => 2
[5,2,3,1,4] => [1,5,4,2,3] => [2,5,1,4,3] => [2,5,1,4,3] => 2
[5,2,3,4,1] => [1,5,2,3,4] => [2,4,5,1,3] => [2,5,4,1,3] => 2
[5,2,4,1,3] => [1,5,3,4,2] => [2,1,4,5,3] => [2,1,5,4,3] => 2
[5,2,4,3,1] => [1,5,2,3,4] => [2,4,5,1,3] => [2,5,4,1,3] => 2
[5,3,1,2,4] => [1,5,4,2,3] => [2,5,1,4,3] => [2,5,1,4,3] => 2
[5,3,1,4,2] => [1,5,2,3,4] => [2,4,5,1,3] => [2,5,4,1,3] => 2
[5,3,2,1,4] => [1,5,4,2,3] => [2,5,1,4,3] => [2,5,1,4,3] => 2
[5,3,2,4,1] => [1,5,2,3,4] => [2,4,5,1,3] => [2,5,4,1,3] => 2
[5,3,4,1,2] => [1,5,2,3,4] => [2,4,5,1,3] => [2,5,4,1,3] => 2
[5,3,4,2,1] => [1,5,2,3,4] => [2,4,5,1,3] => [2,5,4,1,3] => 2
[5,4,1,2,3] => [1,5,3,2,4] => [2,5,4,1,3] => [2,5,4,1,3] => 2
[5,4,1,3,2] => [1,5,2,4,3] => [2,4,1,5,3] => [2,5,1,4,3] => 2
[5,4,2,1,3] => [1,5,3,2,4] => [2,5,4,1,3] => [2,5,4,1,3] => 2
[5,4,2,3,1] => [1,5,2,4,3] => [2,4,1,5,3] => [2,5,1,4,3] => 2
[5,4,3,1,2] => [1,5,2,4,3] => [2,4,1,5,3] => [2,5,1,4,3] => 2
[5,4,3,2,1] => [1,5,2,4,3] => [2,4,1,5,3] => [2,5,1,4,3] => 2
search for individual values
searching the database for the individual values of this statistic
/ search for generating function
searching the database for statistics with the same generating function
Description
The maximal modular displacement of a permutation.
This is $\max_{1\leq i \leq n} \left(\min(\pi(i)-i\pmod n, i-\pi(i)\pmod n)\right)$ for a permutation $\pi$ of $\{1,\dots,n\}$.
Map
cycle-as-one-line notation
Description
Return the permutation obtained by concatenating the cycles of a permutation, each written with minimal element first, sorted by minimal element.
Map
Simion-Schmidt map
Description
The Simion-Schmidt map sends any permutation to a $123$-avoiding permutation.
Details can be found in [1].
In particular, this is a bijection between $132$-avoiding permutations and $123$-avoiding permutations, see [1, Proposition 19].
Map
Kreweras complement
Description
Sends the permutation $\pi \in \mathfrak{S}_n$ to the permutation $\pi^{-1}c$ where $c = (1,\ldots,n)$ is the long cycle.