Identifier
Values
[1] => [1] => [1] => [1] => 0
[1,2] => [1,2] => [1,2] => [1,2] => 0
[2,1] => [2,1] => [2,1] => [2,1] => 0
[1,2,3] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => [1,3,2] => [1,3,2] => 0
[2,1,3] => [2,1,3] => [2,1,3] => [2,1,3] => 0
[2,3,1] => [2,3,1] => [3,2,1] => [3,2,1] => 0
[3,1,2] => [2,3,1] => [3,2,1] => [3,2,1] => 0
[3,2,1] => [3,2,1] => [3,2,1] => [3,2,1] => 0
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 0
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 0
[1,3,4,2] => [1,3,4,2] => [1,4,3,2] => [1,4,3,2] => 0
[1,4,2,3] => [1,3,4,2] => [1,4,3,2] => [1,4,3,2] => 0
[1,4,3,2] => [1,4,3,2] => [1,4,3,2] => [1,4,3,2] => 0
[2,1,3,4] => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 0
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 0
[2,3,1,4] => [2,3,1,4] => [3,2,1,4] => [3,2,1,4] => 0
[2,3,4,1] => [2,3,4,1] => [4,2,3,1] => [4,2,3,1] => 0
[2,4,1,3] => [4,3,1,2] => [4,3,2,1] => [4,3,2,1] => 0
[2,4,3,1] => [2,4,3,1] => [4,3,2,1] => [4,3,2,1] => 0
[3,1,2,4] => [2,3,1,4] => [3,2,1,4] => [3,2,1,4] => 0
[3,1,4,2] => [2,3,4,1] => [4,2,3,1] => [4,2,3,1] => 0
[3,2,1,4] => [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 0
[3,2,4,1] => [3,2,4,1] => [4,2,3,1] => [4,2,3,1] => 0
[3,4,1,2] => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 0
[3,4,2,1] => [3,4,2,1] => [4,3,2,1] => [4,3,2,1] => 0
[4,1,2,3] => [2,3,4,1] => [4,2,3,1] => [4,2,3,1] => 0
[4,1,3,2] => [2,4,3,1] => [4,3,2,1] => [4,3,2,1] => 0
[4,2,1,3] => [3,2,4,1] => [4,2,3,1] => [4,2,3,1] => 0
[4,2,3,1] => [3,4,2,1] => [4,3,2,1] => [4,3,2,1] => 0
[4,3,1,2] => [3,4,2,1] => [4,3,2,1] => [4,3,2,1] => 0
[4,3,2,1] => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 0
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 0
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 0
[1,2,4,5,3] => [1,2,4,5,3] => [1,2,5,4,3] => [1,2,5,4,3] => 0
[1,2,5,3,4] => [1,2,4,5,3] => [1,2,5,4,3] => [1,2,5,4,3] => 0
[1,2,5,4,3] => [1,2,5,4,3] => [1,2,5,4,3] => [1,2,5,4,3] => 0
[1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 0
[1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 0
[1,3,4,2,5] => [1,3,4,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => 0
[1,3,4,5,2] => [1,3,4,5,2] => [1,5,3,4,2] => [1,5,3,4,2] => 0
[1,3,5,2,4] => [1,5,4,2,3] => [1,5,4,3,2] => [1,5,4,3,2] => 0
[1,3,5,4,2] => [1,3,5,4,2] => [1,5,4,3,2] => [1,5,4,3,2] => 0
[1,4,2,3,5] => [1,3,4,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => 0
[1,4,2,5,3] => [1,3,4,5,2] => [1,5,3,4,2] => [1,5,3,4,2] => 0
[1,4,3,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => 0
[1,4,3,5,2] => [1,4,3,5,2] => [1,5,3,4,2] => [1,5,3,4,2] => 0
[1,4,5,2,3] => [1,5,4,3,2] => [1,5,4,3,2] => [1,5,4,3,2] => 0
[1,4,5,3,2] => [1,4,5,3,2] => [1,5,4,3,2] => [1,5,4,3,2] => 0
[1,5,2,3,4] => [1,3,4,5,2] => [1,5,3,4,2] => [1,5,3,4,2] => 0
[1,5,2,4,3] => [1,3,5,4,2] => [1,5,4,3,2] => [1,5,4,3,2] => 0
[1,5,3,2,4] => [1,4,3,5,2] => [1,5,3,4,2] => [1,5,3,4,2] => 0
[1,5,3,4,2] => [1,4,5,3,2] => [1,5,4,3,2] => [1,5,4,3,2] => 0
[1,5,4,2,3] => [1,4,5,3,2] => [1,5,4,3,2] => [1,5,4,3,2] => 0
[1,5,4,3,2] => [1,5,4,3,2] => [1,5,4,3,2] => [1,5,4,3,2] => 0
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 number of crossings of a signed permutation.
A crossing of a signed permutation $\pi$ is a pair $(i, j)$ of indices such that
  • $i < j \leq \pi(i) < \pi(j)$, or
  • $-i < j \leq -\pi(i) < \pi(j)$, or
  • $i > j > \pi(i) > \pi(j)$.
Map
to signed permutation
Description
The signed permutation with all signs positive.
Map
Demazure product with inverse
Description
This map sends a permutation $\pi$ to $\pi^{-1} \star \pi$ where $\star$ denotes the Demazure product on permutations.
This map is a surjection onto the set of involutions, i.e., the set of permutations $\pi$ for which $\pi = \pi^{-1}$.
Map
catalanization
Description
The catalanization of a permutation.
For a permutation $\sigma$, this is the product of the reflections corresponding to the inversions of $\sigma$ in lex-order.
A permutation is $231$-avoiding if and only if it is a fixpoint of this map. Also, for every permutation there exists an index $k$ such that the $k$-fold application of this map is $231$-avoiding.