Identifier
Values
[.,.] => [1] => [1] => [1] => 0
[.,[.,.]] => [2,1] => [2,1] => [2,1] => 0
[[.,.],.] => [1,2] => [1,2] => [1,2] => 0
[.,[.,[.,.]]] => [3,2,1] => [3,2,1] => [3,2,1] => 0
[.,[[.,.],.]] => [2,3,1] => [3,1,2] => [3,1,2] => 0
[[.,.],[.,.]] => [1,3,2] => [1,3,2] => [1,3,2] => 0
[[.,[.,.]],.] => [2,1,3] => [2,1,3] => [2,1,3] => 0
[[[.,.],.],.] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[.,[.,[.,[.,.]]]] => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 0
[.,[.,[[.,.],.]]] => [3,4,2,1] => [4,3,1,2] => [4,3,1,2] => 1
[.,[[.,.],[.,.]]] => [2,4,3,1] => [4,1,3,2] => [4,1,3,2] => 0
[.,[[.,[.,.]],.]] => [3,2,4,1] => [4,2,1,3] => [4,2,1,3] => 0
[.,[[[.,.],.],.]] => [2,3,4,1] => [4,1,2,3] => [4,1,2,3] => 0
[[.,.],[.,[.,.]]] => [1,4,3,2] => [1,4,3,2] => [1,4,3,2] => 0
[[.,.],[[.,.],.]] => [1,3,4,2] => [1,4,2,3] => [1,4,2,3] => 0
[[.,[.,.]],[.,.]] => [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 0
[[[.,.],.],[.,.]] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 0
[[.,[.,[.,.]]],.] => [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 0
[[.,[[.,.],.]],.] => [2,3,1,4] => [3,1,2,4] => [3,1,2,4] => 0
[[[.,.],[.,.]],.] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 0
[[[.,[.,.]],.],.] => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 0
[[[[.,.],.],.],.] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[[.,.],[.,[.,[.,.]]]] => [1,5,4,3,2] => [1,5,4,3,2] => [1,5,4,3,2] => 0
[[.,.],[.,[[.,.],.]]] => [1,4,5,3,2] => [1,5,4,2,3] => [1,5,4,2,3] => 1
[[.,.],[[.,.],[.,.]]] => [1,3,5,4,2] => [1,5,2,4,3] => [1,5,2,4,3] => 0
[[.,.],[[.,[.,.]],.]] => [1,4,3,5,2] => [1,5,3,2,4] => [1,5,3,2,4] => 0
[[.,.],[[[.,.],.],.]] => [1,3,4,5,2] => [1,5,2,3,4] => [1,5,2,3,4] => 0
[[[.,.],.],[.,[.,.]]] => [1,2,5,4,3] => [1,2,5,4,3] => [1,2,5,4,3] => 0
[[[.,.],.],[[.,.],.]] => [1,2,4,5,3] => [1,2,5,3,4] => [1,2,5,3,4] => 0
[[[.,.],[.,.]],[.,.]] => [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 0
[[[[.,.],.],.],[.,.]] => [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 0
[[[.,.],[.,[.,.]]],.] => [1,4,3,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => 0
[[[.,.],[[.,.],.]],.] => [1,3,4,2,5] => [1,4,2,3,5] => [1,4,2,3,5] => 0
[[[[.,.],.],[.,.]],.] => [1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 0
[[[[.,.],[.,.]],.],.] => [1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 0
[[[[[.,.],.],.],.],.] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 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
inverse
Description
Sends a permutation to its inverse.
Map
to 312-avoiding permutation
Description
Return a 312-avoiding permutation corresponding to a binary tree.
The linear extensions of a binary tree form an interval of the weak order called the Sylvester class of the tree. This permutation is the minimal element of this Sylvester class.
Map
to signed permutation
Description
The signed permutation with all signs positive.