Identifier
Values
[.,.] => [1] => [1] => [1] => 0
[.,[.,.]] => [2,1] => [2,1] => [2,1] => 0
[[.,.],.] => [1,2] => [1,2] => [1,2] => 0
[.,[.,[.,.]]] => [3,2,1] => [3,1,2] => [3,1,2] => 0
[.,[[.,.],.]] => [2,3,1] => [3,2,1] => [3,2,1] => 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] => [4,1,2,3] => [4,1,2,3] => 0
[.,[.,[[.,.],.]]] => [3,4,2,1] => [4,1,3,2] => [4,1,3,2] => 0
[.,[[.,.],[.,.]]] => [2,4,3,1] => [4,2,1,3] => [4,2,1,3] => 0
[.,[[.,[.,.]],.]] => [3,2,4,1] => [4,3,2,1] => [4,3,2,1] => 0
[.,[[[.,.],.],.]] => [2,3,4,1] => [4,2,3,1] => [4,2,3,1] => 1
[[.,.],[.,[.,.]]] => [1,4,3,2] => [1,4,2,3] => [1,4,2,3] => 0
[[.,.],[[.,.],.]] => [1,3,4,2] => [1,4,3,2] => [1,4,3,2] => 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] => [3,1,2,4] => [3,1,2,4] => 0
[[.,[[.,.],.]],.] => [2,3,1,4] => [3,2,1,4] => [3,2,1,4] => 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] => [1,5,2,3,4] => [1,5,2,3,4] => 0
[[.,.],[.,[[.,.],.]]] => [1,4,5,3,2] => [1,5,2,4,3] => [1,5,2,4,3] => 0
[[.,.],[[.,.],[.,.]]] => [1,3,5,4,2] => [1,5,3,2,4] => [1,5,3,2,4] => 0
[[.,.],[[.,[.,.]],.]] => [1,4,3,5,2] => [1,5,4,3,2] => [1,5,4,3,2] => 0
[[.,.],[[[.,.],.],.]] => [1,3,4,5,2] => [1,5,3,4,2] => [1,5,3,4,2] => 1
[[[.,.],.],[.,[.,.]]] => [1,2,5,4,3] => [1,2,5,3,4] => [1,2,5,3,4] => 0
[[[.,.],.],[[.,.],.]] => [1,2,4,5,3] => [1,2,5,4,3] => [1,2,5,4,3] => 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] => [1,4,2,3,5] => [1,4,2,3,5] => 0
[[[.,.],[[.,.],.]],.] => [1,3,4,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => 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 occurrences of a type-B 231 pattern in a signed permutation.
For a signed permutation $\pi\in\mathfrak H_n$, a triple $-n \leq i < j < k\leq n$ is an occurrence of the type-B $231$ pattern, if $1 \leq j < k$, $\pi(i) < \pi(j)$ and $\pi(i)$ is one larger than $\pi(k)$, i.e., $\pi(i) = \pi(k) + 1$ if $\pi(k) \neq -1$ and $\pi(i) = 1$ otherwise.
Map
first fundamental transformation
Description
Return the permutation whose cycles are the subsequences between successive left to right maxima.
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.