Identifier
Values
{{1}} => [1] => [1] => [1] => 0
{{1,2}} => [2,1] => [2,1] => [2,1] => 0
{{1},{2}} => [1,2] => [1,2] => [1,2] => 0
{{1,2,3}} => [2,3,1] => [3,2,1] => [3,2,1] => 0
{{1,2},{3}} => [2,1,3] => [2,1,3] => [2,1,3] => 0
{{1,3},{2}} => [3,2,1] => [2,3,1] => [2,3,1] => 1
{{1},{2,3}} => [1,3,2] => [1,3,2] => [1,3,2] => 0
{{1},{2},{3}} => [1,2,3] => [1,2,3] => [1,2,3] => 0
{{1,2,3,4}} => [2,3,4,1] => [4,3,2,1] => [4,3,2,1] => 0
{{1,2,3},{4}} => [2,3,1,4] => [3,2,1,4] => [3,2,1,4] => 0
{{1,2,4},{3}} => [2,4,3,1] => [3,4,2,1] => [3,4,2,1] => 1
{{1,2},{3,4}} => [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 0
{{1,2},{3},{4}} => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 0
{{1,3,4},{2}} => [3,2,4,1] => [2,4,3,1] => [2,4,3,1] => 2
{{1,3},{2,4}} => [3,4,1,2] => [4,1,3,2] => [4,1,3,2] => 0
{{1,3},{2},{4}} => [3,2,1,4] => [2,3,1,4] => [2,3,1,4] => 1
{{1,4},{2,3}} => [4,3,2,1] => [3,2,4,1] => [3,2,4,1] => 1
{{1},{2,3,4}} => [1,3,4,2] => [1,4,3,2] => [1,4,3,2] => 0
{{1},{2,3},{4}} => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 0
{{1,4},{2},{3}} => [4,2,3,1] => [2,3,4,1] => [2,3,4,1] => 2
{{1},{2,4},{3}} => [1,4,3,2] => [1,3,4,2] => [1,3,4,2] => 1
{{1},{2},{3,4}} => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 0
{{1},{2},{3},{4}} => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
{{1},{2,3,4,5}} => [1,3,4,5,2] => [1,5,4,3,2] => [1,5,4,3,2] => 0
{{1},{2,3,4},{5}} => [1,3,4,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => 0
{{1},{2,3,5},{4}} => [1,3,5,4,2] => [1,4,5,3,2] => [1,4,5,3,2] => 1
{{1},{2,3},{4,5}} => [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 0
{{1},{2,3},{4},{5}} => [1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 0
{{1},{2,4,5},{3}} => [1,4,3,5,2] => [1,3,5,4,2] => [1,3,5,4,2] => 2
{{1},{2,4},{3,5}} => [1,4,5,2,3] => [1,5,2,4,3] => [1,5,2,4,3] => 0
{{1},{2,4},{3},{5}} => [1,4,3,2,5] => [1,3,4,2,5] => [1,3,4,2,5] => 1
{{1},{2,5},{3,4}} => [1,5,4,3,2] => [1,4,3,5,2] => [1,4,3,5,2] => 1
{{1},{2},{3,4,5}} => [1,2,4,5,3] => [1,2,5,4,3] => [1,2,5,4,3] => 0
{{1},{2},{3,4},{5}} => [1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 0
{{1},{2,5},{3},{4}} => [1,5,3,4,2] => [1,3,4,5,2] => [1,3,4,5,2] => 2
{{1},{2},{3,5},{4}} => [1,2,5,4,3] => [1,2,4,5,3] => [1,2,4,5,3] => 1
{{1},{2},{3},{4,5}} => [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 0
{{1},{2},{3},{4},{5}} => [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
click to show known generating functions       
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
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
to permutation
Description
Sends the set partition to the permutation obtained by considering the blocks as increasing cycles.
Map
to signed permutation
Description
The signed permutation with all signs positive.