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] => [3,1,2] => [3,1,2] => 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] => [4,1,2,3] => [4,1,2,3] => 0
{{1,2,3},{4}} => [2,3,1,4] => [3,1,2,4] => [3,1,2,4] => 0
{{1,2,4},{3}} => [2,4,3,1] => [3,4,1,2] => [3,4,1,2] => 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] => [2,4,1,3] => [2,4,1,3] => 1
{{1,3},{2,4}} => [3,4,1,2] => [3,1,4,2] => [3,1,4,2] => 1
{{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] => [1,4,2,3] => [1,4,2,3] => 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] => [1,5,2,3,4] => [1,5,2,3,4] => 0
{{1},{2,3,4},{5}} => [1,3,4,2,5] => [1,4,2,3,5] => [1,4,2,3,5] => 0
{{1},{2,3,5},{4}} => [1,3,5,4,2] => [1,4,5,2,3] => [1,4,5,2,3] => 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] => [1,3,5,2,4] => [1,3,5,2,4] => 1
{{1},{2,4},{3,5}} => [1,4,5,2,3] => [1,4,2,5,3] => [1,4,2,5,3] => 1
{{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] => [1,2,5,3,4] => [1,2,5,3,4] => 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
to signed permutation
Description
The signed permutation with all signs positive.
Map
to permutation
Description
Sends the set partition to the permutation obtained by considering the blocks as increasing cycles.
Map
inverse first fundamental transformation
Description
Let $\sigma = (i_{11}\cdots i_{1k_1})\cdots(i_{\ell 1}\cdots i_{\ell k_\ell})$ be a permutation given by cycle notation such that every cycle starts with its maximal entry, and all cycles are ordered increasingly by these maximal entries.
Maps $\sigma$ to the permutation $[i_{11},\ldots,i_{1k_1},\ldots,i_{\ell 1},\ldots,i_{\ell k_\ell}]$ in one-line notation.
In other words, this map sends the maximal entries of the cycles to the left-to-right maxima, and the sequences between two left-to-right maxima are given by the cycles.