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] => [2,3,1] => [2,3,1] => 1
{{1,2},{3}} => [2,1,3] => [2,1,3] => [2,1,3] => 0
{{1,3},{2}} => [3,2,1] => [3,2,1] => [3,2,1] => 0
{{1},{2,3}} => [1,3,2] => [3,1,2] => [3,1,2] => 0
{{1},{2},{3}} => [1,2,3] => [1,2,3] => [1,2,3] => 0
{{1,2,3,4}} => [2,3,4,1] => [2,3,4,1] => [2,3,4,1] => 2
{{1,2,3},{4}} => [2,3,1,4] => [2,3,1,4] => [2,3,1,4] => 1
{{1,2,4},{3}} => [2,4,3,1] => [4,2,3,1] => [4,2,3,1] => 1
{{1,2},{3,4}} => [2,1,4,3] => [2,4,1,3] => [2,4,1,3] => 1
{{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] => [3,2,4,1] => [3,2,4,1] => 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] => [3,2,1,4] => [3,2,1,4] => 0
{{1,4},{2,3}} => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 0
{{1},{2,3,4}} => [1,3,4,2] => [3,4,1,2] => [3,4,1,2] => 1
{{1},{2,3},{4}} => [1,3,2,4] => [3,1,2,4] => [3,1,2,4] => 0
{{1,4},{2},{3}} => [4,2,3,1] => [2,4,3,1] => [2,4,3,1] => 2
{{1},{2,4},{3}} => [1,4,3,2] => [4,3,1,2] => [4,3,1,2] => 0
{{1},{2},{3,4}} => [1,2,4,3] => [4,1,2,3] => [4,1,2,3] => 0
{{1},{2},{3},{4}} => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
{{1},{2},{3},{4},{5}} => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
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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 Foata bijection
Description
The inverse of Foata's bijection.
See Mp00067Foata bijection.