Identifier
Values
{{1}} => [1] => [1] => [1] => 0
{{1,2}} => [2,1] => [1,2] => [1,2] => 0
{{1},{2}} => [1,2] => [2,1] => [2,1] => 0
{{1,2,3}} => [2,3,1] => [3,1,2] => [3,1,2] => 0
{{1,2},{3}} => [2,1,3] => [3,2,1] => [3,2,1] => 0
{{1,3},{2}} => [3,2,1] => [1,2,3] => [1,2,3] => 0
{{1},{2,3}} => [1,3,2] => [2,1,3] => [2,1,3] => 0
{{1},{2},{3}} => [1,2,3] => [2,3,1] => [2,3,1] => 1
{{1,2,3,4}} => [2,3,4,1] => [3,4,1,2] => [3,4,1,2] => 2
{{1,2,3},{4}} => [2,3,1,4] => [3,4,2,1] => [3,4,2,1] => 1
{{1,2,4},{3}} => [2,4,3,1] => [3,1,2,4] => [3,1,2,4] => 0
{{1,2},{3,4}} => [2,1,4,3] => [3,2,1,4] => [3,2,1,4] => 0
{{1,2},{3},{4}} => [2,1,3,4] => [3,2,4,1] => [3,2,4,1] => 1
{{1,3,4},{2}} => [3,2,4,1] => [4,3,1,2] => [4,3,1,2] => 1
{{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] => [4,3,2,1] => [4,3,2,1] => 0
{{1,4},{2,3}} => [4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 0
{{1},{2,3,4}} => [1,3,4,2] => [2,4,1,3] => [2,4,1,3] => 1
{{1},{2,3},{4}} => [1,3,2,4] => [2,4,3,1] => [2,4,3,1] => 1
{{1,4},{2},{3}} => [4,2,3,1] => [1,4,2,3] => [1,4,2,3] => 0
{{1},{2,4},{3}} => [1,4,3,2] => [2,1,3,4] => [2,1,3,4] => 0
{{1},{2},{3,4}} => [1,2,4,3] => [2,3,1,4] => [2,3,1,4] => 1
{{1},{2},{3},{4}} => [1,2,3,4] => [2,3,4,1] => [2,3,4,1] => 2
{{1,5},{2,3,4}} => [5,3,4,2,1] => [1,5,2,3,4] => [1,5,2,3,4] => 0
{{1,5},{2,3},{4}} => [5,3,2,4,1] => [1,5,4,2,3] => [1,5,4,2,3] => 1
{{1,5},{2,4},{3}} => [5,4,3,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,5},{2},{3,4}} => [5,2,4,3,1] => [1,4,2,3,5] => [1,4,2,3,5] => 0
{{1,5},{2},{3},{4}} => [5,2,3,4,1] => [1,4,5,2,3] => [1,4,5,2,3] => 2
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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
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
Lehmer code rotation
Description
Sends a permutation $\pi$ to the unique permutation $\tau$ (of the same length) such that every entry in the Lehmer code of $\tau$ is cyclically one larger than the Lehmer code of $\pi$.