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] => [3,2,1] => [3,2,1] => 0
{{1},{2,3}} => [1,3,2] => [2,3,1] => [2,3,1] => 1
{{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] => [4,2,3,1] => [4,2,3,1] => 0
{{1,2},{3,4}} => [2,1,4,3] => [3,2,4,1] => [3,2,4,1] => 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] => [4,2,1,3] => [4,2,1,3] => 0
{{1,3},{2,4}} => [3,4,1,2] => [1,4,2,3] => [1,4,2,3] => 0
{{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] => [2,4,1,3] => [2,4,1,3] => 1
{{1},{2,3},{4}} => [1,3,2,4] => [2,3,1,4] => [2,3,1,4] => 1
{{1,4},{2},{3}} => [4,2,3,1] => [4,1,3,2] => [4,1,3,2] => 0
{{1},{2,4},{3}} => [1,4,3,2] => [3,4,2,1] => [3,4,2,1] => 1
{{1},{2},{3,4}} => [1,2,4,3] => [2,3,4,1] => [2,3,4,1] => 2
{{1},{2},{3},{4}} => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
{{1,2,4},{3,5}} => [2,4,5,1,3] => [1,3,5,2,4] => [1,3,5,2,4] => 1
{{1,3,4},{2,5}} => [3,5,4,1,2] => [1,5,3,4,2] => [1,5,3,4,2] => 0
{{1,3},{2,4},{5}} => [3,4,1,2,5] => [1,4,2,3,5] => [1,4,2,3,5] => 0
{{1,4},{2,3,5}} => [4,3,5,1,2] => [1,5,3,2,4] => [1,5,3,2,4] => 0
{{1,4},{2,5},{3}} => [4,5,3,1,2] => [1,5,4,2,3] => [1,5,4,2,3] => 1
{{1,4},{2},{3,5}} => [4,2,5,1,3] => [1,4,5,2,3] => [1,4,5,2,3] => 2
{{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 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 permutation
Description
Sends the set partition to the permutation obtained by considering the blocks as increasing cycles.
Map
major-index to inversion-number bijection
Description
Return the permutation whose Lehmer code equals the major code of the preimage.
This map sends the major index to the number of inversions.
Map
to signed permutation
Description
The signed permutation with all signs positive.