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] => [1,3,2] => [1,3,2] => 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] => [4,1,3,2] => [4,1,3,2] => 0
{{1,2},{3,4}} => [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 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] => [4,2,1,3] => [4,2,1,3] => 1
{{1,3},{2,4}} => [3,4,1,2] => [3,4,1,2] => [3,4,1,2] => 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] => [1,4,2,3] => [1,4,2,3] => 2
{{1},{2,3},{4}} => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 1
{{1,4},{2},{3}} => [4,2,3,1] => [4,2,3,1] => [4,2,3,1] => 0
{{1},{2,4},{3}} => [1,4,3,2] => [1,4,3,2] => [1,4,3,2] => 1
{{1},{2},{3,4}} => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 2
{{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] => 3
{{1},{2,3,4},{5}} => [1,3,4,2,5] => [1,4,2,3,5] => [1,4,2,3,5] => 2
{{1},{2,3,5},{4}} => [1,3,5,4,2] => [1,5,2,4,3] => [1,5,2,4,3] => 2
{{1},{2,3},{4,5}} => [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 3
{{1},{2,3},{4},{5}} => [1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 1
{{1},{2,4,5},{3}} => [1,4,3,5,2] => [1,5,3,2,4] => [1,5,3,2,4] => 3
{{1},{2,4},{3,5}} => [1,4,5,2,3] => [1,4,5,2,3] => [1,4,5,2,3] => 2
{{1},{2,4},{3},{5}} => [1,4,3,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => 1
{{1},{2,5},{3,4}} => [1,5,4,3,2] => [1,5,4,3,2] => [1,5,4,3,2] => 2
{{1},{2},{3,4,5}} => [1,2,4,5,3] => [1,2,5,3,4] => [1,2,5,3,4] => 4
{{1},{2},{3,4},{5}} => [1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 2
{{1},{2,5},{3},{4}} => [1,5,3,4,2] => [1,5,3,4,2] => [1,5,3,4,2] => 1
{{1},{2},{3,5},{4}} => [1,2,5,4,3] => [1,2,5,4,3] => [1,2,5,4,3] => 2
{{1},{2},{3},{4,5}} => [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 3
{{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 alignments of type EN of a signed permutation.
An alignment of type EN of a signed permutation π∈Hn is a pair −n≤i≤j≤n, i,j≠0, such that one of the following conditions hold:
  • $-i < 0 < -\pi(i) < \pi(j) < j$
  • $i \leq\pi(i) < \pi(j) < j$.
Map
to permutation
Description
Sends the set partition to the permutation obtained by considering the blocks as increasing cycles.
Map
inverse
Description
Sends a permutation to its inverse.
Map
to signed permutation
Description
The signed permutation with all signs positive.