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] => [3,-2,1] => 0
{{1,2},{3}} => [2,1,3] => [2,1,3] => [-2,1,3] => 0
{{1,3},{2}} => [3,2,1] => [3,2,1] => [-2,-3,1] => 1
{{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] => [2,3,4,1] => [3,4,-2,1] => 0
{{1,2,3},{4}} => [2,3,1,4] => [2,3,1,4] => [3,-2,1,4] => 0
{{1,2,4},{3}} => [2,4,3,1] => [2,4,3,1] => [3,-2,-4,1] => 2
{{1,2},{3,4}} => [2,1,4,3] => [2,1,4,3] => [-2,1,-4,3] => 2
{{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] => [-2,4,-3,1] => 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] => [-2,-3,1,4] => 1
{{1,4},{2,3}} => [4,3,2,1] => [4,3,2,1] => [-2,-3,-4,1] => 3
{{1},{2,3,4}} => [1,3,4,2] => [1,3,4,2] => [1,4,-3,2] => 1
{{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] => [2,3,-4,1] => 2
{{1},{2,4},{3}} => [1,4,3,2] => [1,4,3,2] => [1,-3,-4,2] => 3
{{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] => [1,3,4,5,2] => [1,4,5,-3,2] => 1
{{1},{2,3,4},{5}} => [1,3,4,2,5] => [1,3,4,2,5] => [1,4,-3,2,5] => 1
{{1},{2,3,5},{4}} => [1,3,5,4,2] => [1,3,5,4,2] => [1,4,-3,-5,2] => 4
{{1},{2,3},{4,5}} => [1,3,2,5,4] => [1,3,2,5,4] => [1,-3,2,-5,4] => 4
{{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] => [1,4,3,5,2] => [1,-3,5,-4,2] => 3
{{1},{2,4},{3},{5}} => [1,4,3,2,5] => [1,4,3,2,5] => [1,-3,-4,2,5] => 3
{{1},{2,5},{3,4}} => [1,5,4,3,2] => [1,5,4,3,2] => [1,-3,-4,-5,2] => 6
{{1},{2},{3,4,5}} => [1,2,4,5,3] => [1,2,4,5,3] => [1,2,5,-4,3] => 2
{{1},{2},{3,4},{5}} => [1,2,4,3,5] => [1,2,4,3,5] => [1,2,-4,3,5] => 2
{{1},{2},{3,5},{4}} => [1,2,5,4,3] => [1,2,5,4,3] => [1,2,-4,-5,3] => 5
{{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 nesting alignments of a signed permutation.
A nesting alignment of a signed permutation $\pi\in\mathfrak H_n$ is a pair $1\leq i, j \leq n$ such that
  • $-i < -j < -\pi(j) < -\pi(i)$, or
  • $-i < j \leq \pi(j) < -\pi(i)$, or
  • $i < j \leq \pi(j) < \pi(i)$.
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
Foata-Han
Description
The Foata-Han bijection for signed permutations.
This map sends the flag major index St001433The flag major index of a signed permutation. to the flag inversion number St001428The number of B-inversions of a signed permutation..