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Identifier

Mp00080: Set partitions —to permutation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
Mp00190: Signed permutations —Foata-Han⟶ Signed permutations
St001866: Signed permutations ⟶ ℤ

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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$F_{1} = 1$

$F_{2} = 2$

$F_{3} = 3 + 2\ q$

$F_{4} = 5 + 4\ q + 4\ q^{2} + 2\ q^{3}$

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 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..

Map

to signed permutation

Description

The signed permutation with all signs positive.