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] => [-3,1,2] => 0
{{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] => [-4,-3,-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,-4,-2,1] => 1
{{1,2},{3,4}} => [2,1,4,3] => [2,1,4,3] => [-4,-2,1,3] => 0
{{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] => 2
{{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] => [-3,2,-4,1] => 1
{{1},{2,3,4}} => [1,3,4,2] => [1,3,4,2] => [-4,-3,1,2] => 0
{{1},{2,3},{4}} => [1,3,2,4] => [1,3,2,4] => [-3,1,2,4] => 0
{{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] => [3,-4,1,2] => 1
{{1},{2},{3,4}} => [1,2,4,3] => [1,2,4,3] => [-4,1,2,3] => 0
{{1},{2},{3},{4}} => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
{{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 occurrences of the signed pattern 1-2 in a signed permutation.
This is the number of pairs $1\leq i < j\leq n$ such that $0 < \pi(i) < -\pi(j)$.
Map
to permutation
Description
Sends the set partition to the permutation obtained by considering the blocks as increasing cycles.
Map
Foata-Han inverse
Description
The inverse of the Foata-Han bijection for signed permutations.
This map sends the the flag inversion number St001428The number of B-inversions of a signed permutation. to the flag major index St001433The flag major index of a signed permutation.
Map
to signed permutation
Description
The signed permutation with all signs positive.