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
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 permutation
Description
Sends the set partition to the permutation obtained by considering the blocks as increasing cycles.
Map
to signed permutation
Description
The signed permutation with all signs positive.