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Matching statistic: St001433
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Mapping
St001433: Signed permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 0
[-1] => 1
[1,2] => 0
[1,-2] => 3
[-1,2] => 1
[-1,-2] => 4
[2,1] => 2
[2,-1] => 3
[-2,1] => 1
[-2,-1] => 2
[1,2,3] => 0
[1,2,-3] => 5
[1,-2,3] => 3
[1,-2,-3] => 8
[-1,2,3] => 1
[-1,2,-3] => 6
[-1,-2,3] => 4
[-1,-2,-3] => 9
[1,3,2] => 4
[1,3,-2] => 5
[1,-3,2] => 3
[1,-3,-2] => 4
[-1,3,2] => 5
[-1,3,-2] => 6
[-1,-3,2] => 4
[-1,-3,-2] => 5
[2,1,3] => 2
[2,1,-3] => 7
[2,-1,3] => 3
[2,-1,-3] => 8
[-2,1,3] => 1
[-2,1,-3] => 6
[-2,-1,3] => 2
[-2,-1,-3] => 7
[2,3,1] => 4
[2,3,-1] => 5
[2,-3,1] => 3
[2,-3,-1] => 4
[-2,3,1] => 5
[-2,3,-1] => 6
[-2,-3,1] => 4
[-2,-3,-1] => 5
[3,1,2] => 2
[3,1,-2] => 7
[3,-1,2] => 3
[3,-1,-2] => 8
[-3,1,2] => 1
[-3,1,-2] => 6
[-3,-1,2] => 2
[-3,-1,-2] => 7
Description
The flag major index of a signed permutation. The flag major index of a signed permutation $\sigma$ is: $$\operatorname{fmaj}(\sigma)=\operatorname{neg}(\sigma)+2\cdot \sum_{i\in \operatorname{Des}_B(\sigma)}{i} ,$$ where $\operatorname{Des}_B(\sigma)$ is the $B$-descent set of $\sigma$; see [1, Eq.(10)]. This statistic is equidistributed with the $B$-inversions ([[St001428]]) and with the negative major index on the groups of signed permutations (see [1, Corollary 4.6]).
Matching statistic: St001428
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Mapping
Mp00190: Signed permutations —Foata-Han⟶ Signed permutations
St001428: Signed permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => 0
[-1] => [-1] => 1
[1,2] => [1,2] => 0
[1,-2] => [1,-2] => 3
[-1,2] => [-1,2] => 1
[-1,-2] => [-1,-2] => 4
[2,1] => [-2,1] => 2
[2,-1] => [-2,-1] => 3
[-2,1] => [2,1] => 1
[-2,-1] => [2,-1] => 2
[1,2,3] => [1,2,3] => 0
[1,2,-3] => [1,2,-3] => 5
[1,-2,3] => [1,-2,3] => 3
[1,-2,-3] => [1,-2,-3] => 8
[-1,2,3] => [-1,2,3] => 1
[-1,2,-3] => [-1,2,-3] => 6
[-1,-2,3] => [-1,-2,3] => 4
[-1,-2,-3] => [-1,-2,-3] => 9
[1,3,2] => [1,-3,2] => 4
[1,3,-2] => [3,1,-2] => 5
[1,-3,2] => [-3,1,2] => 3
[1,-3,-2] => [1,3,-2] => 4
[-1,3,2] => [-1,-3,2] => 5
[-1,3,-2] => [3,-1,-2] => 6
[-1,-3,2] => [-3,-1,2] => 4
[-1,-3,-2] => [-1,3,-2] => 5
[2,1,3] => [-2,1,3] => 2
[2,1,-3] => [-2,1,-3] => 7
[2,-1,3] => [-2,-1,3] => 3
[2,-1,-3] => [-2,-1,-3] => 8
[-2,1,3] => [2,1,3] => 1
[-2,1,-3] => [2,1,-3] => 6
[-2,-1,3] => [2,-1,3] => 2
[-2,-1,-3] => [2,-1,-3] => 7
[2,3,1] => [3,-2,1] => 4
[2,3,-1] => [3,-2,-1] => 5
[2,-3,1] => [3,2,1] => 3
[2,-3,-1] => [3,2,-1] => 4
[-2,3,1] => [-3,-2,1] => 5
[-2,3,-1] => [-3,-2,-1] => 6
[-2,-3,1] => [-3,2,1] => 4
[-2,-3,-1] => [-3,2,-1] => 5
[3,1,2] => [3,1,2] => 2
[3,1,-2] => [1,-3,-2] => 7
[3,-1,2] => [3,-1,2] => 3
[3,-1,-2] => [-1,-3,-2] => 8
[-3,1,2] => [1,3,2] => 1
[-3,1,-2] => [-3,1,-2] => 6
[-3,-1,2] => [-1,3,2] => 2
[-3,-1,-2] => [-3,-1,-2] => 7
Description
The number of B-inversions of a signed permutation. The number of B-inversions of a signed permutation $\sigma$ of length $n$ is $$ \operatorname{inv}_B(\sigma) = \big|\{ 1 \leq i < j \leq n \mid \sigma(i) > \sigma(j) \}\big| + \big|\{ 1 \leq i \leq j \leq n \mid \sigma(-i) > \sigma(j) \}\big|, $$ see [1, Eq. (8.2)]. According to [1, Eq. (8.4)], this is the Coxeter length of $\sigma$.