FlagMajorIndex on SignedPermutation

The number of negative entries plus twice the sum of the descents, a negative first entry counting as a descent at 0.

On SignedPermutation

The descent at 0 adds 0 to the sum, so only the descents between entries weigh.

Called as CombinatorialStat(x, "FlagMajorIndex"). The head's page has its examples.

Live

Distribution

Over SignedPermutations at each size, tallied here in your browser; sizes past 6 are not offered.

Sizes 1 to

FindStat St001433

The flag major index of a signed permutation.

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)].

Object	Value
[1]	0
[-1]	1
[1,2]	0
[2,1]	2
[-1,2]	1
[-2,1]	1
[1,-2]	3
[2,-1]	3

Data from FindStat, the combinatorial statistic finder; the page at findstat.org/St001433 has the rest.