SignedSortingIndex on SignedPermutation

The sum, over the signed transpositions that sort p by moving n, then n - 1, and so on into place, of j - i - 1 when i < 0 and j - i otherwise.

On SignedPermutation

FindStat's St001821. The fold carries the total and the word, as SortingIndex does.

Called as CombinatorialStat(x, "SignedSortingIndex"). 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 St001821

The sorting index of a signed permutation.

The sorting index of a signed permutation. A signed permutation $\sigma = [\sigma(1),\ldots,\sigma(n)]$ can be sorted $[1,\ldots,n]$ by signed transpositions in the following way: First move $\pm n$ to its position and swap the sign if needed, then $\pm (n-1), \pm (n-2)$ and so on.

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

Data from FindStat, the combinatorial statistic finder; the page at findstat.org/St001821 has the rest.