StasinskiVollLength on SignedPermutation

Half the pairs -n <= i < j <= n of opposite parity with p(i) > p(j), where p(-k) = -p(k) and p(0) = 0.

On SignedPermutation

FindStat's St001823. The map (i, j) -> (-j, -i) pairs the counted pairs off, so counting those with i + j > 0, i = j - 1, j - 3, ..., 1 - j, gives the half.

Called as CombinatorialStat(x, "StasinskiVollLength"). 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 St001823

The Stasinski-Voll length of a signed permutation.

The Stasinski-Voll length of a signed permutation. The Stasinski-Voll length of a signed permutation $\sigma$ is

$$L(\sigma) = \frac{1}{2} \#\{(i,j) ~\mid -n \leq i < j \leq n,~ i \not\equiv j \operatorname{mod} 2,~ \sigma(i) > \sigma(j)\},$$

where $n$ is the size of $\sigma$.

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

Data from FindStat, the combinatorial statistic finder; the page at findstat.org/St001823 has the rest.