TotalSquaredDisplacement on Permutation

The sum of (p(i) - i)^2.

On Permutation

Twice InversionSum, though neither definition shows it. FindStat calls it Spearman's rho of p against the identity.

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

Live

Distribution

Over SymmetricGroup at each size, tallied here in your browser; sizes past 8 are not offered.

Sizes 1 to

FindStat St000828

The spearman's rho of a permutation and the identity permutation.

The spearman's rho of a permutation and the identity permutation. This is, for a permutation $\pi$ of $n$, given by $\sum_{i=1}^n (\pi(i)−i)^2$. This is also twice the inversion sum of a permutation (St000055).

Object	Value
[1]	0
[1,2]	0
[2,1]	2
[1,2,3]	0
[1,3,2]	2
[2,1,3]	2
[2,3,1]	6
[3,1,2]	6

Data from FindStat, the combinatorial statistic finder; the page at findstat.org/St000828 has the rest.