Inversions in Random Permutations Under the Ewens Sampling Distribution With and Without a Prescribed Number of Fixed Points

TL;DR

This paper derives exact and asymptotic formulas for the inversion statistic in random permutations under Ewens distributions, analyzing effects of parameters and fixed points.

Contribution

It provides new exact formulas and asymptotic analysis for inversions in Ewens-distributed permutations, including fixed point conditioning.

Findings

Expected inversions decrease with increasing theta.

Probability of inversion for distant pairs decreases with theta.

Asymptotic behaviors vary with parameters and fixed points.

Abstract

In the first part of the paper, we study the inversion statistic of random permutations under the family $(\mathbb{P}_\theta^{(n)})_{\theta \ge 0}$ of Ewens sampling distributions on $S_n$. We obtain a rather simple exact formula for the expected number of inversions under $\mathbb{P}_\theta^{(n)}$. In particular, we show that this expected number of inversions is decreasing in the tilting parameter $\theta$ for any $n$ and that it is convex in $\theta$ for $n \not \in \{3,4\}$ only. Furthermore, we derive an exact formula for the probability that a specific pair of indices $(i,j) \in \{1,\dots,n\}^2$ is inverted and show that this probability is decreasing in $\theta$ if and only if $|j-i| \ge 2$ holds. We also exhibit the asymptotic behavior of these quantities as $n \to \infty$ and $\theta \to \infty$. In the second part of our paper, we analyze the inversion statistic of random permutations under~$(\mathbb{P}_\theta^{(n)})_{\theta > 0}$ conditioned on having a prescribed number of fixed points. Again, we obtain exact formulas for the expected number of inversions and for the probability that a specific pair of indices is inverted. Since, as expected, the resulting formulas are rather complicated, we focus on the asymptotic behavior of these quantities as $n \to \infty$, $\theta \to \infty$ and $\theta \to 0$.