On Inversions in Random Permutations Under the Ewens Sampling Distribution

Abstract: We study the inversion statistic of random permutations under the family~$(\p_\theta){\theta \ge 0}$ of Ewens sampling distributions on~$S_n$. We obtain an exact formula for the expected number of inversions under~$\p\theta$. In particular, we show that this expected number of inversions is decreasing in the tilting parameter~$\theta$ and that it is convex in~$\theta$ for~$n \ge 5$ only. Further, 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~$\abs j-i\abs \ge 2$ holds.