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

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 $n\in ̸\{3,4\}$ 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 $\mathrm{\mid }j-i\mathrm{\mid }\ge 2$ holds. We also exhibit the asymptotic behavior of these quantities as $n\to \mathrm{\infty }$ and $\theta \to \mathrm{\infty }$. 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 $n\to \mathrm{\infty }$, $\theta \to \mathrm{\infty }$ and $\theta \to 0$.