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

TL;DR

This work analyzes the inversion statistic for random permutations under the Ewens sampling distribution, both unconditionally and conditioned on a prescribed number of fixed points. Using a Chinese restaurant construction, the authors derive exact formulas for the probability that a pair forms an inversion and for the total number of inversions, and they establish monotonicity and convexity properties with respect to the tilting parameter $\theta$. The paper extends these results to the conditional setting $D_{n,m}$, providing exact expressions and asymptotic behavior as $n\to\infty$ and as $\theta$ varies, with attention to parity effects in the conditioned regime. The results connect to broader themes in permutation statistics and population genetics, offering precise quantitative descriptions of inversions under Ewens tilting and fixed-point conditioning, and complementing known cycle-based analyses under Mallows-type models.

Abstract

In the first part of the paper, we study the inversion statistic of random permutations under the family $(\mathbb{P}_θ^{(n)})_{θ\ge 0}$ of Ewens sampling distributions on $S_n$. We obtain a rather simple exact formula for the expected number of inversions under $\mathbb{P}_θ^{(n)}$. In particular, we show that this expected number of inversions is decreasing in the tilting parameter $θ$ for any $n$ and that it is convex in $θ$ 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 $θ$ if and only if $|j-i| \ge 2$ holds. We also exhibit the asymptotic behavior of these quantities as $n \to \infty$ and $θ\to \infty$. In the second part of our paper, we analyze the inversion statistic of random permutations under~$(\mathbb{P}_θ^{(n)})_{θ> 0}$ 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 $n \to \infty$, $θ\to \infty$ and $θ\to 0$.

Theorems & Definitions (18)

• Theorem 1
• Corollary 2
• Remark 3
• Proposition 4
• Theorem 5
• Remark 6
• Remark 7
• Theorem 8
• Proposition 9
• proof : Proof of Theorem \ref{['thm:uncond']}