2025/10/23 by Ross G. Pinsky, Pinsky, Ross G., Dominic T. Schickentanz +1
Computer Science · Mathematics · #05A05 #60C05 #Advanced Combinatorial Mathematics #Bayesian Methods and Mixture Models #Combinatorics (math.CO) #FOS: Mathematics #Probability (math.PR) #Random Matrices and Applications
paper · pdf · doi:10.48550/arxiv.2510.20654
openalex publication_date 2025/10/23 · openalex created_date 2025/10/25 · openalex updated_date 2026/07/28
In the first part of the paper, we study the inversion statistic of random permutations under the family (ℙθ(n))θ≥ 0 of Ewens sampling distributions on Sn. We obtain a rather simple exact formula for the expected number of inversions under ℙθ(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 ∈ \3,4\ only. Furthermore, we derive an exact formula for the probability that a specific pair of indices (i,j) ∈ \1,…,n\2 is inverted and show that this probability is decreasing in θ if and only if |j-i| ≥ 2 holds. We also exhibit the asymptotic behavior of these quantities as n → ∞ and θ→ ∞. In the second part of our paper, we analyze the inversion statistic of random permutations under~(ℙθ(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 → ∞, θ→ ∞ and θ→ 0.