The inversion statistic in derangements and in other permutations with a prescribed number of fixed points
Ross G. Pinsky
TL;DR
The work analyzes how the inversion statistic in a random permutation shifts when conditioning on a fixed-point structure, providing exact formulas for the mean and for fixed-point order probabilities under $P_n^{(k)}$. Using a Chinese restaurant process construction, the authors derive precise expressions for $E_n^{(k)}I_n$ and $P_n^{(k)}(\sigma_i^{-1}<\sigma_j^{-1})$ across all $k$, revealing a threshold at $k=1$ with a $+\tfrac{1}{12}$ correction and a linear-in-$n$ bias for $k=0$ and $k\ge2$. In particular, derangements ($k=0$) yield $E_n^{(0)}I_n=\frac{n(n-1)}{4}+\frac{n}{6}+\frac{1}{12}+O\left(\frac{1}{(n-1)!}\right)$, while for $k\ge2$, $E_n^{(k)}I_n=\frac{n(n-1)}{4}-\frac{k-1}{6}n-\frac{k^2-k-1}{12}+O\left(\frac{1}{(n-k)!}\right)$. The results illuminate how fixed-point constraints modulate inversion structure and connect fixed-point Poisson-$(1)$ behavior to inversion counts, with proofs relying on a CR-based, consistent permutation construction.
Abstract
We study how the inversion statistic is influenced by fixed points in a permutation. %The expected number of inversions in a uniformly random permutation in $S_n$ is $\frac{n(n-1)}4$. For each $n\in\mathbb{N}$, and each $k\in\{0,1,\cdots, n\}$, let $P_n^{(k)}$ denote the uniform probability measure on the set of permutations in $S_n$ with exactly $k$ fixed points. We obtain an exact formula for the expected number of inversions under the measure $P_n^{(k)}$ as well as for $P_n^{(k)}(σ^{-1}_i<σ^{-1}_j)$, for $1\le i<j\le n$, the $P_n^{(k)}$-probability that the number $i$ precedes the number $j$. In particular, up to a super-exponentially small correction as $n\to\infty$, the expected number of inversions in a random derangement $(k=0)$ is $\frac16n+\frac1{12}$ more than the value $\frac{n(n-1)}4$ that one obtains for a uniformly random general permutation in $S_n$. On the other hand, up to a super-exponentially small correction, for $k\ge2$, the expected number of inversions in a random permutation with $k$ fixed points is $\frac{k-1}6n+\frac{k^2-k-1}{12}$ less than $\frac{n(n-1)}4$. In the borderline case, $k=1$, up to a super-exponentially small correction, the expected number of inversions in a random permutation with one fixed point is $\frac1{12}$ more than $\frac{n(n-1)}4$. The proofs make strategic and perhaps novel use of the Chinese restaurant construction for a uniformly random permutation.