The Inversion Statistic in Derangements and in other Permutations with a Prescribed Number of Fixed Points

Abstract

We study how the inversion statistic is influenced by fixed points in a permutation. 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 $P_n^{(k)}(\sigma^{-1}_i<\sigma^{-1}_j)$, 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 $\frac{n(n-1)}4$. We can also let $k$ and the pair $i,j$ depend on $n$. As corollaries of the theorems we obtain the asymptotic behavior of the expected number of inversions under $P_n^{(k_n)}$ when $k_n\to\infty$, for various regimes of $k_n$, and the asymptotic behavior of $P_n^{(k_n)}(\sigma^{-1}_{i_n}<\sigma^{-1}_{j_n})$, for various regimes of $k_n$ and of $j_n-i_n$. The proofs make strategic and perhaps novel use of the Chinese restaurant construction for a uniformly random permutation.