The Inversion Statistic in Derangements and in other Permutations with a Prescribed Number of Fixed Points
Ross G. Pinsky · The Electronic Journal of Combinatorics · 2026
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