Counting Deranged Matchings

Let $\mathrm{p}\mathrm{m}(G)$ denote the number of perfect matchings of a graph $G$, and let $K_{r\times 2n\mathrm{/}r}$ denote the complete $r$-partite graph where each part has size $2n\mathrm{/}r$. Johnson, Kayll, and Palmer conjectured that for any perfect matching $M$ of $K_{r\times 2n\mathrm{/}r}$, we have for $2n$ divisible by $r$ $$\frac{\mathrm{p}\mathrm{m}(K_{r\times 2n\mathrm{/}r}-M)}{\mathrm{p}\mathrm{m}(K_{r\times 2n\mathrm{/}r})}\sim e^{-r\mathrm{/}(2r-2)}\mathrm{.}$$ This conjecture can be viewed as a common generalization of counting the number of derangements on $n$ letters, and of counting the number of deranged matchings of $K_{2n}$. We prove this conjecture. In fact, we prove the stronger result that if $R$ is a uniformly random perfect matching of $K_{r\times 2n\mathrm{/}r}$, then the number of edges that $R$ has in common with $M$ converges to a Poisson distribution with parameter $\frac{r}{2r-2}$.