Counting Deranged Matchings

TL;DR

This paper proves a conjecture relating the ratio of perfect matchings in a complete r-partite graph minus a perfect matching to an exponential term, generalizing derangement counts and showing a Poisson distribution convergence.

Contribution

The paper proves a conjecture on the ratio of perfect matchings in complete r-partite graphs, extending derangement counting and establishing a Poisson convergence result.

Findings

Confirmed the conjecture for all r and n divisible by r.

Established Poisson distribution for the number of edges in common with a random perfect matching.

Unified derangement counts with matchings in complete multipartite graphs.

Abstract

Let $\mathrm{pm}(G)$ denote the number of perfect matchings of a graph $G$, and let $K_{r\times 2n/r}$ denote the complete $r$-partite graph where each part has size $2n/r$. Johnson, Kayll, and Palmer conjectured that for any perfect matching $M$ of $K_{r\times 2n/r}$, we have for $2n$ divisible by $r$ \[\frac{\mathrm{pm}(K_{r\times 2n/r}-M)}{\mathrm{pm}(K_{r\times 2n/r})}\sim e^{-r/(2r-2)}.\] 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 $K_{r\times 2n/r}$, then the number of edges that $R$ has in common with $M$ converges to a Poisson distribution with parameter $\frac{r}{2r-2}$.