Counting deranged matchings

Abstract

Let pm(G) denote the number of perfect matchings of a graph G, and let Kr×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 Kr×2n/r, we have for 2n divisible by rpm(Kr×2n/r−M)pm(Kr×2n/r)∼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 K2n. We prove this conjecture. In fact, we prove the stronger result that if R is a uniformly random perfect matching of Kr×2n/r, then the number of edges that R has in common with M converges to a Poisson distribution with parameter r2r−2.