2025/10/10 by Luciano, Antoine
#Combinatorics (math.CO) #FOS: Mathematics #History and Overview (math.HO)
paper · doi:10.48550/arxiv.2510.21753
We give a concise historical background to Montmort's matching problem and its modern variants such as the hat-check problem, then develop a unified counting framework for fixed-point-free allocations. Using elementary recurrence and inclusion-exclusion arguments, we derive closed forms for derangements, rectangular injections, and partial l-matchings, and we combine them into a single formula. We also provide exact counts for the number of fixed points and Poisson limit laws. This note thus offers a compact, self-contained synthesis linking classical results with their two principal generalizations in a single scheme.