Derangements and Generalizations: A Counting Note on the Matching Problem
History and Overview
2025-10-28 v1 Combinatorics
Abstract
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.
Keywords
Cite
@article{arxiv.2510.21753,
title = {Derangements and Generalizations: A Counting Note on the Matching Problem},
author = {Antoine Luciano},
journal= {arXiv preprint arXiv:2510.21753},
year = {2025}
}