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Derangements and Generalizations: A Counting Note on the Matching Problem

Antoine Luciano

TL;DR

The paper studies the matching problem and its generalizations by unifying fixed-point-free allocations through recurrence and inclusion-exclusion. It derives closed forms for derangements, rectangular injections, and partial $\ell$-matchings, and it analyzes the distribution of fixed points, establishing Poisson limit laws in natural asymptotic regimes. The results connect classic results (derangements, hat-check problem) with two principal generalizations in a single counting framework, highlighting structural parallels across models. This unified treatment provides exact counts, distributional insights, and asymptotic behavior useful for applications in combinatorics and probability theory.

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.

Derangements and Generalizations: A Counting Note on the Matching Problem

TL;DR

The paper studies the matching problem and its generalizations by unifying fixed-point-free allocations through recurrence and inclusion-exclusion. It derives closed forms for derangements, rectangular injections, and partial -matchings, and it analyzes the distribution of fixed points, establishing Poisson limit laws in natural asymptotic regimes. The results connect classic results (derangements, hat-check problem) with two principal generalizations in a single counting framework, highlighting structural parallels across models. This unified treatment provides exact counts, distributional insights, and asymptotic behavior useful for applications in combinatorics and probability theory.

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.
Paper Structure (11 sections, 45 equations, 2 figures)

This paper contains 11 sections, 45 equations, 2 figures.

Table of Contents

  1. Initial Problem: Permutations and Derangements
  2. Recursive Approach
  3. Inclusion–Exclusion Approach
  4. Distribution of the Number of Fixed Points (Encounter Numbers)
  5. First Generalization: Rectangular Case — Fixed-Point-Free Injections
  6. Inclusion–Exclusion Approach (Poincaré Sieve)
  7. Distribution of the Number of Fixed Points (Encounter Numbers)
  8. Second Generalization: Partial Matchings — Fixed-Point-Free $\ell$-Matchings
  9. Inclusion–Exclusion Approach (Poincaré Sieve)
  10. Distribution of the Number of Fixed Points (Encounter Numbers)
  11. Unified Framework: Rectangular $\ell$-Matchings

Figures (2)

  • Figure 1: Engraving by Sébastien Leclerc from the Preface of Pierre Rémond de Montmort's Essay d'Analyse sur les Jeux de Hazarddemontmort1708essay.
  • Figure 2: Illustration of the general case: $n$ people (left), $m$ hats (right), and $\ell$ matchings (here: $n=5$, $m=7$, $\ell=3$).