Matchings in permutations

TL;DR

This paper investigates large families of permutations that avoid certain matchings, providing characterizations and Hilton-Milner type results, especially focusing on derangements.

Contribution

It offers a characterization of the largest s-matching-free permutation families and extends results to derangements, advancing combinatorial understanding.

Findings

Characterization of largest s-matching-free families

Hilton--Milner type results for permutation families

Results specific to derangements

Abstract

We say that two permutations $[n]\to [n]$ intersect if they map some element $x$ to the same element $y$. A matching in a family of permutations is a collection of pairwise disjoint permutations. In this paper, we study families of permutations with no matchings of size $s$. In particular, we obtain a characterization of the largest $s$-matching-free families and a Hilton--Milner type result. We also obtain results for the families of derangements.