Matchings in permutations

2026-05-06

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.

Problem

• permutations
• matchings
• intersecting permutations
• derangements
• extremal combinatorics

Innovation

• permutation families
• matchings
• intersecting permutations
• Hilton–Milner theorem
• derangements