Matchings in permutations
This is a theoretical contribution to extremal combinatorics, providing structural results for permutation families with forbidden matchings.
The paper studies families of permutations with no matchings of size s, characterizing the largest such families and proving a Hilton–Milner type result, with additional results for derangements.
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.