Maximum size of reverse-free sets of permutations

TL;DR

This paper investigates the maximum size of reverse-free sets of permutations and words, establishing bounds that relate to the structure of permutations and their reverse relations.

Contribution

It provides new bounds on the size of reverse-free sets of permutations and words, advancing understanding of their combinatorial limits.

Findings

Established bounds for F(n,k) in the case n >= k

Derived upper bounds for sets of permutations with reverse relations

Connected bounds to the maximum size of permutation sets with reverse pairs

Abstract

Two words have a reverse if they have the same pair of distinct letters on the same pair of positions, but in reversed order. A set of words no two of which have a reverse is said to be reverse-free. Let F(n,k) be the maximum size of a reverse-free set of words from [n]^k where no letter repeats within a word. We show the following lower and upper bounds in the case n >= k: F(n,k) \in n^k k^{-k/2 + O(k/log k)}. As a consequence of the lower bound, a set of n-permutations each two having a reverse has size at most n^{n/2 + O(n/log n)}.