Large infinite antichains of permutations

TL;DR

This paper constructs infinite antichains of permutations with large growth rates, showing every proper permutation class is contained in one with a rational generating function, leveraging the Marcus-Tardos theorem.

Contribution

It introduces a method to construct infinite antichains with arbitrarily large growth rates and demonstrates their application to permutation class containment.

Findings

Existence of infinite antichains with arbitrarily large growth rates

Every proper permutation class is contained in a class with a rational generating function

Utilizes the Marcus-Tardos theorem in the proof

Abstract

Infinite antichains of permutations have long been used to construct interesting permutation classes and counterexamples. We prove the existence and detail the construction of infinite antichains with arbitrarily large growth rates. As a consequence, we show that every proper permutation class is contained in a class with a rational generating function. While this result implies the conclusion of the Marcus-Tardos theorem, that theorem is used in our proof.