Pin classes II: Small pin classes

TL;DR

This paper investigates the growth rates of pin permutation classes, revealing a phase transition at approximately 3.28277 and classifying pin classes below this threshold based on their structural properties.

Contribution

It identifies a critical growth rate threshold for pin classes and classifies all pin classes with growth rates below this value by their periodic structure.

Findings

Existence of a phase transition at growth rate ~3.28277

Uncountably many pin classes at the threshold growth rate

Pin classes below the threshold are characterized by periodic structures

Abstract

Pin permutations play an important role in the structural study of permutation classes, most notably in relation to simple permutations and well-quasi-ordering, and in enumerative consequences arising from these. In this paper, we continue our study of pin classes, which are permutation classes that comprise all the finite subpermutations contained in an infinite pin permutation. We show that there is a phase transition at $\mu\approx 3.28277$: there are uncountably many different pin classes whose growth rate is equal to $\mu$, yet only countably many below $\mu$. Furthermore, by showing that all pin classes with growth rate less than $\mu$ are essentially defined by pin permutations that possess a periodic structure, we classify the set of growth rates of pin classes up to $\mu$.