Growth rates of permutation classes: categorization up to the uncountability threshold

TL;DR

This paper characterizes the growth rates of permutation classes below a specific algebraic number, establishing their structure and the nature of accumulation points, with implications for finitely based classes.

Contribution

It provides a complete classification of growth rates less than the algebraic number , showing they are achieved by finitely based classes and clarifying accumulation point properties.

Findings

 is the least accumulation point from above of growth rates

All growth rates  or less are achieved by finitely based classes

Refutes Klazar's suggestion that  is an accumulation point of finitely based classes

Abstract

In the antecedent paper to this it was established that there is an algebraic number $\xi\approx 2.30522$ such that while there are uncountably many growth rates of permutation classes arbitrarily close to $\xi$, there are only countably many less than $\xi$. Here we provide a complete characterization of the growth rates less than $\xi$. In particular, this classification establishes that $\xi$ is the least accumulation point from above of growth rates and that all growth rates less than or equal to $\xi$ are achieved by finitely based classes. A significant part of this classification is achieved via a reconstruction result for sum indecomposable permutations. We conclude by refuting a suggestion of Klazar, showing that $\xi$ is an accumulation point from above of growth rates of finitely based permutation classes.