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

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.