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.
We are grateful to David Bevan for his many insightful comments on this work.
These two programs constitute the computer proofs of Propositions 7.1 - 7.3 in Section 7. They require the PermPy library.