# Jay Pantone

## Growth Rates of Permutation Classes: Categorization up to the Uncountability Threshold

with Vincent Vatter

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.

• What real numbers between $$\xi$$ and $$\lambda_B$$ are growth rates of permutation classes?