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 ξ « 2.30522 such that while there are uncountably many growth rates of permutation classes arbitrarily close to ξ, there are only countably many less than ξ. Here we provide a complete characterization of the growth rates less than ξ. In particular, this classification establishes that ξ is the least accumulation point from above of growth rates and that all growth rates less than or equal to ξ 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 ξ is an accumulation point from above of growth rates of finitely based permutation classes.

Growth rates of permutation classes: from countable to uncountable

“…After establishing several delicate results about sum indecomposable permutations in a subsequent paper with Pantone , we are able to provide a complete characterization of growth rates under , leading to the number line shown in Figure of the introduction. Together with Bevan's Theorem , which shows that every real number greater than is the growth rate of a permutation class, this leaves us tantalizingly close to the complete characterization of growth rates of permutation classes — the gap between the two results is approximately 0.05176.…”

“…In order to extend the approach used here and in to complete this characterization, it seems that one would first want to establish the following. Conjecture Every upper growth rate of a permutation class is the growth rate of a sum closed class.…”

“…The proof of Theorem is completed in Section 9, after we have built up a collection of tools to analyze the classes of interest. In , we establish the complete list of growth rates between and , and in doing so establish that each of these growth rates is achieved by a finitely based class, whereas there are growth rates arbitrarily close to which cannot be achieved by finitely based classes (because there are uncountably many such growth rates). This characterization and Bevan's Theorem leaves us tantalizingly close to the complete determination of the set of possible growth rates of permutation classes, as shown in Figure .…”

“…The set of all growth rates of permutation classes between 2 and 2.5, as it stands after the characterization in Pantone and Vatter .…”

“…In Section 9, we show that all growth rates below are growth rates of sum closed classes. Thus, our characterization of these growth rates in reduces to a characterization of the possible sequences of sum indecomposable classes, though that effort is far from trivial. It is not a coincidence that is the growth rate of a sum closed class whose sum indecomposable members have the enumeration (where denotes an infinite sequence of 4s), though it is perhaps a coincidence that the same growth rate is achieved by a sum closed class whose sum indecomposable members have the enumeration 1,1,2,4,3,3,2,1.…”