Growth rates of permutation classes: from countable to uncountable

Abstract

We establish 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. Central to the proof are various structural notions regarding generalized grid classes and a new property of permutation classes called concentration. The classification of growth rates up to ξ is completed in a subsequent paper.

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

“…In [20] it was proved that ξ « 2.30522 represents the phase transition from countably to uncountably many growth rates of permutation classes and that every growth rate below ξ is achieved by a sum closed class. Here we have used this result to determine the complete list of growth rates below ξ, establishing that ξ is the least accumulation point of growth rates from above, and showing that each of these growth rates is achieved by a finitely based class, while there are growth rates arbitrarily close to ξ which cannot be achieved by finitely based classes.…”

“…Given that Bevan [6] has shown that every real number at least λ B « 2.35698 is the growth rate of a permutation class, it is natural to try to close this gap of approximately 0.05176. The first challenge would be to extend the results of [20] to this range, i.e., to show that every growth rate between ξ and λ B is achieved by a sum closed class. More generally, [20] presents a conjecture that every growth rate of a permutation class is achieved by a sum closed class; this is known to be true for growth rates up to ξ by [20] and for growth rates between λ B and approximately 3.79 by [6,21].…”

“…Most recently, in the antecedent paper to this, Vatter [20] established that while there are uncountably many growth rates of permutation classes in every neighborhood of the algebraic integer ξ " the unique positive root of x 5´2 x 4´x2´x´1 « 2.30522, there are only countably many growth rates under ξ. In this work we completely determine these growth rates.…”

“…While our work builds on [20], we use only one of its results. To state this result we need a few definitions.…”

“…It follows from the supermultiplicative version of Fekete's Lemma that every sum closed permutation class has a proper growth rate (a fact first observed by Arratia [4]). The only result we use from [20] is the following. Theorem 1.1 (Vatter [20,Theorem 9.7]).…”