Intervals of permutation class growth rates

TL;DR

This paper demonstrates that permutation class growth rates form an infinite sequence of intervals and include all values above a certain threshold, using analysis of non-integer base expansions to improve previous results.

Contribution

It establishes new intervals of permutation class growth rates and refutes a prior conjecture about their distribution below a specific value.

Findings

Identified an infinite sequence of growth rate intervals with a lower bound of approximately 2.35526.

Proved that all growth rates above approximately 2.35698 are attainable.

Extended results on expansions in non-integer bases to sets of allowed digits.

Abstract

We prove that the set of growth rates of permutation classes includes an infinite sequence of intervals whose infimum is $\theta_B\approx2.35526$, and that it also contains every value at least $\lambda_B\approx2.35698$. These results improve on a theorem of Vatter, who determined that there are permutation classes of every growth rate at least $\lambda_A\approx2.48187$. Thus, we also refute his conjecture that the set of growth rates below $\lambda_A$ is nowhere dense. Our proof is based upon an analysis of expansions of real numbers in non-integer bases, the study of which was initiated by R\'enyi in the 1950s. In particular, we prove two generalisations of a result of Pedicini concerning expansions in which the digits are drawn from sets of allowed values.