Intervals of permutation class growth rates

Abstract

We prove that the set of growth rates of permutation classes includes an infinite sequence of intervals whose infimum is ${\theta }_B\approx 2.35526$, and that it also contains every value at least ${\lambda }_B\approx 2.35698$. These results improve on a theorem of Vatter, who determined that there are permutation classes of every growth rate at least ${\lambda }_A\approx 2.48187$. 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 Rényi 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.

20 pages, 10 figures, ancillary files containing computer-aided calculations included

References in corpus (2)

• Small permutation classes
• Permutation classes of every growth rate above 2.48188