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arXiv · 1410.3679

Intervals of permutation class growth rates

David Bevan
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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.

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David Bevan. 2014-10-14. Intervals of permutation class growth rates. https://arxiv.org/abs/1410.3679

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