Intervals of Permutation Class Growth Rates
We prove that the set of growth rates of permutation classes includes an infinite sequence of intervals whose infimum is θ B ≈ 2:35526, and that it also contains every value at least θ B ≈ 2:35698. These results improve on a theorem of Vatter, who determined that there are permutation classes of every growth rate at least λ A ≈ 2:48187. Thus, we also refute his conjecture that the set of growth rates below λ 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.
Mathematics Subject Classification (2000)05A05 05A16
Unable to display preview. Download preview PDF.
- V. Komornik: Expansions in noninteger bases, Integers, 11B, 2011.Google Scholar
- V. Vatter: Permutation classes, in: The Handbook of Enumerative Combinatorics (Miklós Bóna, editor), CRC Press, 2015.Google Scholar
- Wolfram Research, Inc.: Mathematica, Version 10.0, www.wolfram.com/mathematica, 2014.Google Scholar