Intervals of permutation class growth rates

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We prove that the set of growth rates of permutation classes includes an infinite sequence of intervals whose infimum is θB ≈ 2.35526, and that it also contains every value at least λB ≈ 2.35698. These results improve on a theorem of Vatter, who determined that there are permutation classes of every growth rate at least λA ≈ 2.48187. Thus, we also refute his conjecture that the set of growth rates below λA 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.
Original languageEnglish
Pages (from-to)279-303
Number of pages25
Issue number2
Early online date1 Mar 2017
Publication statusPublished - 30 Apr 2018


  • permutation classes
  • growth rates
  • expansions in noninteger bases


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