# Intervals of permutation class growth rates

Research output: Research - peer-review › Article

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 λ_{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.Original language | English |
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Pages (from-to) | 279-303 |

Number of pages | 25 |

Journal | Combinatorica |

Volume | 38 |

Issue number | 2 |

Early online date | 1 Mar 2017 |

DOIs | |

State | Published - 30 Apr 2018 |

- permutation classes, growth rates, expansions in noninteger bases

### Research areas

## Publications

(1)## On the growth of permutation classes

Research output: Research › Doctoral Thesis