### Abstract

_{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 |
---|---|

Pages (from-to) | 279-303 |

Number of pages | 25 |

Journal | Combinatorica |

Volume | 38 |

Issue number | 2 |

Early online date | 1 Mar 2017 |

DOIs | |

Publication status | Published - 30 Apr 2018 |

### Fingerprint

### Keywords

- permutation classes
- growth rates
- expansions in noninteger bases

### Cite this

*Combinatorica*,

*38*(2), 279-303. https://doi.org/10.1007/s00493-016-3349-2

}

*Combinatorica*, vol. 38, no. 2, pp. 279-303. https://doi.org/10.1007/s00493-016-3349-2

**Intervals of permutation class growth rates.** / Bevan, David.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Intervals of permutation class growth rates

AU - Bevan, David

PY - 2018/4/30

Y1 - 2018/4/30

N2 - 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.

AB - 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.

KW - permutation classes

KW - growth rates

KW - expansions in noninteger bases

U2 - 10.1007/s00493-016-3349-2

DO - 10.1007/s00493-016-3349-2

M3 - Article

VL - 38

SP - 279

EP - 303

JO - Combinatorica

JF - Combinatorica

SN - 0209-9683

IS - 2

ER -