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.

This is a preview of subscription content, access via your institution.


  1. [1]

    M. H. Albert and S. A. Linton: Growing at a perfect speed, Combin. Probab. Comput. 18 (2009), 301–308.

    MathSciNet  Article  MATH  Google Scholar 

  2. [2]

    M. H. Albert, N. Ruškuc and V. Vatter: In ations of geometric grid classes of permutations, Israel J. Math. 205 (2015), 73–108.

    MathSciNet  Article  MATH  Google Scholar 

  3. [3]

    J. Balogh, B. Bollobás and R. Morris: Hereditary properties of ordered graphs, in: Topics in Discrete Mathematics, volume 26 of Algorithms Combin., 179–213. Springer, 2006.

    Google Scholar 

  4. [4]

    D. Bevan: Calculating intervals of permutation class growth rates,, 2015.

    Google Scholar 

  5. [5]

    P. Flajolet and R. Sedgewick: Analytic Combinatorics, Cambridge University Press, 2009.

    Google Scholar 

  6. [6]

    S. Huczynska and V. Vatter: Grid classes and the Fibonacci dichotomy for restricted permutations, Electron. J. Combin. 13 Research paper 54, 2006.

    MathSciNet  MATH  Google Scholar 

  7. [7]

    T. Kaiser and M. Klazar: On growth rates of closed permutation classes, Electron. J. Combin. 9 Research paper 10, 2003.

    MathSciNet  MATH  Google Scholar 

  8. [8]

    M. Klazar: On the least exponential growth admitting uncountably many closed permutation classes, Theoret. Comput. Sci. 321 (2004), 271–281.

    MathSciNet  Article  MATH  Google Scholar 

  9. [9]

    M. Klazar: Some general results in combinatorial enumeration, in: Permutation Patterns, volume 376 of London Math. Soc. Lecture Note Ser., 3–40, Cambridge Univ. Press, 2010.

    Google Scholar 

  10. [10]

    V. Komornik: Expansions in noninteger bases, Integers, 11B, 2011.

  11. [11]

    A. Marcus and G. Tardos: Excluded permutation matrices and the Stanley-Wilf conjecture, J. Combin. Theory Ser. A 107 (2004), 153–160.

    MathSciNet  Article  MATH  Google Scholar 

  12. [12]

    M. Pedicini: Greedy expansions and sets with deleted digits, Theoret. Comput. Sci. 332 (2005), 313–336.

    MathSciNet  Article  MATH  Google Scholar 

  13. [13]

    A. Rényi: Representations for real numbers and their ergodic properties, Acta Math. Acad. Sci. Hungar. 8 (1957), 477–493.

    MathSciNet  Article  MATH  Google Scholar 

  14. [14]

    V. Vatter: Permutation classes of every growth rate above 2.48188, Mathematika 56 (2010), 182–192.

    MathSciNet  Article  MATH  Google Scholar 

  15. [15]

    V. Vatter: Small permutation classes, Proc. Lond. Math. Soc. 103 (2011), 879–921.

    MathSciNet  Article  MATH  Google Scholar 

  16. [16]

    V. Vatter: Permutation classes, in: The Handbook of Enumerative Combinatorics (Miklós Bóna, editor), CRC Press, 2015.

    Google Scholar 

  17. [17]

    Wolfram Research, Inc.: Mathematica, Version 10.0,, 2014.

Download references

Author information



Corresponding author

Correspondence to David Bevan.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Bevan, D. Intervals of Permutation Class Growth Rates. Combinatorica 38, 279–303 (2018).

Download citation

Mathematics Subject Classification (2000)

  • 05A05
  • 05A16