Hostname: page-component-7b7567c8c-lpg2g Total loading time: 0 Render date: 2026-11-15T16:43:17.233Z Has data issue: false hasContentIssue false
  • English
  • Français

PERMUTATION CLASSES OF EVERY GROWTH RATE ABOVE 2.48188

Part of:CombinatoricsOrdered sets

Published online by Cambridge University Press:  10 December 2009

Vincent Vatter
Vincent Vatter*
Affiliation:
Department of Mathematics, Dartmouth College, Hanover, NH 03755, U.S.A. (email: vincent.vatter@dartmouth.edu)

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

We prove that there are permutation classes (hereditary properties of permutations) of every growth rate (Stanley–Wilf limit) at least λ≈2.48187, the unique real root of x5−2x4−2x2−2x−1, thereby establishing a conjecture of Albert and Linton.

MSC classification

Secondary:05A05: Permutations, words, matrices05A15: Exact enumeration problems, generating functions06A06: Partial order, general

Information

Type
Research Article
Information
Mathematika , Volume 56 , Issue 1 , January 2010, pp. 182 - 192
Copyright
Copyright © University College London 2010

References

[1]Albert, M. H., On the length of the longest subsequence avoiding an arbitrary pattern in a random permutation. Random Structures Algorithms 31(2) (2007), 227–238.CrossRefGoogle Scholar
[2]Albert, M. H. and Atkinson, M. D., Simple permutations and pattern restricted permutations. Discrete Math. 300(1–3) (2005), 1–15.CrossRefGoogle Scholar
[3]Albert, M. H. and Linton, S., Growing at a perfect speed. Combin. Probab. Comput. 18 (2009), 301–308.CrossRefGoogle Scholar
[4]Arratia, R., On the Stanley–Wilf conjecture for the number of permutations avoiding a given pattern. Electron. J. Combin. 6 (1999), 4, Note, N1.CrossRefGoogle Scholar
[5]Balogh, J., Bollobás, B. and Morris, R., Hereditary properties of ordered graphs. In Topics in Discrete Mathematics (Algorithms and Combinatorics 26) (eds M. Klazar, J. Kratochvíl, M. Loebl, J. Matoušek, R. Thomas and P. Valtr), Springer (Berlin, 2006), 179–213.CrossRefGoogle Scholar
[6]Bollobás, B., Hereditary and monotone properties of combinatorial structures. In Surveys in Combinatorics 2007 (London Mathematical Society Lecture Note Series 346) (eds A. Hilton and J. Talbot), Cambridge University Press (Cambridge, 2007), 1–39.Google Scholar
[7]Flajolet, P. and Sedgewick, R., Analytic Combinatorics, Cambridge University Press (Cambridge, 2009).CrossRefGoogle Scholar
[8]Kaiser, T. and Klazar, M., On growth rates of closed permutation classes. Electron. J. Combin. 9(2) (2003), 20, Research paper 10.CrossRefGoogle Scholar
[9]Klazar, M., Overview of some general results in combinatorial enumeration. In Permutation Patterns, St Andrews 2007 (London Mathematical Society Lecture Note Series) (eds. S. Linton, N. Ruškuc and V. Vatter), Cambridge University Press (Cambridge), (to appear).Google Scholar
[10]Marcus, A. and Tardos, G., Excluded permutation matrices and the Stanley–Wilf conjecture. J. Combin. Theory Ser. A 107(1) (2004), 153–160.CrossRefGoogle Scholar
[11]Vatter, V., Small permutation classes. arXiv:0712.4006v2 [math.CO].Google Scholar