Asymptotic enumeration of permutations avoiding generalized patterns

Motivated by the recent proof of the Stanley-Wilf conjecture, we study the asymptotic behavior of the number of permutations avoiding a generalized pattern. Generalized patterns allow the requirement that some pairs of letters must be adjacent in an occurrence of the pattern in the permutation, and consecutive patterns are a particular case of them. We determine the asymptotic behavior of the number of permutations avoiding a consecutive pattern, showing that they are an exponentially small proportion of the total number of permutations. For some other generalized patterns we give partial results, showing that the number of permutations avoiding them grows faster than for classical patterns but more slowly than for consecutive patterns.

[1]  Toufik Mansour,et al.  Restricted Motzkin permutations, Motzkin paths, continued fractions, and Chebyshev polynomials , 2005, Discret. Math..

[2]  Sergi Elizalde Torrent.  Consecutive patterns and statistics on restricted permutations , 2004 .

[3]  Miklós Bóna.  Exact Enumeration of 1342-Avoiding Permutations: A Close Link with Labeled Trees and Planar Maps , 1997, J. Comb. Theory, Ser. A.

[4]  Sergey Kitaev.  Partially ordered generalized patterns , 2005, Discret. Math..

[5]  M. Fekete.  Über die Verteilung der Wurzeln bei gewissen algebraischen Gleichungen mit ganzzahligen Koeffizienten , 1918 .

[6]  Toufik Mansour,et al.  Enumerating Permutations Avoiding A Pair Of Babson-Steingrimsson Patterns , 2005, Ars Comb..

[7]  Richard M. Wilson,et al.  A course in combinatorics , 1992 .

[8]  Noga Alon,et al.  On the Number of Permutations Avoiding a Given Pattern , 2000, J. Comb. Theory, Ser. A.

[9]  Gábor Tardos,et al.  Excluded permutation matrices and the Stanley-Wilf conjecture , 2004, J. Comb. Theory, Ser. A.

[10]  Anders Claesson.  GENERALISED PATTERN AVOIDANCE , 2000 .

[11]  Toufik Mansour.  Continued Fractions and Generalized Patterns , 2002, Eur. J. Comb..

[12]  Martin Klazar,et al.  The Füredi-Hajnal Conjecture Implies the Stanley-Wilf Conjecture , 2000 .

[13]  Richard Arratia,et al.  On the Stanley-Wilf Conjecture for the Number of Permutations Avoiding a Given Pattern , 1999, Electron. J. Comb..

[14]  Eric Babson,et al.  Generalized permutation patterns and a classification of the Mahonian statistics , 2000 .

[15]  Anders Claesson,et al.  Generalized Pattern Avoidance , 2001, Eur. J. Comb..

[16]  Miklós Bóna.  The Solution of a Conjecture of Stanley and Wilf for All Layered Patterns , 1999, J. Comb. Theory, Ser. A.

[17]  Sergey Kitaev.  Multi-avoidance of generalised patterns , 2003, Discret. Math..

[18]  Miklós Bóna.  The limit of a Stanley-Wilf sequence is not always rational, and layered patterns beat monotone patterns , 2005, J. Comb. Theory, Ser. A.

[19]  R. Warlimont,et al.  Permutations avoiding consecutive patterns, II , 2005 .

[20]  Marc Noy,et al.  Consecutive patterns in permutations , 2003, Adv. Appl. Math..

[21]  Amitai Regev,et al.  Asymptotic values for degrees associated with strips of young diagrams , 1981 .