Longest Increasing Subsequences in Pattern-Restricted Permutations

Inspired by the results of Baik, Deift and Johansson on the limiting distribution of the lengths of the longest increasing subsequences in random permutations, we find those limiting distributions for pattern-restricted permutations in which the pattern is any one of the six patterns of length 3. We show that the (132)-avoiding case is identical to the distribution of heights of ordered trees, and that the (321)-avoiding case has interesting connections with a well known theorem of Erd\H os-Szekeres.

[1]  Eric Todd Quinto,et al.  Analysis, geometry, number theory : the mathematics of Leon Ehrenpreis , 2000 .

[2]  Robert G. Rieper,et al.  Continued Fractions and Catalan Problems , 2000, Electron. J. Comb..

[3]  Philippe Flajolet,et al.  Mellin Transforms and Asymptotics: Harmonic Sums , 1995, Theor. Comput. Sci..

[4]  Kendra Killpatrick,et al.  An Area-to-Inv Bijection Between Dyck Paths and 312-avoiding Permutations , 2001, Electron. J. Comb..

[5]  Ac Univie,et al.  COMMENT ON JASON BANDLOW AND KENDRA KILLPATRICK: “AN AREA-TO-INV BIJECTION BETWEEN DYCK PATHS AND 312-AVOIDING PERMUTATIONS” , .

[6]  C. Schensted.  Longest Increasing and Decreasing Subsequences , 1961, Canadian Journal of Mathematics.

[7]  Astrid Reifegerste.  On the diagram of 132-avoiding permutations , 2003, Eur. J. Comb..

[8]  Andrew Odlyzko,et al.  On Longest Increasing Subsequences in Random Permutations , 1998 .

[9]  T. Inui,et al.  The Symmetric Group , 1990 .

[10]  Philippe Flajolet,et al.  The Average Height of Binary Trees and Other Simple Trees , 1982, J. Comput. Syst. Sci..

[11]  J. Baik,et al.  On the distribution of the length of the longest increasing subsequence of random permutations , 1998, math/9810105.

[12]  A. Rényi,et al.  On the height of trees , 1967, Journal of the Australian Mathematical Society.

[13]  B. Logan,et al.  A Variational Problem for Random Young Tableaux , 1977 .

[14]  Julian West,et al.  Forbidden subsequences and Chebyshev polynomials , 1999, Discret. Math..

[15]  Markus Fulmek.  Enumeration of permutations containing a prescribed number of occurrences of a pattern of length three , 2003, Adv. Appl. Math..

[16]  Christian Krattenthaler,et al.  Permutations with Restricted Patterns and Dyck Paths , 2000, Adv. Appl. Math..

[17]  de Ng Dick Bruijn,et al.  THE AVERAGE HEIGHT OF PLANTED PLANE TREES , 1972 .