Upper bounds for the Stanley-Wilf limit of 1324 and other layered patterns

Authors:
Anders Claesson;VíT JelíNek;Einar SteingríMsson
Affiliations:
Department of Computer and Information Sciences, University of Strathclyde, Glasgow G1 1XH, UK;Computer Science Institute, Faculty of Mathematics and Physics, Charles University, Malostranské námstí 25, Prague 1, 118 00, Czech Republic;Department of Computer and Information Sciences, University of Strathclyde, Glasgow G1 1XH, UK
Venue:
Journal of Combinatorial Theory Series A
Year:
2012

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Abstract

We prove that the Stanley-Wilf limit of any layered permutation pattern of length @? is at most 4@?^2, and that the Stanley-Wilf limit of the pattern 1324 is at most 16. These bounds follow from a more general result showing that a permutation avoiding a pattern of a special form is a merge of two permutations, each of which avoids a smaller pattern. We also conjecture that, for any k=0, the set of 1324-avoiding permutations with k inversions contains at least as many permutations of length n+1 as those of length n. We show that if this is true then the Stanley-Wilf limit for 1324 is at most e^@p^2^/^3~13.001954.