Avoidance of boxed mesh patterns on permutations

  • Authors:

  • Sergey Avgustinovich;Sergey Kitaev;Alexandr Valyuzhenich

  • Affiliations:

  • Sobolev Institute of Mathematics, 4 Acad. Koptyug Ave, 630090 Novosibirsk, Russia;Department of Computer and Information Sciences, University of Strathclyde, Livingstone Tower, 26 Richmond Street, Glasgow G1 1XH, United Kingdom;Novosibirsk State University, 2 Pirogova Street, 630090 Novosibirsk, Russia

  • Venue:
  • Discrete Applied Mathematics
  • Year:
  • 2013

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Abstract

We introduce the notion of a boxed mesh pattern and study avoidance of these patterns on permutations. We prove that the celebrated former Stanley-Wilf conjecture is not true for all but eleven boxed mesh patterns; for seven out of the eleven patterns the former conjecture is true, while we do not know the answer for the remaining four (length-four) patterns. Moreover, we prove that an analogue of a well-known theorem of Erdos and Szekeres does not hold for boxed mesh patterns of lengths larger than 2. Finally, we discuss enumeration of permutations avoiding simultaneously two or more length-three boxed mesh patterns, where we meet generalized Catalan numbers.