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Avoidance of boxed mesh patterns on permutations

Avgustinovich, Sergey and Kitaev, Sergey and Valyuzhenich, Alexander (2013) Avoidance of boxed mesh patterns on permutations. Discrete Applied Mathematics, 161 (1-2). pp. 43-51. ISSN 0166-218X

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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 Erdős 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.