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arXiv · 1111.5736

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

Anders Claesson·Vít Jelínek·Einar Steingrímsson
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Abstract

We prove that the Stanley-Wilf limit of any layered permutation pattern of length $\ell$ is at most $4\ell^2$, 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. If the conjecture is true that the maximum Stanley-Wilf limit for patterns of length $\ell$ is attained by a layered pattern then this implies an upper bound of $4\ell^2$ for the Stanley-Wilf limit of any pattern of length $\ell$. We also conjecture that, for any $k\ge 0$, 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 $e^{\pi\sqrt{2/3}} \simeq 13.001954$.

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Anders Claesson, Vít Jelínek, Einar Steingrímsson. 2011-11-24. Upper bounds for the Stanley-Wilf limit of 1324 and other layered patterns. https://doi.org/10.1016/j.jcta.2012.05.006

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