Algorithmic coincidence classification of mesh patterns

Bean, Christian; Guðmundsson, Bjarki Ágúst; Magnússon, Tómas Ken; Ulfarsson, Henning

doi:10.48550/arxiv.1910.08127

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2020

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“…Of particular interest in permutation patterns is the enumeration of the avoidance or containment class of particular permutations. Mesh patterns have been studied extensively since their introduction, see e.g., [AKV13,Ten13,CTU15,JKR15,BGMU19]. The first systematic study of the avoidance of mesh patterns was conducted in [HJS + 15], where enumeration results were given for the avoidance of 25 patterns of length 2.…”

Section: Introductionmentioning

confidence: 99%

Asymptotic Behaviour of the Containment of Certain Mesh Patterns

Govc¹,

Smith²

2020

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We present some results on the proportion of permutations of length n containing certain mesh patterns as n grows large, and give exact enumeration results in some cases. In particular, we focus on mesh patterns where entire rows and columns are shaded. We prove some general results which apply to mesh patterns of any length, and then consider mesh patterns of length four. An important consequence of these results is to show that the proportion of permutations containing a mesh pattern can take a wide range of values between 0 and 1.

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