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arXiv · 2208.12845

Singleton mesh patterns in multidimensional permutations

Sergey Avgustinovich·Sergey Kitaev·Jeffrey Liese·Vladimir Potapov·Anna Taranenko
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Abstract

This paper introduces the notion of mesh patterns in multidimensional permutations and initiates a systematic study of singleton mesh patterns (SMPs), which are multidimensional mesh patterns of length 1. A pattern is avoidable if there exist arbitrarily large permutations that do not contain it. As our main result, we give a complete characterization of avoidable SMPs using an invariant of a pattern that we call its rank. We show that determining avoidability for a $d$-dimensional SMP $P$ of cardinality $k$ is an $O(d\cdot k)$ problem, while determining rank of $P$ is an NP-complete problem. Additionally, using the notion of a minus-antipodal pattern, we characterize SMPs which occur at most once in any $d$-dimensional permutation. Lastly, we provide a number of enumerative results regarding the distributions of certain general projective, plus-antipodal, minus-antipodal and hyperplane SMPs.

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Sergey Avgustinovich, Sergey Kitaev, Jeffrey Liese, Vladimir Potapov, Anna Taranenko. 2022-08-26. Singleton mesh patterns in multidimensional permutations. https://doi.org/10.1016/j.jcta.2023.105801

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