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.

本文介绍了多维排列中网格模式的概念，并启动了对单例网格模式（SMP）的系统研究，单例网格模式是长度为 1 的多维网格模式。如果存在不包含该模式的任意大排列，则该模式是可以避免的。作为我们的主要结果，我们使用我们称之为等级的模式不变量给出了可避免的 SMP 的完整特征。我们证明，确定基数为k的d维 SMP P的可避免性是$氧（d\cdot k）$问题，而确定P的秩是一个 NP 完全问题。此外，使用负对映模式的概念，我们描述了在任何d维排列中最多出现一次的 SMP 。最后，我们提供了一些关于某些一般射影、正对映体、负对映体和超平面 SMP 分布的枚举结果。