Singleton mesh patterns in multidimensional permutations

TL;DR

This paper gives a complete characterization of avoidable SMPs using an invariant of a pattern that is called its rank, and shows that determining avoidability for a d-dimensional SMP of cardinality k is an O(d\cdot k) problem, while determining rank of P is an NP-complete problem.

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.

Figures (2)

• Figure 1: The graph of $\pi = 471569283$
• Figure 2: The graph of $\Pi = (12534,51243)$

Theorems & Definitions (41)

• Definition 1
• Definition 2
• Remark 1
• Proposition 2
• proof
• Proposition 3
• proof
• Theorem 4
• Lemma 5
• proof