Permutation Patterns in Latin Squares

TL;DR

This paper explores pattern avoidance in Latin Squares, providing enumeration, characterization, and generalizations of classical permutation results, and discusses open questions and techniques for future research.

Contribution

It introduces the study of pattern avoidance in Latin Squares, extending permutation pattern concepts to two dimensions and analyzing their structural implications.

Findings

Enumerated Latin Squares avoiding patterns of length three

Generalized Erd ext{"o}s-Szekeres theorem for Latin Squares

Identified equivalence classes among longer patterns

Abstract

In this paper we study pattern avoidance in Latin Squares, which gives us a two dimensional analogue of the well studied notion of pattern avoidance in permutations. Our main results include enumerating and characterizing the Latin Squares which avoid patterns of length three and a generalization of the Erd\H{o}s-Szekeres theorem. We also discuss equivalence classes among longer patterns, and conclude by describing open questions of interest both in light of pattern avoidance and their potential to reveal information about the structure of Latin Squares. Along the way, we show that classical results need not trivially generalize, and demonstrate techniques that may help answer future questions.