Pattern avoidance of $[4,k]$-pairs in circular permutations

TL;DR

This paper investigates the avoidance of specific pattern pairs in circular permutations, providing bounds on equivalence classes and characterizations for certain pattern pairs, advancing understanding of circular permutation pattern avoidance.

Contribution

It introduces bounds on Wilf equivalence classes for [4,k]-pairs and characterizes avoidance classes for [4,5]-pairs, extending prior work on pattern avoidance.

Findings

Upper bounds for Wilf equivalence classes of [4,k]-pairs.

Proved tight bounds for the pattern [1342].

Complete characterization of [4,5]-pair avoidance classes.

Abstract

The study of pattern avoidance in linear permutations has been an active area of research for almost half a century now, starting with the work of Knuth in 1973. More recently, the question of pattern avoidance in circular permutations has gained significant attention. In 2002-03, Callan and Vella independently characterized circular permutations avoiding a single permutation of size $4$. Building on their results, Domagalski et al. studied circular pattern avoidance for multiple patterns of size $4$. In this article, our main aim is to study circular pattern avoidance of $[4,k]$-pairs, i.e., circular permutations avoiding one pattern of size 4 and another of size $k$. We do this by using well-studied combinatorial objects to represent circular permutations avoiding a single pattern of size $4$. In particular, we obtain upper bounds for the number of Wilf equivalence classes of $[4,k]$-pairs. Moreover, we prove that the obtained bound is tight when the pattern of size $4$ in consideration is $[1342]$. Using ideas from our general results, we also obtain a complete characterization of the avoidance classes for $[4,5]$-pairs.