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

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.