On a question about pattern avoidance of cyclic permutations

TL;DR

This paper resolves an open problem in pattern avoidance of cyclic permutations by providing explicit formulas for the case where the pattern is 1432, using structural analysis and Dilworth's theorem.

Contribution

It completes the classification of pattern avoidance for cyclic permutations avoiding a decreasing pattern and another pattern of length 4, specifically solving the case for 1432.

Findings

Explicit formulas for cyclic permutations avoiding 1432

Structural analysis of cycle forms

Application of Dilworth's theorem

Abstract

Recently, Archer et al.\ studied cyclic permutations that avoid the decreasing pattern $\delta_k=k(k-1)\cdots21$ in one-line notation and avoid another pattern $\tau$ of length $4$ in all their cycle forms. There are three cases in total to consider, namely, $\tau=1324, 1342$ and $1432$. They determined two of them, leaving the case $\tau=1432$ as an open question. In this paper, we resolve this case by deriving explicit formulas based on an analysis of the structure of cycle forms and an application of Dilworth's theorem.