On a question about pattern avoidance of cyclic permutations

Recently, Archer et al.\ studied cyclic permutations that avoid the decreasing pattern ${\delta }_k=k(k-1)\cdots 21$ 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.