On a question about pattern avoidance of cyclic permutations

Abstract

Recently, Archer et al.\ studied cyclic permutations that avoid the decreasing pattern $\hat{I}{´}_k=k(k-1)\cdots 21$ in one-line notation and avoid another pattern $\ddot{I}„$ of length $4$ in all their cycle forms. There are three cases in total to consider, namely, $\ddot{I}„=1324,1342$ and $1432$. They determined two of them, leaving the case $\ddot{I}„=1432$ 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.

12 pages