On a conjecture about pattern avoidance of cycle permutations

Abstract: Let $\pi$ be a cycle permutation that can be expressed as one-line $\pi = \pi_1\pi_2 \cdot\cdot\cdot \pi_n$ and a cycle form $\pi = (c_1,c_2, ..., c_n)$. Archer et al. introduced the notion of pattern avoidance of one-line and all cycle forms for a cycle permutation $\pi$, defined as $\pi_1\pi_2 \cdot\cdot\cdot \pi_n$ and its arbitrary cycle form $c_ic_{i+1}\cdot\cdot\cdot c_nc_1c_2\cdot\cdot\cdot c_{i-1}$ avoid a given pattern. Let $\mathcal{A}^\circ_n(\sigma; \tau)$ denote the set of cyclic permutations in the symmetric group $S_n$ that avoid $\sigma$ in their one-line form and avoid $\tau$ in their all cycle forms. In this note, we prove that $|\mathcal{A}^\circ_n(2431; 1324)|$ is the $(n-1)^{\rm{st}}$ Pell number for any positive integer $n$. Thereby, we give a positive answer to a conjecture of Archer et al.