On a conjecture about pattern avoidance of cycle permutations

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 $\pi =(c_1,c_2,\mathrm{.}\mathrm{.}\mathrm{.},c_n)$. 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 $c_i{c}_{i+1}\cdot \cdot \cdot c_n{c}_1{c}_2\cdot \cdot \cdot c_{i-1}$ avoid a given pattern. Let ${\mathcal{A}}_n^{\circ }(\sigma ;\tau )$ 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 $\mathrm{\mid }{\mathcal{A}}_n^{\circ }(2431;1324)\mathrm{\mid }$ is the $(n-1{)}^{\mathrm{s}\mathrm{t}}$ Pell number for any positive integer $n$. Thereby, we give a positive answer to a conjecture of Archer et al.