Result: On a Conjecture about Pattern Avoidance of Cyclic Permutations: On a conjecture about pattern avoidance of cyclic permutations

Let $π$ be a cycle permutation that can be expressed as one-line $π= π_1π_2 \cdot\cdot\cdot π_n$ and a cycle form $π= (c_1,c_2, ..., c_n)$. Archer et al. introduced the notion of pattern avoidance of one-line and all cycle forms for a cycle permutation $π$, defined as $π_1π_2 \cdot\cdot\cdot π_n$ 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 $\mathcal{A}^\circ_n(σ; τ)$ denote the set of cyclic permutations in the symmetric group $S_n$ that avoid $σ$ in their one-line form and avoid $τ$ 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.