On two conjectures about pattern avoidance of cyclic permutations

Abstract

Let $π$ be a cyclic permutation that can be expressed in its one-line form as $π= π_1π_2 \cdot\cdot\cdot π_n$ and in its standard cycle form as $π= (c_1,c_2, ..., c_n)$ where $c_1=1$. Archer et al. introduced the notion of pattern avoidance of one-line and the standard cycle form for a cyclic permutation $π$, defined as both $π_1π_2 \cdot\cdot\cdot π_n$ and its standard cycle form $c_1c_2\cdot\cdot\cdot c_{n}$ avoiding a given pattern. Let $\mathcal{A}_n(σ_1,...,σ_k; τ)$ denote the set of cyclic permutations in the symmetric group $S_n$ that avoid each pattern of $\{σ_1,...,σ_k\}$ in their one-line forms and avoid $τ$ in their standard cycle forms. In this paper, we obtain some results about the cyclic permutations avoiding patterns in both one-line and cycle forms. In particular, we resolve two conjectures of Archer et al.