On two conjectures about pattern avoidance of cyclic permutations

Abstract

Let $\ddot{I}€$ be a cyclic permutation that can be expressed in its one-line form as $\ddot{I}€=\ddot{I}{€}_1\ddot{I}{€}_2\cdot \cdot \cdot \ddot{I}{€}_n$ and in its standard cycle form as $\ddot{I}€=(c_1,c_2,...,c_n)$ 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 $\ddot{I}€$, defined as both $\ddot{I}{€}_1\ddot{I}{€}_2\cdot \cdot \cdot \ddot{I}{€}_n$ and its standard cycle form $c_1{c}_2\cdot \cdot \cdot c_n$ avoiding a given pattern. Let ${\mathcal{A}}_n(\ddot{I}{ƒ}_1,...,\ddot{I}{ƒ}_k;\ddot{I}„)$ denote the set of cyclic permutations in the symmetric group $S_n$ that avoid each pattern of $\{\ddot{I}{ƒ}_1,...,\ddot{I}{ƒ}_k\}$ in their one-line forms and avoid $\ddot{I}„$ 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.