Cyclic permutations avoiding patterns in both one-line and cycle forms

Abstract

Let $\ddot{I}ƒ$ and $\ddot{I}„$ be patterns of length three; that is $\ddot{I}ƒ,\ddot{I}„\in \{123,132,213,231,312,321\}$. In this paper, we enumerate the set of cyclic permutations in ${\mathcal{S}}_n$ that avoid $\ddot{I}ƒ$ in their one-line notation and another pattern $\ddot{I}„$ in their cycle notation.