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

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