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

Abstract: 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.