  Given permutations $\sigma \in S_k$ and $\pi \in S_n$ with $k<n$, the
\emph{pattern matching} problem is to decide whether $\pi$ matches $\sigma$ as
an order-isomorphic subsequence. We give a linear-time algorithm in case both
$\pi$ and $\sigma$ avoid the two size-$3$ permutations $213$ and $231$. For the
special case where only $\sigma$ avoids $213$ and $231$, we present a
$O(max(kn^2,n^2\log(\log(n)))$ time algorithm. We extend our research to
bivincular patterns that avoid $213$ and $231$ and present a $O(kn^4)$ time
algorithm. Finally we look at the related problem of the longest subsequence
which avoids $213$ and $231$.
