Given permutations σ ∈ S k and π ∈ Sn with k < n, the pattern matching problem is to decide whether σ occurs π as an order-isomorphic subsequence. We give a linear-time algorithm in case both π and σ avoid the two size-3 permutations 213 and 231. For the special case where only σ 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.