Pattern matching in $(213, 231)$-avoiding permutations

Given permutations σ ∈ Sk and π ∈ Sn with k < n, the pattern matching problem is to decide whether π matches σ as an orderisomorphic 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 aO(max(kn, n log(log(n))) time algorithm. We extend our research to bivincular patterns that avoid 213 and 231 and present a O(kn) time algorithm. Finally we look at the related problem of the longest subsequence which avoids 213 and 231.