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

Abstract

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 $O(max(k{n}^2,n^2\log (\log (n)))$ time algorithm. We extend our research to bivincular patterns that avoid $213$ and $231$ and present a $O(k{n}^4)$ time algorithm. Finally we look at the related problem of the longest subsequence which avoids $213$ and $231$.