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

TL;DR

This paper develops efficient algorithms for pattern matching in permutations avoiding the patterns 213 and 231, including extensions to bivincular patterns and longest subsequence problems.

Contribution

It introduces linear and polynomial-time algorithms for pattern matching in specific pattern-avoiding permutations, extending to bivincular patterns and longest subsequence analysis.

Findings

Linear-time algorithm for pattern matching when both permutations avoid 213 and 231

Polynomial-time algorithms for special and bivincular pattern cases

Results on longest subsequence avoiding 213 and 231

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 $O(max(kn^2,n^2\log(\log(n)))$ 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 $231$.