Fast Algorithms for Finding Pattern Avoiders and Counting Pattern Occurrences in Permutations

TL;DR

This paper introduces efficient algorithms for generating and counting permutations avoiding specific patterns, significantly improving over previous exponential-time methods and enabling large-scale enumeration and analysis.

Contribution

The authors develop polynomial-time algorithms for pattern avoidance enumeration and counting, extending to vincular patterns and sets closed under standardization, with practical applications in permutation analysis.

Findings

Generated permutation counts for sizes 5 to 16 for complex pattern sets.

Discovered potentially novel pattern-avoidance conjectures.

Algorithms outperform previous exponential-time methods.

Abstract

Given a set $\Pi$ of permutation patterns of length at most $k$, we present an algorithm for building $S_{\le n}(\Pi)$, the set of permutations of length at most $n$ avoiding the patterns in $\Pi$, in time $O(|S_{\le n - 1}(\Pi)| \cdot k + |S_{n}(\Pi)|)$. Additionally, we present an $O(n!k)$-time algorithm for counting the number of copies of patterns from $\Pi$ in each permutation in $S_n$. Surprisingly, when $|\Pi| = 1$, this runtime can be improved to $O(n!)$, spending only constant time per permutation. Whereas the previous best algorithms, based on generate-and-check, take exponential time per permutation analyzed, all of our algorithms take time at most polynomial per outputted permutation. If we want to solve only the enumerative variant of each problem, computing $|S_{\le n}(\Pi)|$ or tallying permutations according to $\Pi$-patterns, rather than to store information about every permutation, then all of our algorithms can be implemented in $O(n^{k+1}k)$ space. Using our algorithms, we generated $|S_5(\Pi)|, \ldots, |S_{16}(\Pi)|$ for each $\Pi \subseteq S_4$ with $|\Pi| > 4$, and analyzed OEIS matches. We obtained a number of potentially novel pattern-avoidance conjectures. Our algorithms extend to considering permutations in any set closed under standardization of subsequences. Our algorithms also partially adapt to considering vincular patterns.