Computational Approaches to Consecutive Pattern Avoidance in Permutations

TL;DR

This paper introduces two computational methods for counting permutations avoiding specific consecutive patterns, along with a Maple package, a Wilf-equivalence theorem, and tools for asymptotic analysis.

Contribution

The paper presents two novel algorithms for consecutive pattern avoidance, a Wilf-equivalence theorem, and a Maple package for practical implementation and analysis.

Findings

Algorithms effectively count pattern-avoiding permutations.

The Wilf-equivalence theorem provides a sufficient condition for pattern set equivalence.

The CAV package aids in approximating asymptotic constants.

Abstract

In recent years, there has been increasing interest in consecutive pattern avoidance in permutations. In this paper, we introduce two approaches to counting permutations that avoid a set of prescribed patterns consecutively. These algoritms have been implemented in the accompanying Maple package CAV, which can be downloaded from the author's website. As a byproduct of the first algorithm, we have a theorem giving a sufficient condition for when two pattern sets are strongly (consecutively) Wilf-Equivalent. For the implementation of the second algorithm, we define the cluster tail generating function and show that it always satisfies a certain functional equation. We also explain how the CAV package can be used to approximate asymptotic constants for single pattern avoidance.