Generalizations of the Goulden-Jackson Cluster Method

TL;DR

This paper extends the Goulden-Jackson Cluster method to incorporate weighted probabilities and recursive approaches for generating functions that count words avoiding specific forbidden patterns, with implementations in Maple.

Contribution

It introduces several modifications to the original cluster method, including weighted and recursive versions, and provides practical Maple implementations.

Findings

Enhanced methods for weighted word avoidance analysis

Recursive approach improves computational efficiency

Maple package facilitates practical application

Abstract

We give several modifications of the Goulden-Jackson Cluster method for finding generating functions for words avoiding a given set of forbidden words. Our modifications include functions which can take into account various 'weights' on words, including single letter probability distributions, double letter (i.e. pairwise) probability distributions, and triple letter probability distributions. We also describe an alternative, recursive approach to finding such generating functions. We describe Maple implementations of the various modifications. The accompanying Maple package is available at the website for this paper.