An application of the Goulden-Jackson cluster theorem

TL;DR

This paper applies the Goulden-Jackson cluster theorem to derive a reciprocal series representation for words avoiding certain subwords, linking combinatorial word enumeration with lattice M"obius functions.

Contribution

It introduces a novel application of the Goulden-Jackson cluster theorem to formal power series in noncommuting variables for word avoidance problems.

Findings

The sum of words avoiding subwords is the reciprocal of a series with coefficients 0, 1, or -1.

Establishes a connection between word avoidance series and lattice M"obius functions.

Provides a new perspective on combinatorial word enumeration using algebraic series.

Abstract

Let A be an alphabet and let F be a set of words with letters in A. We show that the sum of all words with letters in A with no consecutive subwords in F, as a formal power series in noncommuting variables, is the reciprocal of a series with all coefficients 0, 1 or -1. We also explain how this result is related to a result of Curtis Greene on lattices with M\"obius function 0, 1, or -1.