A generalized Goulden-Jackson cluster method and lattice path enumeration

TL;DR

This paper extends the Goulden-Jackson cluster method to monoid networks and applies it to enumerate Motzkin paths with various subword occurrence statistics, providing new generating function formulas.

Contribution

It introduces a generalized cluster method for monoid networks, broadening the combinatorial framework beyond free monoids.

Findings

Derived closed-form generating functions for Motzkin paths

Obtained continued fraction representations of generating functions

Counted paths with bounded and unbounded height using subword statistics

Abstract

The Goulden-Jackson cluster method is a powerful tool for obtaining generating functions for counting words in a free monoid by occurrences of a set of subwords. We introduce a generalization of the cluster method for monoid networks, which generalize the combinatorial framework of free monoids. As a sample application of the generalized cluster method, we compute bivariate and multivariate generating functions counting Motzkin paths---both with height bounded and unbounded---by statistics corresponding to the number of occurrences of various subwords, yielding both closed-form and continued fraction formulae.