A generalized Goulden-Jackson cluster method and lattice path enumeration

The Goulden-Jackson cluster method is a powerful tool for obtaining generating functions 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 formulas.