The Umbral Transfer-Matrix Method: V. The Goulden-Jackson Cluster Method for Infinitely Many Mistakes

This is the fifth, and last, installment of the saga on the Umbral Transfer-Matrix method, based on Gian-Carlo Rota’s seminal notion of the umbra. Here we extend the powerful GouldenJackson Cluster method, that enables one to compute generating functions enumerating words that avoid, as factors, any given finite set of “mistakes”, to the case where there are infinitely many mistakes. This infinite set of mistakes should be a union of finitely many finite-parameter families, given symbolically in frequency notation. We illustrate the method by introducing a new ‘toy model’ for self-avoiding walks, that is much more interesting and complex than finite-memory approximations, yet much simpler than the (probably intractable) “real thing”.