A Combinatorial Theory of Assembly Systems via Generating Functions

This document presents a combinatorial framework for analyzing assembly systems using generating functions. We explore the theory through concrete examples, such as linear polymers, and develop recursive equations to characterize valid assemblies. The framework accommodates various levels of description, from atoms to complex molecular structures, and provides combinatorial interpretations for each generating function. This approach enables the study of equilibrium behavior and the computation of equilibrium probabilities in polymer ensembles.