A Combinatorial Theory of Assembly Systems via Generating Functions
Andrés Ortiz-Muñoz
TL;DR
The paper addresses the combinatorial explosion in assembly systems by developing a generating-function framework that yields exact recursions for valid assemblies and enables equilibrium computations. It first demonstrates the approach on linear polymers, deriving a species generating function, an assembly recursion, and expressions for concentrations and ensemble probabilities, then extends to polymers with rings to show how cycles modify generating functions and recursions. Key contributions include a coherent pipeline from syntax to equilibrium statistics, formal derivative and integral interpretations, and ensemble- level recursions grounded in combinatorial operations. The framework offers a flexible, principled way to analyze diverse assembly systems and provides a foundation for future work on more complex architectures and dynamics, with potential applications to biopolymer assembly and related molecular systems.
Abstract
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.