Partition functions encode all the thermodynamics of a system, but for most systems of practical importance, they cannot be calculated exactly. In this work we present a new hierarchical method for calculating partition functions to arbitrary precision. We discuss the algorithmic details of our implementation, including elements of shape-matching and entropy calculation for on-lattice and off-lattice systems. We highlight computational trade-offs between speed and accuracy, showing how this varies with temperature, and demonstrate its utility in studying nanoscale self-assembly for a system of model patchy particles.