Exact expression for the number of energy states in lattice models

We derive a closed-form combinatorial expression for the number of energy states in canonical systems with discrete energy levels. The expression results from the exact low-temperature power series expansion of the partition function. The approach provides interesting insights into basis of statistical mechanics. In particular, it is shown that in some cases the logarithm of the partition function may be considered the generating function for the number of internal states of energy clusters, which characterize system's microscopic configurations. Insights provided by the method allow one to understand the circumstances under which the widespread distributions for the energy, such as the Poisson and exponential distributions, arise. Apart from elementary examples, the framework is validated against the one-dimensional Ising model in zero field.