Exact expression for the number of states in lattice models

We derive a closed-form combinatorial expression for the number of 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. Apart from elementary examples including the Poisson, geometric and negative binomial probability distributions for the energy, the framework is also validated against the one-dimensional Ising model.