  In this paper, we provide the exact expression for the coefficients in the
low-temperature series expansion of the partition function of the
two-dimensional Ising model on the infinite square lattice. This is equivalent
to exact determination of the number of spin configurations at a given energy.
With these coefficients, we show that the ferromagnetic--to--paramagnetic phase
transition in the square lattice Ising model can be explained through
equivalence between the model and the perfect gas of energy clusters model, in
which the passage through the critical point is related to the complete change
in the thermodynamic preferences on the size of clusters. The combinatorial
approach reported in this article is very general and can be easily applied to
other lattice models.
