  Through applying Bell polynomials to the integral representation of the free
energy of the Ising model for the triangular and hexagonal lattices we obtain
the exact combinatorial formulas for the number of spin configurations at a
given energy (i.e. low-temperature series expansion of the partition function
or, alternatively, the number of states). We also generalize this approach to
the wider class of the (chequered) Utiyama graphs. Apart from the presented
exact formulas, our technique allows one to establish the correspondence
between the perfect gas of clusters and the Ising model on the lattices which
have positive coefficients in the low-temperature expansion (e.g. square
lattice, hexagonal lattice). However it is not always the case -- we present
that for the triangular lattice the coefficients could be negative and the
perfect gas of clusters interpretation is problematic.
