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Bell polynomials in the series expansions of the Ising model

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.