Derivation of low-temperature expansions for Ising model. II. General theory

The enumeration problem that arises in the derivation of low-temperature and high-field expansions for the Ising model of a ferromagnet and antiferromagnet is studied. The method of partial generating functions (complete codes) is developed and a principle of complete code balance is explicitly stated. The detailed application of the method to a number of lattices is described and substitutions given that interpret the generating functions of certain lattices on the corresponding shadow lattice. It is shown that in zero-field and two dimensions some of these substitutions reduce to the well-known star triangle and magnetic-moment results.