Abstract
This preliminary report considers all the clusters with n less than or equal to 10 and l less than or equal to 10 on a regular square lattice and uses the representations of the symmetric group to calculate their partition functions as a series in powers of the reciprocal temperature 1/T up to twentieth order. The contribution of these clusters to the free energy are computed for the infinite regular square lattice in zero external magnetic field. The contribution of the external field can be calculated exactly to any desired power in the reciprocal temperature. This calculation is carried out for the linear lattice. The complete contribution to twentieth order for an infinite square lattice must include graphs that cover remaining clusters up to n = 20 and l = 20. The contributions of these graphs will be found in a subsequent publication. However, the expression for the linear lattice is complete. It should be mentioned that the computations are completely free of rounding errors. 4 figs., 3 tables.
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