High-Temperature Ising Partition Function and Related Noncrossing Polygons for the Simple Cubic Lattice

We have found the high-temperature expansion of the partition function for the simple-cubic lattice Ising problem up to the term involving u14, where u is the high-temperature variable tanhw/2kT. Comment is passed on an odd feature of the coefficients in this expansion and the corresponding expressions for the specific heat in terms of both u and w/2kT are presented. The calculations involve machine counts of the numbers pn of non-self-crossing lattice polygons; the method of obtaining these is described, and the value of p16 reported. The paper ends with a brief discussion of the trend of the numbers pn and of the closely related noncrossing chain numbers cn (using the data of Sykes).