Bill Pletsch E_mail: bpletsch@twi.cc.nm.us Title: The Generation of Polya Polynomials and Some Experiments using Macsyma A theorem by George Polya can be used to determine the number of double cosets HgK in a group G that are generated by subgroups H and K. In order to apply the theory, the cycle polynomial of each group must be computed. This can be an extremely cumbersome process. In this paper, it will be shown how Polya polynomials of two classes of groups, the Young group and the first order wreath product group, can be generated by the CAS Macsyma. In addition, via the vehicle of the computer experiment, instances of great simpification have been discovered and will be presented.