The power group enumeration theorem

Abstract

AbstractThe number of orbits of a permutation group was determined in a fundamental result of Burnside. This was extended in a classical paper by Pólya to a solution of the problem of enumerating the equivalence classes of functions with a given weight from a set X into a set Y, subject to the action of the permutation group A acting on X. A generalization by de Bruijn solved the counting problem when two permutation groups are involved, A acting on X and B on Y. Thus the Pólya formula is the special case of the de Bruijn result in which B is the identity group. The Power Group Enumeration Theorem achieves the same result using only one permutation group: the power group, BA, acting on the set YX of functions. de Bruijn's method was used to count self-complementary graphs by R. C. Read and finite automata by M. Harrison. These results as well as the number of self-converse directed graphs and others are easily obtained by the proper use of the Power Group Enumeration Theorem