The goal of this paper is to develop extensions of Polya enumeration methods which count orbits of functions. De Bruijn, Harary, and Palmer all worked on this problem and created generalizations which involve permuting the codomain and domain of functions simultaneously. We cover their results and specifically extend them to the case where the group of permutations need not be a direct product of groups. In this situation, we develop a way of breaking the orbits into subclasses based on a characteristic of the functions involved. Additionally, we develop a formula for the number of orbits made up of bijective functions. As a final extension, we also expand the set we are acting on to be the set of all relations between finite sets. Then we show how to count the orbits of relations.
College and Department
Physical and Mathematical Sciences; Mathematics
BYU ScholarsArchive Citation
Green, Shawn Jeffrey, "Extensions of the Power Group Enumeration Theorem" (2019). Theses and Dissertations. 7526.
Polya enumeration, De Bruijn, power group, cycle index