George Pólya and Robert Redfield independently developed a theory of generating functions that describe the action of a group G on colorings of a set S by a set T when we know the action of G on S. Pó...George Pólya and Robert Redfield independently developed a theory of generating functions that describe the action of a group G on colorings of a set S by a set T when we know the action of G on S. Pólya’s work on the subject is very accessible in its exposition, and so the subject has become popularly known as Pólya theory, though Pólya-Redfield theory would be a better name. In this section we develop the elements of this theory.