Hi. Do you know if there exist algebraic studies of the ring of the power series which emerge when using the theory of Polya for enumeration of sets with certain symmetries? For instance if some ideals have nice properties, and similar. I have seen a brief account on the chapter "Algebraic Enumeration" by Gessel and Stanley in the book "Handbook of Combinatorics", but not more much. Thanks in advance!
The article Formal Power Series by Ivan Niven The American Mathematical Monthly Vol. 76, No. 8 (Oct., 1969), pp. 871-889 is well written. I'm not sure if that is what you are looking for though.
The book Analytic Combinatorics is available online and worth looking at.