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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 in the chapter "Algebraic Enumeration" by Gessel and Stanley in the "Handbook of Combinatorics," but not much more.

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Could you be a little more specific? – J.C. Ottem May 26 '11 at 17:37
I don't think this question is well posed. Polya theory is a tool for counting orbits of structures under a group action. What group do you have in mind? (And note that we need a permutation group - the action must be specified.) – Chris Godsil May 26 '11 at 22:57
Might the question be related to the relation of the Burnside ring and combinatorial species? – Martin Rubey May 27 '11 at 5:52
What I had in mind was, for instance, having found a generating function for some enumeration problem, i.e. the number of nonisomorphic simple graphs with given number of nodes and edges, whether this has been considered as the Hilbert function of some graded algebra, and whether this graded algebra could be obtained from a simpler algebra (like a polynomial algebra) through e.g. quotients. An example in a similar spirit is the Stanley-Reisner ring of a simplicial complex. – Camilo Sarmiento Jul 22 '11 at 15:32
up vote 3 down vote accepted

Pólya's theorem arises in the theory of invariants of finite groups. See Section 10 of

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thanks, this is exactly what i was looking for four years ago – Camilo Sarmiento Aug 4 '15 at 8:24

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.

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