Rational and algebraic series in combinatorial enumeration

Let A be a class of objects, equipped with an integer size such that for all n the number an of objects of size n is finite. We are interested in the case where the generating function ∑ n ant n is rational, or more generally algebraic. This property has a practical interest, since one can usually say a lot on the numbers an, but also a combinatorial one: the rational or algebraic nature of the generating function suggests that the objects have a (possibly hidden) structure, similar to the linear structure of words in the rational case, and to the branching structure of trees in the algebraic case. We describe and illustrate this combinatorial intuition, and discuss its validity. While it seems to be satisfactory in the rational case, it is probably incomplete in the algebraic one. We conclude with open questions. Mathematics Subject Classification (2000). Primary 05A15; Secondary 68Q45.