# Rational and algebraic series in combinatorial enumeration

### Mireille Bousquet-Mélou

Université de Bordeaux I, Talence, France

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## Abstract

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 Σ_n_ *an* *tn* 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.

This enables to show that a priori different discrete models deﬁne the same curves in the scaling limit and exhibit some surprising symmetries. It gives also a way to tie links between these concrete measures on curves and conformal ﬁeld theory. Important roles in this theory are played by Brownian loops and by the Schramm–Loewner Evolutions (SLE).

Most of the results described in this paper were derived in joint work with Greg Lawler, and Oded Schramm.