Rational and algebraic series in combinatorial enumeration

TL;DR

This paper explores the properties and implications of rational and algebraic generating functions in combinatorial enumeration, linking their algebraic nature to underlying structural characteristics of combinatorial objects.

Contribution

It provides a detailed analysis of how rational and algebraic generating functions reflect the structural features of combinatorial classes, highlighting differences and open questions.

Findings

Rational generating functions indicate linear structure in objects.

Algebraic generating functions suggest branching structures.

The connection between generating function type and object structure is complex and not fully understood.

Abstract

Let A be a class of objects, equipped with an integer size such that for all n the number a(n) of objects of size n is finite. We are interested in the case where the generating fucntion sum_n a(n) t^n is rational, or more generally algebraic. This property has a practical interest, since one can usually say a lot on the numbers a(n), 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.