Introduction

This chapter contains the basic concepts of the combinatorial theory of species of structures. It is an indispensable starting point for the developments and applications presented in the subsequent chapters. We begin with some general considerations on the notion of structure, everywhere present in mathematics and theoretical computer science. These preliminary considerations lead us in a natural manner to the fundamental concept of species of structures.

The definition puts the emphasis on the transport of structures along bijections and is due to C. Ehresmann [87], but it is A. Joyal [158] who showed its effectiveness in the combinatorial treatment of formal power series and for the enumeration of labeled structures as well as unlabeled (isomorphism types of) structures.

We introduce in Section 1.2 some of the first power series that can be associated to species: generating series, types generating series, cycle index series. They serve to encode all the information concerning labeled and unlabeled enumeration.

Sections 1.3 and 1.4 form an introduction to the algebra of species of structures. Various combinatorial operations on species of structures are used to produce new ones, in general more complex. The operations introduced here are addition, multiplication, substitution, and differentiation of species of structures. They constitute a combinatorial lifting of the corresponding operations on formal power series. The problems of specification, classification, and enumeration of structures are then greatly simplified, using this algebra of species.