ABSTRACT
Following a slow (and forced) evolution in the history of mathematics, the modern notion of function (due to Dirichlet, 1837) has been made independent of any actual description format. A similar process led Andre´ Joyal to introduce the notion of “species” in combinatorics [Joyal 81] to make the description of structures independent of any specific format. On one side, the theory serves as an elegant “explanation” for the surprising power of generating functions in the solution of structure enumeration. On another side, it makes clear and natural much of Po´lya’s theory for the enumeration of structures up to isomorphism. Moreover, species are naturally linked to the study of “polynomial functors”, which give a classification of polynomial representations of the general linear group. We refer to [Bergeron et al. 98] for an in-depth reference to the theory of species. Our particular reason for recalling the basic notions of species is that it allows us to generalize many of the questions related to the central material of this book.
