ANALYTIC COMBINATORICS aims at predicting precisely the properties of large structured combinatorial configurations, through an approach based extensively on analytic methods. Generating functions are the central objects of the theory.

Analytic combinatorics starts from an exact enumerative description of combinatorial structures by means of generating functions, which make their first appearance as purely formal algebraic objects. Next, generating functions are interpreted as analytic

objects, that is, as mappings of the complex plane into itself. Singularities determine a function’s coefficients in asymptotic form and lead to precise estimates for counting sequences. This chain applies to a large number of problems of discrete

mathematics relative to words, trees, permutations, graphs, and so on. A suitable adaptation of the methods also opens the way to the quantitative analysis of characteristic parameters of large random structures, via a perturbational approach.

Analytic combinatorics can accordingly be organized based on three components:

SymbolicMethods develops systematic relations between some of the major constructions of discrete mathematics and operations on generating functions which exactly encode counting sequences.

Complex Asymptotics elaborates a collection of methods by which one can extract asymptotic counting information from generating functions, once these are viewed as analytic transformations of the complex domain. Singularities then appear to be a key determinant of asymptotic behaviour.

Random Structures concerns itself with probabilistic properties of large random structures. Which properties hold with high probability? Which laws govern randomness in large objects? In the context of analytic combinatorics,

these questions are treated by a deformation (adding auxiliary variables) and a perturbation (examining the effect of small variations of such auxiliary variables) of the standard enumerative theory.