This paper develops a unified enumerative and asymptotic theory of directed 2-dimensional lattice paths in half-planes and quarter-planes. The lattice paths are speci ed by a finite set of rules that are both time and space homogeneous, and have a privileged direction of increase. (They are then essentially 1-dimensional objects.) The theory relies on a specific "kernel method" that provides an important decomposition of the algebraic generating functions involved, as well as on a generic study of singularities of an associated algebraic curve. Consequences are precise computable estimates for the number of lattice paths of a given length under various constraints (bridges, excursions, meanders) as well as a characterization of the limit laws associated to several basic parameters of paths.

A considerable number of asymptotic distributions arising in random combinatorics and analysis of algorithms are of the exponential-quadratic type, that is, Gaussian. We exhibit a class of "universal" phenomena that are of the exponential-cubic type, corresponding to distributions that involve the Airy function. In this paper, such Airy phenomena are related to the coalescence of saddle points and the confluence of singularities of generating functions. For about a dozen types of random planar maps, a common Airy distribution (equivalently, a stable law of exponent 3/2) describes the sizes of cores and of largest (multi)connected components. Consequences include the analysis and fine optimization of random generation algorithms for multiply connected planar graphs. Based on an extension of the singularity analysis framework suggested by the Airy case, the paper also presents a general classification of compositional schemas in analytic combinatorics.

Abstract. We study the asymptotic behaviour of the number Nk,n of nodes of given degree k in unlabeled random trees, when the tree size n and the node degree k both tend to infinity. It is shown that Nk,n is asymptotically normal if ENk,n → ∞ and asymptotically Poisson distributed if ENk,n → C> 0. If ENk,n → 0, then the distribution degenerates. The same holds for rooted, unlabeled trees and forests. 1.

We derive asymptotic results on the distribution of the number of descendants in simply generated trees. Our method is based on a generating function approach and complex contour integration. 1.

This talk presents a general theorem which can be used to identify the limiting distribution for a class of combinatorial schemata. For example, many parameters in random mappings can be covered in this way. 1. Methods We consider the general working scheme "Symbolic Structures A or fA; !g ! Generating Functions a(z) or a(u; z) ! a n or a n;k ". Then by Cauchy's formula, we get for structures A a(z) = X ff2A z jffj jffj! = X n0 a n z n n! =) a n n! = 1 2i I a(z) dz z n+1 : When considering marked structures with parameters fA; !g, (! is a mapping A ! N), we have a(u; z) = X ff2A u !(ff) z jffj jffj! = X n;k a n;k u k z n n! : In this case, a n;k can be obtained by double Cauchy inversion, or by Cauchy inversion and Continuity Theorem. Table 1 gives some examples of translation of marked combinatorial structures to generating functions. The mark is represented by character "ffl" and translated to parameter u. By a classical theorem about characterist...