L. Bruce Richmond

Central and local limit theorems applied to asymptotic enumeration II: Multivariate generating functions

Abstract Let a multivariate sequence a n (k) ⩾ 0 be given. Multivariate central and local limit theorems are proved for a n (k) as n → ∞ that are based on examining the generating function. Applications are made to permutations with rises and falls, ordered partitions of sets, Tutte polynomials of recursive families, and dissections of polygons.

Generalized Digital Trees and Their Difference- Differential Equations

Consider a tree partitioning process in which n elements are split into b at the root of a tree (b a design parameter), the rest going recursively into two subtrees with a binomial probability distribution. This extends some familiar tree data structures of computer science like the digital trie and the digital search tree. The exponential generating function for the expected size of the tree satisfies a difference-differential equation of order b, The solution involves going to ordinary (rather than exponential) generating functions, analyzing singularities by means of Mellin transforms and contour integration. The method is of some general interest since a large number of related problems on digital structures can be treated in this way via singularity analysis of ordinary generating functions. 0 1992 John Wiley & Sons, Inc.

The asymptotic number of rooted maps on a surface II: enumeration by vertices and faces

Abstract We extend some of the earlier results on the enumeration of rooted maps on a surface by number of edges to simultaneous enumeration by vertices and faces. In particular, (i) an asymptotic formula is obtained, (ii) the generating functions on orientable surfaces are shown to be rational functions of the parameterizations of Arques and of Tutte, and (iii) the generating function for rooted maps on the projective plane is given.

A survey of the asymptotic behaviour of maps

Abstract A survey is given of the asymptotic enumeration of maps. The asymptotic formulas for both rooted and unrooted maps on surfaces are discussed. All results known to use concerning random maps are described. The techniques used to derive these results are briefly discribed. The survey concludes with open problems and areas for future research.

The Distribution of Heights of Binary Trees and Other Simple Trees

The number of binary trees of fixed size and given height is estimated asymptotically near the peak of the distribution. There, a local limit theorem with convergence to a theta law is established. Large deviation bounds corresponding to large heights and small heights are also derived. The methods based on the analysis of singular iterations apply to any simple family of trees.

Central and local limit theorems applied to asymptotic enumeration IV: multivariate generating functions

Abstract Flajolet and Soria (1989, 1990) discussed some general combinatorial structures in which central limit theorem and exponential tail results hold. In this paper, we shall use Flajolet and Odlyzkos “transfer theorems” (1990) to extend Bender and Richmonds (1983) central and local limit theorems to a wider class of generating functions which will cover the above-mentioned combinatorial structures. The local limit theorem provides more accurate asymptotic information and implies the superexponential tail results.

Submaps of maps. I: General 0–1 laws

Abstract Let M n be the set of n edge maps of some class on a surface of genus g. When g = 0 (planar maps) we show how to prove that limn → ∞ | M n|1/n exists for many classes of maps. Let P be a particular map that can appear as a submap of maps in our class. There is often a strong 0–1 law for the property that P is a submap of a randomly chosen map in M n: If P is planar, then almost all M n contain at least cn disjoint copies of P for small enough c; while if P is not planar, almost no M n contain a copy of P. We show how to establish this for various classes of maps. For planar P, the existence of limn → ∞ | M n| 1 n suffices. For nonplanar P, we require more detailed asymptotic information.

The asymptotic enumeration of rooted convex polyhedra

An asymptotic formula is obtained for the number of rooted c-nets with m vertices and n edges as m, n → ∞ with 12 + e 0.

Smallest components in decomposable structures: Exp-log class

The smallest size of components in random decomposable combinatorial structures is studied in a general framework. Our results include limit distribution and local theorems for the size of therth smallest component of an object of sizen. Expectation, variance and higher moments of therth smallest component are also derived. The results apply to several combinatorial structures in the exp-log class for both labelled and unlabelled objects. We exemplify with several combinatorial structures like permutations and polynomials over finite fields.

Central and local limit theorems applied to asymptotic enumeration. III. Matrix recursions

Abstract Let a multivariate sequence an(k) be obtained by a matrix recursion. It is shown that it is usually easy to establish central and local limit theorems for an(k). The proof requires a lemma on multisection of multivariate series which appears to be new. The applications of the limit theorems include covering by polyominoes, enumeration of plane animals, occupancy problems, 0–1 matrices, and nonexistence of critical phenomena.