Edward A. Bender

The asymptotic number of labeled graphs with given degree sequences

Abstract Asymptotics are obtained for the number of n × n symmetric non-negative integer matrices subject to the following constraints: (i) each row sum is specified and bounded, (ii) the entries are bounded, and (iii) a specified “sparse” set of entries must be zero. The result can be interpreted in terms of incidence matrices for labeled graphs.

Asymptotic Methods in Enumeration

This is an expository paper dealing with those tools in asymptotic analysis which are especially useful in obtaining asymptotic results in enumeration problems. Emphasis is on tools which are general, are easily applied, and give estimates of the form

Central and local limit theorems applied to asymptotic enumeration

a_n \sim f(n)

Enumeration of plane partitions

. Many examples are given to illustrate the usage of the various tools. It is assumed that a summation or a generating function for

The asymptotic number of tree-rooted maps on a surface

a_n

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

is explicitly or implicitly given.

The asymptotic number of labeled connected graphs with a given number of vertices and edges

Abstract Let a double sequence an(k) ⩾ 0 be given. We prove a simple theorem on generating functions which can be used to establish the asymptotic normality of an(k) as a function of k. Next we turn our attention to local limit theorems in order to obtain asymptotic formulas for an(k). Applications include constant coefficient recursions, Stirling numbers, and Eulerian numbers.

The number of rooted maps on an orientable surface

Abstract Using some recent results involving Young tableaux and matrices of non-negative integers [10], it is possible to enumerate various classes of plane partitions by actual construction. One of the results is a simple proof of MacMahons [12] generating function for plane partitions. Previous results of this type [12, 4, 3, 8, 7] involved complicated algebraic methods which did not reveal any intrinsic “reason” why the corresponding generating functions have such a simple form.

A theoretical analysis of backtracking in the graph coloring problem

Let S be a surface. We asymptotically enumerate two classes of n-edged maps on S as N → ∞: tree-rooted and tree-rooted smooth. These results are based on a system of equations enumerating single vertex maps and on a relation found by Walsh and Lehman for the case of orientable surfaces between tree-rooted and single vertex maps.

The Number of Degree-Restricted Rooted Maps on the Sphere

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.