Preliminaries.- Bell polynomials, composite structures and Gibbs partitions.- Exchangeable random partitions.- Sequential constructions of random partitions.- Poisson constructions of random partitions.- Coagulation and fragmentation processes.- Random walks and random forests.- The Brownian forest.- Brownian local times, branching and Bessel processes.- Brownian bridge asymptotics for random mappings.- Random forests and the additive coalescent.

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Combinatorial Stochastic Processes

In this chapter, we give a brief introduction to a relatively new subject, called the umbral calculus. This is an algebraic theory used to study certain types of polynomial functions that play an important role in applied mathematics. We give only a brief introduction to the subject — emphasizing the algebraic aspects rather than the applications. For more on the umbral calculus, we suggest The Umbral Calculus, by Roman [1984].

The Umbral Calculus

The set of rooted trees, generated by random bifurcation at the terminal nodes, is considered with the aims of enumerating it and of determining its probability distribution. The account of enumeration collates much previous work and attempts a complete perspective of the problems and their solutions. Asymptotic and numerical results are given, and some unsolved problems are pointed out. The problem of ascertaining the probability distribution is solved by obtaining its governing recurrence equation, and numerical results are given. The difficult problem of determining the most probable tree-shape of given size is considered, and for labelled trees a conjecture at its solution is offered. For unlabelled shapes the problem remains open. These mathematical problems arise in attempting to reconstruct evolutionary trees by the statistical approach of Cavalli-Sforza and Edwards.

The probabilities of rooted tree-shapes generated by random bifurcation

This paper is a survey on Extremal Graph Theory, primarily focusing on the case when one of the excluded graphs is bipartite. On one hand we give an introduction to this field and also describe many important results, methods, problems, and constructions.

The History of Degenerate (Bipartite) Extremal Graph Problems

Un code pour les graphes planaires et ses applications.

Enumeration of Triangulations of the Disk

In (2), Tutte has shown that the number, Bn , of rooted non-separable planar maps having n edges is [2(3n — 3)!]/[n! (2n — 1)!]. Rooting was accomplished by designating one edge as the root, orienting it, and distinguishing between its sides as left and right. We shall here compute the number, Bn,m , of rooted non-separable planar maps having n edges and such that the face to the left of the root is incident with exactly m edges, which maps will be said to be of type [n, m].

Enumeration of Non-Separable Planar Maps

A dissection of the disk will be a cell complex (1, p. 39) K with polyhedron the closed disk B 2. It will further be required that: (a) every edge of K be incident with two distinct vertices (called its ends) ; (b) no two edges have the same ends ; and (b) no two edges have the same ends ; and

Enumeration of quadrangular dissections of the disk

On the existence of square roots in certain rings of power series

ISSN: 0002-9890 (Print) 1930-0972 (Online) Journal homepage: https://www.tandfonline.com/loi/uamm20 Historical Note on a Recurrent Combinatorial Problem William G. Brown To cite this article: William G. Brown (1965) Historical Note on a Recurrent Combinatorial Problem, The American Mathematical Monthly, 72:9, 973-977, DOI: 10.1080/00029890.1965.11970654 To link to this article: https://doi.org/10.1080/00029890.1965.11970654

William G. Brown

Papers

Enumeration of Triangulations of the Disk

Enumeration of Non-Separable Planar Maps

Enumeration of quadrangular dissections of the disk

On the existence of square roots in certain rings of power series

Historical Note on a Recurrent Combinatorial Problem