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 NRL Plasma Formulary originated over twenty five years ago and has been revised several times during this period

NRL: Plasma Formulary

This paper studies the multiplicity-correction effect of standard Bayesian variable-selection priors in linear regression. Our first goal is to clarify when, and how, multiplicity correction happens automatically in Bayesian analysis, and to distinguish this correction from the Bayesian Ockham’s-razor effect. Our second goal is to contrast empirical-Bayes and fully Bayesian approaches to variable selection through examples, theoretical results and simulations. Considerable differences between the two approaches are found. In particular, we prove a theorem that characterizes a surprising aymptotic discrepancy between fully Bayes and empirical Bayes. This discrepancy arises from a different source than the failure to account for hyperparameter uncertainty in the empirical-Bayes estimate. Indeed, even at the extreme, when the empirical-Bayes estimate converges asymptotically to the true variable-inclusion probability, the potential for a serious difference remains.

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Bayes and empirical-Bayes multiplicity adjustment in the variable-selection problem

(2002). The Curious History of Faa di Bruno's Formula. The American Mathematical Monthly: Vol. 109, No. 3, pp. 217-234.

/pdf/the-curious-history-of-faa-di-bruno-s-formula-20k360pdyj.pdf

The Curious History of Faà di Bruno's Formula

Restricted permutations are those constrained by having to avoid subsequences ordered in various prescribed ways. They have functioned as a convenient descriptor for several sets of permutations which arise naturally in combinatorics and computer science. We study the partial order on permutations that underlies the idea of restriction and which gives rise to sets of sequences closed under taking subsequences. In applications, the question of whether a closed set has a finite basis is often considered. Several constructions that respect the finite basis property are given. A family of closed sets, called profile-closed sets, is introduced and used to solve some instances of the inverse problem:– describing a closed set from its basis. Some enumeration results are also given.

Restricted permutations

The $q$-Stirling numbers of first and second kinds

(1956). Some Generalizations of Vandermonde's Convolution. The American Mathematical Monthly: Vol. 63, No. 2, pp. 84-91.

Some Generalizations of Vandermonde's Convolution

Explicit Formulas for Bernoulli Numbers

This book is a unique work which provides an in-depth exploration into the mathematical expertise, philosophy, and knowledge of H W Gould. It is written in a style that is accessible to the reader with basic mathematical knowledge, and yet contains material that will be of interest to the specialist in enumerative combinatorics. This book begins with exposition on the combinatorial and algebraic techniques that Professor Gould uses for proving binomial identities. These techniques are then applied to develop formulas which relate Stirling numbers of the second kind to Stirling numbers of the first kind. Professor Gould's techniques also provide connections between both types of Stirling numbers and Bernoulli numbers. Professor Gould believes his research success comes from his intuition on how to discover combinatorial identities.This book will appeal to a wide audience and may be used either as lecture notes for a beginning graduate level combinatorics class, or as a research supplement for the specialist in enumerative combinatorics.

H. W. Gould

Papers

The $q$-Stirling numbers of first and second kinds

Some Generalizations of Vandermonde's Convolution

Explicit Formulas for Bernoulli Numbers

Combinatorial Identities for Stirling Numbers: The Unpublished Notes of H W Gould

Some new inverse series relations