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Combinatorics: Ancient and Modern

Robin Wilson and John J. Watkins, editors
Oxford University Press
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The Basic Library List Committee recommends this book for acquisition by undergraduate mathematics libraries.

[Reviewed by
Daniel J. Otero
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When does a branch of mathematics become recognizable as such? Granted, questions like this are oily things, hard to answer, and harder even to clarify. But the book under review is offered as just the sort of resource that might shed light on when and how combinatorics emerged as a categorized branch of mathematics. In fact, the preface of this book suggests that it “is perhaps the first book-length survey of the history of combinatorics” in that it brings together “for the first time in a single source research… that would otherwise be inaccessible to the general reader.” And no less a light in the field than Ron Graham confirms as much in its foreword. One wishes that Graham might have been persuaded to contribute one of the fourteen chapters that comprise the book, but rest assured: the editors, Robin Wilson (Emeritus Professor at the Open University in the UK and former Professor of Geometry at Gresham College, a chair first held by Henry Briggs of logarithms fame and later by the likes of Robert Hooke, Karl Pearson, and Roger Penrose) and John J. Watkins (Emeritus Professor at Colorado College), convinced Donald E. Knuth to provide a 40-page overview that begins this survey. Moreover, Wilson and Watkins have assembled a collection of expert writers with ample skills to inform readers about the history and development of the theories of counting, ordering, cataloging and partitioning discrete objects.

The multifarious manifestations in the ancient world of what we call today combinations and permutations give evidence for the ubiquity of these patterns. These discrete structures were discovered and rediscovered numerous times and in many different places around the world: in South Asia and in China in the early centuries of the Common Era, in the writings of Muslim and Jewish scholars of the Middle Ages, and in the European Renaissance. Eberhard Knobloch engagingly summarizes the mystical combinatorial philosophy of Ramon Llull from the thirteenth century, as well as the mathematics of some more familiar lights of later years — familiar to workaday modern mathematicians, that is: Cardano, Mersenne, Descartes, and Leibniz (about whom few other people are more knowledgeable than Knobloch). Pascal’s 1654 Traité du Triangle Arithmétique is identified by statistician A. W. F. Edwards as pivotal in the history of combinatorics, marking the transition to the modern era.

By the eighteenth century, we find an explosion of new ideas. Graph theory is born in the wake of Euler’s work on the Königsberg bridge problem and his formulation of the polyhedron formula \(F+V=E+2\), relating the number of vertices \(V\), edges \(E\), and faces \(F\) of a (convex and simply connected) polyhedron. The subject received a further boost in the nineteenth century, first, when chemists drew attention to problems of isomer enumeration, leading to the theory of trees and their enumeration; and then in the aftermath of De Morgan’s communication to Hamilton of the now-famous four color problem. Graph coloring and graph factorization problems, largely investigated in the twentieth century, are now fundamental elements of a broad literature that grew out of this playing around with dots and lines.

George Andrews gives a précis of the remarkable story of integer partition theory, in which the marvelous Srinivasa Ramanujan is only the most interesting of many actors; Andrews also marks the vital contributions of J. J. Sylvester, Percy MacMahon, and L. J. Rogers. And an endless fascination with the magical properties of latin squares down through history becomes formalized and finds important applications in geometry in the search for finite projective planes, and in the experimental sciences in seminal work on block designs by the statistician R. A. Fisher in his Design of Experiments (1935).

Another trajectory towards modern combinatorics traces an evolution from patterns noticed centuries ago in algebraic binomial expansions, formulated succinctly in nineteenth century “combinatorial analysis”, and the powerful techniques of generating functions pioneered by Euler, which were greatly expanded in the cycle index methods of J. Howard Redfield and George Pölya in the 1920s and 1930s. The deep insights of Frank Ramsey, Paul Erdős, George Szekeres, and Marshall Hall augmented the mathematical tools that emerged in the twentieth century from the use of the humble pigeonhole principle; these have been used to study a great variety of fiendishly hard (in fact, NP-hard!) combinatorial optimization problems.

To close, Peter J. Cameron bookends this survey with a brief essay on the present and future of combinatorics, in which he muses that “unexpected connections” between exciting new developments within combinatorics and other branches of mathematics and science, and especially the science of the mind, “cannot be predicted. … [D]eep links in mathematics often reveal themselves in combinatorial patterns.”

As one may expect with surveys of this nature, it is both possible to read the book from beginning to end to get a full sweep of the immense scope of the history of combinatorial mathematics, or to jump in wherever one pleases to learn about a particular subcategory. In particular, the editors have done an admirable job of maintaining a consistent level of discourse from one chapter to the next, despite multiple their authorship. It allows anyone with a background equivalent to a good introductory college course in graph theory or combinatorics the ability to follow along without difficulty. This fine book will surely soon become a staple of every comprehensive mathematics library, and its existence is testament enough that combinatorics has come of age as a well-defined branch of mathematics. And it only took 2000 years.

Daniel E. Otero ( is professor of mathematics at Xavier University in Cincinnati, OH. Otero is co-founder with Daniel J. Curtin (Northern Kentucky University) of the ORESME (Ohio River Early Sources in Mathematics Exposition) Reading Group. He is currently chair of HOMSIGMAA, the History of Mathematics Special Interest Group of the MAA, and is on the editorial board for the MAA Spectrum series.

Two Thousand Years of Combinatorics, DONALD E. KNUTH
1. Indian Combinatorics, TAKANORI KUSUBA and KIM PLOFKER
3. Islamic Combinatorics, AHMED DJEBBAR
4. Jewish Combinatorics, VICTOR J. KATZ
5. Renaissance Combinatorics, EBERHARD KNOBLOCH
6. The Origins of Modern Combinatorics, EBERHARD KNOBLOCH
7. The Arithmetical Triangle, A. W. F. EDWARDS
8. Early Graph Theory, ROBIN WILSON
9. Partitions, GEORGE E. ANDREWS
10. Block Designs, NORMAN BIGGS and ROBIN WILSON
12. Enumeration (18th-20th Centuries), E. KEITH LLOYD
13. Combinatorial Set Theory, IAN ANDERSON
14. Modern Graph Theory, LOWELL BEINEKE and ROBIN WILSON
A Personal View of Combinatroics, PETER J. CAMERON