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Combinatorics: The Rota Way

Cambridge University Press
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Through a wondrous confluence of circumstances I’ve had occasion of late to review in this very venue the charming book Recountings, a compendium of interviews with mainstays of the MIT mathematics department. It is noted in different places in this book that the ineffable but undeniable ésprit of MIT’s mathematics department, unique and singular, is traceable to a handful of major players in the not all that distant past, namely, Norbert Wiener, Norman Levinson, and Gian-Carlo Rota. And Rota, who only passed away in 1999 at the early age of 66, was distinguished, among many other things, by the fact that as a pure mathematician of the finest water he was bivouacked on the floor devoted to applied mathematics. And this surely attests to the peculiar status of his field of combinatorics in the mathematical pantheon: a condition a number theorist can of course relate to.

Continuing along with synchronistic events, I did meet Rota once, at the house of a physicist/electrical engineer from the Feynman circle, at a party thrown by his wife, a colleague of mine from the Department of English. I only had occasion to chat with Rota briefly, but was struck by his grace, obvious kindness, and manners: he was some one impossible not to like right off.

And so the book under review, Combinatorics: The Rota Way, based on notes from his “Combinatorics 18:315” course and on conversations with Rota and offered by two of his former students, Joseph P. S. Kung and Catherine H. Yan, purposely reflects Rota’s unique way of pursuing the art of counting. In a fitting acknowledgement of his great influence Kung and Yan list Rota alphabetically as a co-author.

In their preface to the book the two authors note that their use of the word “way” in the title is layered: “The word ‘way’ resonates with the word ‘cammin’ in the first line of Dante’s Divina Commedia, ‘Nel mezzo del cammin di nostra vita.’ It also resonates as the character ‘tao’ in Chinese. In both senses the way has to be struggled for and sought individually.” This, I think, is a truly marvelous pedagogical observation, particularly in today’s climate of pedagogy as a science or an engineering discipline. I think that cammin or tao or, in Japanese, do (my own favorite phrasing, given my avocation as a judo teacher), is truly the more apt description of how doing mathematics should engender a trajectory from apprenticeship to individual practice, with one’s own style and touch being revealed in the process. There is no universal judo except as a taxonomy of techniques: every judoka discovers his own judo. Similarly the act of doing mathematics is unique to the doer, to the individual artisan or artist, and Kung and Yan note that “Rota’s way is but one way of doing combinatorics. After ‘seeing through’ Rota’s way, the reader will seek his or her own way.” Irresistible.

And what makes the book under review even more tantalizing, to me at least, is the authors’ comment that “[t]o convey Rota’s thinking, which involves all of mathematics, we must go against any idée reçue of textbook writing: the prerequisites for this book are, in a sense all mathematics. However, it is the ideas, not the technical details, that matter. Thus, in a different sense, there are no prerequisites to this book…” What an adventure must lie ahead! Not only will you learn some combinatorics, you’ll discover something about yourself. Again, irresistible.

This leaves me the task of describing the dojo curriculum (dojo = place to learn the way = (approx.) martial arts studio). The trajectory is this: [1] Sets, functions, and relations; [2] Matching theory; [3] Partially ordered sets and lattices; [4] Generating functions and the umbral calculus; [5] Symetric functions and Baxter algebras; [6] Determinants, matrices, and polynomials. In this connection let me single out one example of the excellent taste the authors display throughout the book. On p. 218 ff. they deal with Riemann’s zeta function, bringing in right away the angle of probability distributions — all this on the heels of their discussion of generating functions and the umbral calculus. The discussion is crisp and elegant, culminating in the passage, “[b]y Möbius inversion, we obtain [that the probability that the joint kernel of an s-tuple coincides with the cyclic group of order n is 1/nsζ(s) ]. This yields a probabilistic interpretation of the reciprocal of the Riemann zeta function for positive integers s.” Beautiful.

Additionally, the exercise sets in Combinatorics:The Rota Way uniformly display a sense of depth, with many important and fascinating themes represented. For instance, problem 1.4.8 deals with the Boltzmann entropy, problem 4.3.6 with abstract Poisson processes, and problem 6.6.2 with the Orlik-Solomon algebra of a geometric lattice. Appropriately, there are “Selected Solutions” offered, too.

Finally, I also want to note that Kung and Yan pepper the pages of the book with a lot of useful and interesting mathematical history, too, all attesting to their ecumenical perspective, obviously in keeping with Rota’s example. For example, it was very interesting to learn that 1n 1942 R. P. Dilworth learned about lattices from nothing less than Dedekind’s 1900 Mathematische Annalen article “Über die von drei Moduln erzeugte Dualgruppe.” The authors note, appropriatley, that Emmy Nöther herself was apt to state again and again that “everything is already in Dedekind.” What fun!

All in all, then, I find Combinatorics: The Rota Way simply too good to pass up. Would that every day had 48 hours…

Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.

Date Received: 
Wednesday, March 18, 2009
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Joseph P. S. Kung, Gian-Carlo Rota, and Catherine H. Yan
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Michael Berg
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1. Sets, functions, and relations; 2. Matching theory; 3. Partially ordered sets and lattices; 4. Generating functions and the umbral calculus; 5. Symmetric functions and Baxter algebras; 6. Determinants, matrices, and polynomials; 7. Selected solutions.

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Sunday, May 31, 2009