Faculty research can contribute in many ways to the development of our University and the enhancement of our mission. It would be a mistake, I believe, to let the need to promote student research undermine the importance of faculty research. The two are interrelated and mutually supportive, but it is important not to forget that faculty research also stands independently of student research. To force faculty research to defer to the requirements of student research cripples the character and possibility of faculty research. Moreover, faculty research itself can directly contribute to the promoting of the University within the larger academy. I would like to give one distinctive example of this point: a review of Martin J. Erikson's book, "Introduction to Combinatorics."

Introduction to Combinatorics By Martin J. Erickson

NEW YORK: JOHN WILEY AND SONS, 1996. xi  195PP. US $59.95 (paperback). ISBN 0-471-15408-3

REVIEWED BY JET WIMP

The regnant institution of higher learning in the obscure hamlet of Kirksville, MO has undergone more transmutations than the legendary Vicar of Bray. Originally, it was North Missouri Normal School and Commercial College; then it became Kirksville Normal College, then First District Normal School, then Northeast Missouri State Teachers College, then Northeast Missouri State University, and now we hope finally, Truman State University. In its reincarnations, it has achieved a sort of flowering, becoming the home of two highly regarded literary periodicals, and it is now rated fourth of all the institutions in the nation by Money Magazine in terms of best education for the buck. It was just a matter of time before textbooks, like this one, began to issue from Truman State University. This excellent text should prove a useful accoutrement for any developing mathematics program.

The problems with writing a text on combinatorics - or any other subject - are basically two: (1) keeping the book both clear and short and (2) making the material current.

I have taught combinatorics several times: choosing a text has always been a source of frustration.Texts, as they meander through several editions, tend to gigantism, ending up engorged with material. What good is 500 pages of information to a harried instructor planning a course of one term or semester? I have noticed, too, that authors as they segue from one edition to another succumb to pretentiousness, so that the statements of basic results become insufferably abstract; some of the available versions of the pigeonhole principle, for example, are ludicrously ethereal. The book I like the best, Wilfs generatigfunctionology, doesnt provide the plenary selection of topics necessary for a well-balanced course, although the book is so well written that I have sometimes made it serve as a text by incorporating additional material.

It appears that with the present book, I can stop looking, at least for awhile. Its short, its sweet, its beautifully written; it contains material on two cutting-edge disciplines, Ramsey Theory and coding theory, that will give the student a hint of whats really going on in combinatorics these days. I think its the best combination of text available. It also has its own peculiar and satisfying fillips. The book sets out with a nonstandard selection of preliminaries: sets and group theory, predictably, but then material on fields, number theory, and linear algebra-unusual. The author prepares sedulously: information on quadratic residues, for example, is important for Ramsey theory. The book is divided into three parts: Part I, Existence; Part II, Enumeration; Part III, Construction. A clever and original trichotomy.

In Chapter 2, the author introduces the pigeonhole principle in an elegant but not unnecessarily abstract formulation. There follow examples. This is one of the books many strengths: each theorem is followed by one, two, sometimes more examples. Chapter 3 deals with sequences and partial orders, including the famous Erdos-Szekeres Theorem-any sequence of n2  1 real numbers contains a monotonic subsequence of n  1 terms. (Actually, the author proves a useful generalization.) The material in this chapter naturally segues into Ramsey theory; as the author asserts.

The Erdos-Szekeres Theorem and Dilworths lemma  guarantee the existence of particular substructures of certain combinatorial configurations. In other words, they say that large random systems contain nonrandom subsystems. We continue this theme by presenting two cornerstones of Ramsey theory (my italics)

Of course, this is true. It had never occurred to me to see Ramsey theory or Erdos-Szekeres result in this light. A good text will always teach us something new. The author begins the discussion of Ramseys theorem with a problem, now famous, from a Putnam competition:

Six points are in general position in space (no three in a line, no four in a plane). The fifteen line segments joining them in pairs are drawn and then printed, some segments red, some blue. Prove that some triangle has all its sides one color.

The author then states and proves Ramseys theorem and a number of related results, including Schurs lemma involving monochromatic solutions to equations involving integers. Part II of the book opens with Chapter 5, The Fundamental Counting Problem. As the author has noticed.

Enumeration is probably the trickiest branch of combinatorics. Sadly, it is common to hear students say, "I just cant count," or "I dont count," or "I count, but I always get two different answers." There is confusion about what is being counted and there are many formulas to remember. Although the situation is fraught with difficulty - even desperation - there is a solution. Surprisingly, one general principle and a few variations suffice for nearly all enumeration problems

The author doesnt claim that his epistemological approach is original, but he advances the view of representing the objects to be counted as set-to-set mappings. This is indeed a novel antidote for the students malady, but it will take an inordinately self-defined student to incorporate this point of view into working conception of combinatorics.

Chapter 6 introduces the meat and potatoes of classical combinatorics, and here the student can take a breather: the inclusion-exclusion principle, Stirling numbers, Bell numbers, recurrence relations, generating functions. Chapter 7discusses Permutations and Tableaux, and Chapter 8, Polyas Theory of Counting, which allows one to do more complicated counting, including non-isomorphic graphs. Part III, Construction, opens with Chapter 9, Codes, and then Chapters 10 and 11, devoted to designs.

The book is stuffed with challenging exercises, and with historical asides. Throughout, the author mentions appealing unsolved problems. Unfortunately, there are no solutions to the exercises, although hints are occasionally given.

Department of Mathematics and Computer Science Drexel University Philadelphia, PA 19104 USA e-mail: jwimp@mcs.drexel.edu

-- Anonymous, June 26, 1999

Answers

Congratulations of Marty Erikson, and to Truman State University! I am particularly gratified by this kind of recognition, because it shows that some people in our professions have taken note of what were are doing. Sometimes I dispair that our reputation is all public-relations drivel, or Money mag fluff, when I know that we have some good people who manage to contribute to the body of knowledge, i.e., they do the things that university folk were always meant to do. I only wish that we had scholars and writers amongst our university administrators. (Maybe we do; but we never hear about it.)

-- Anonymous, August 09, 1999