This is an overwhelmingly complete introductory textbook in combinatorics. It not only covers nearly every topic in the subject, but gives several realistic applications for each topic.
The present book takes a very broad view of combinatorics. It spends about 500 pages on counting problems, which is what most people think of as combinatorics. It also has about 200 pages on existence problems (whether there exist arrangements with certain properties; for example block design problems) and about 150 pages on optimization (such as shortest route problems and the stable marriage problem). These latter two sections deal with techniques that use combinatorial ideas, although usually without explicitly counting things. These sections tend to be very algorithmic, although they typically do not deal with running times and other details as algorithm books do.
Paul Halmos once explained that “Calculus books are bad because there is no such subject as calculus; it is not a subject because it is many subjects.” Is combinatorics a subject, and are combinatorics books bad? I think combinatorics has expanded to the point any reasonably comprehensive text would be horribly fragmented, and that is a problem with the present book. I like this book much better as a reference than as a textbook. The authors state that it is possible to cover the whole book in one year. I believe this, but at the end of the year I think it’s unlikely that the students would retain very much of the hundreds of ideas presented here.
Rummaging through MAA Reviews suggests that all modern introductory texts in combinatorics are at least 450 pages, and most run to 600 or more. The present book has 860 pages and much more breadth than its competitors. I don’t have a good solution for this “combinatorial explosion”. One idea, that would actually work pretty well, is to go back to older, shorter, classic texts such as Ryser’s 1963 Combinatorial Mathematics or Riordan’s 1958 An Introduction to Combinatorial Analysis. These cover most of the important ideas and techniques of combinatorics, but without the myriad variations found in more modern books. Another idea, more promising but more scary, is to build a course based on problems that have a heavy combinatorial flavor, such as those found in Aigner & Ziegler’s Proofs from the BOOK or Iosevich’s A View From the Top. In all of these choices the present book would be valuable as a source of applications and for enrichment reading.
Allen Stenger is a math hobbyist, library propagandist, and retired computer programmer. He volunteers in his spare time at MathNerds.com, a math help site that fosters inquiry learning. His mathematical interests are number theory and classical analysis.