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Nicholas A. Loehr
Publisher: Chapman & Hall/CRC (2011)
Details: 590 pages, Hardcover
Series: Discrete Mathematics and Its Applications
Topics: Enumerative Combinatorics, Combinatorics
This book is in the MAA's basic library list.
MAA Review[Reviewed by Peter Rabinovitch, on 09/22/2011]
A rule I have found to be true is that any book claiming to be suitable for beginners and yet leading to the frontiers of unsolved research problems does neither well. This book is the exception to that rule.
A glance at the table of contents reveals many of the standard combinatorics topics. As the title implies, they are generally explored via bijections. As a user of combinatorics, rather than a dyed in the wool combinatorialist, I find bijections to be the central core of the subject and so I found this book engaging.
The proofs are very clear, and in many cases several proofs are offered. For example, there may be an algebraic proof of an identity, followed by a bijective proof.
This book could serve several purposes. By focussing on the first half of the book, it could be an excellent choice for a first course in cominatorics for senior undergraduates. By selecting topics and/or moving quickly, it could work well for a more mature audience. The book is at a higher level than Stanton & White, but lower than Stanley, thus it also makes a great reference for people who use combinatorics but are not specialists.
There are many exercises. The back cover claims nearly 1000, and although I didn’t count them, I have no reason to doubt this claim. Some are very simple, and some are hard — the back cover claims some are unsolved. Many of the exercises are discussed in an appendix, ranging in detail from mere hints to “draw a diagram” to full but terse solutions. Because there are so many exercises, and because the level of detail of the provided solutions varies so much, an instructor using this text could easily find appropriate problems for assignment for courses of various levels of sophistication.
There are few obvious typos.
On the negative side, the book’s web site is empty, and the author uses “quantum numbers” and “quantum binomial coefficients” etc. rather than the more common “q numbers” and “q binomial coefficients.”
This is a very nice book that deserves serious consideration.
Peter Rabinovitch is a Systems Architect at Research in Motion, and a PhD student in probability. He’s currently thinking about applications of Mallows permutations.
BLL — The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.