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An Introduction to Enumeration

Alan Camina and Barry Lewis
Publication Date: 
Number of Pages: 
Springer Undergraduate Mathematics Series
[Reviewed by
Allen Stenger
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This is an introductory text on counting and combinatorics that has good coverage but is disorganized and lacks motivation and rigor. It is aimed at a sophomore or higher level and has few prerequisites beyond power series. It has extensive coverage for such a short introduction, including a great deal on the use of generating functions and permutation groups, Hall’s marriage theorem (although only in that form and not in the generality of systems of distinct representatives (SDRs)), and a fairly thorough look at Pólya’s theory of counting. There are numerous exercises, and all have brief solutions in the back.

Although I like books with lots of examples, I think this one overdoes it, or perhaps it underdoes the explanations: these are often flimsy and ad hoc rather than systematic. I’ve read the chapter on exponential generating functions twice, but I still don’t understand the book’s rationales (several are given) for using exponential rather than ordinary generating functions. There’s no coherent explanation of how permutation groups are related to counting, and there’s no explanation in the examples of how the symmetry groups were determined for the various geometric objects being counted.

It is unusual to see a book that combines such extensive coverage with so few prerequisites. Some other books with similar coverage include Riordan’s An Introduction to Combinatorial Analysis and Van Lint & Wilson’s A Course in Combinatorics. Both of these assume more background than the present work, but have much better explanations. Another good book is Wilf’s generatingfunctionology that, although nominally about generating functions, does include a fairly complete course in combinatorics.

Allen Stenger is a math hobbyist and retired software developer. He is webmaster and newsletter editor for the MAA Southwestern Section and is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis. He volunteers in his spare time at, a math help site that fosters inquiry learning.