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Combinatorial Mathematics

H. J. Ryser
Mathematical Association of America
Publication Date: 
Number of Pages: 
Electronic Book
Carus Mathematical Monographs 14
[Reviewed by
Allen Stenger
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This classic work is a monograph on combinatorics, that starts from nothing and quickly plunges into very deep waters. The introductory parts deal primarily with combinations and permutations and the inclusion-exclusion principle. There is also a little bit about Ramsey theory and rather more about systems of distinct representatives. The bulk of the book deals with different aspects of combinatorial designs, and it either proves or quotes what were the most advanced results when it was published in 1963.

The book is dated: the matters it treats are still of great interest, but if you were explaining today what combinatorial mathematics is all about, you would not explain it this way. In particular there would be a lot more about generating functions and counting problems.

The Carus Mathematical Monographs are chartered to be “expository presentations of the best thoughts and keenest researches in pure and applied mathematics.” As such they run the danger of being too topical and thereby being made obsolete by new developments. About two-thirds of the Monographs (including this one) are out of print. The ones that have been able to avoid this fate either expounded stable areas of mathematics (for example, Niven’s Irrational Numbers) or have been continually revised (for example, Boas’s A Primer of Real Functions, now in its fourth edition). The present book is still useful if you are interested in the detailed topics presented here, but it is no longer a good way to become introduced to the field.

Allen Stenger is a math hobbyist, library propagandist, and retired computer programmer. He volunteers in his spare time at, a math help site that fosters inquiry learning. His mathematical interests are number theory and classical analysis.

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