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A course in enumeration. (English) Zbl 1123.05001

Graduate Texts in Mathematics 238. Berlin: Springer (ISBN 978-3-540-39032-9/hbk). x, 561 p. (2007).
There are many good textbooks on enumerative combinatorics. The justification of every new text should be a different viewpoint and/or a somewhat different coverage. Martin Aigner’s text passes these requirements with flying colors. The goal of the text is present enumerative combinatorics together with its many applications, including chapters not common in enumerative combinatorics texts, like the ones on hypergeometric summations, on the Tutte polynomial, and on models from statistical physics. The book is organized into three parts: basics, methods, and topics. Further topics include symmetric functions and the ubiquitous Catalan numbers; methods include generating functions, sieve methods, hypergeometric summation and enumeration of objects under permutation group action. A good number of exercises carry additional material, and a number of selected exercises are given a solution at the end. A good degree of maturity is expected from the students, as the treatment prefers generality and natural connections to other mathematical fields pop up all over (graphs, knots, orthogonal polynomials, continued fractions, partially ordered sets, etc.). A nice trend in recent books – closing chapters with some spectacular “book proofs” – is followed and will help at keeping the attention of the students.

MSC:

05-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to combinatorics
05Axx Enumerative combinatorics
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