A Course in Enumeration

Combinatorial enumeration is a readily accessible subject full of easily stated, but sometimes tantalizingly difficult problems. This book leads the reader in a leisurely way from the basic notions to a variety of topics, ranging from algebra to statistical physics. Its aim is to introduce the student to a fascinating field, and to be a source of information for the professional mathematician who wants to learn more about the subject. The book is organized in three parts: Basics, Methods, and Topics. There are 666 exercises, and as a special feature every chapter ends with a highlight, discussing a particularly beautiful or famous result.

• Algebra
• Pólya theory
• combinatorial coefficients
• generating functions
• graph and knot polynomials
• hypergeometric summation
• orthogonal polynomials
• sieve methods
• statisical physics
• statistical physics
• symmetric functions
• combinatorics

“I loved Martin Aigner's Proofs from THE BOOK, a showcase of some of the most elegant and appealing proofs from across mathematics, and it has a similar feel to that. The presentations of ideas and proofs have the kind of clarity and luminousness which makes one feel, after reading them, that they are the natural if not the only ones. Each chapter also ends with a 'highlight', tackling a famous and attractive problem using the tools developed.” (Danny Yee, dannyreviews.com, July, 2018)

“It provides mathematical analysis of combinatorial sets … . Martin Aigner has a reputation as a good expositor of mathematics … and the book does not disappoint. The explanations, while often brief, are quite good. … As the book gets more and more advanced, the explanations grow correspondingly in size. … contains the clearest explanation of graph polynomials that I have ever found. … the book contains good and readable expositions of an interesting and beautiful subject.” (PeterBoothe, SIGACT News, Vol. 41 (2), 2010)

"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. … good number of exercises carry additional material, and a number of selected exercises are given a solution at the end. … 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." (László A. Székely, Zentralblatt MATH, Vol. 1123 (1), 2008)

"The book is divided into three parts … . the structure and topics of this book are well-designed, and there are nearly 700 exercises sprinkled throughout – many with hints and solutions in the back – which make the book far more appealing. I think it would be a good … textbook for any graduate student wishing to learn about enumerative combinatorics." (Darren Glass, MathDL, January, 2008)

"In this graduate textbook on enumerative combinatorics, the author follows the classic structure of basics-methods-special topics. … Each chapter ends with a ‘Highlight’, which is a specific, high-level application of the material learned in that chapter. This will benefit instructors and interested students alike. … the book will broaden access to several special topics and will turn them into more mainstream knowledge. The scope of the book is large, so most readers will find several sections that will teach them many facts, methods and theories." (Miklós Bóna, Mathematical Reviews, Issue 2008 f)

"The techniques one needs to be an expert in enumeration are very involved, sometimes quite genius. … This book moves this important technique much closer to the classrooms than it used to be. … The arguments throughout the book are very clear, many exercises are presented … . This way the lecturers with talented audience will find many ideas how to hold out the beauty behind the dry techniques. We highly recommend this book for anyone related to enumeration … ." (Péter Hajnal, Acta Scientiarum Mathematicarum, Vol. 74, 2008)

• Fachbereich Mathematik und Informatik Institut für Mathematik II, Freie Universität Berlin, Berlin, Germany

  Martin Aigner

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