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This book is an introduction to enumerative combinatorics for graduate students and researchers. It concentrates on the theory and application of generating functions, a fundamental tool in enumerative combinatorics. The four chapters are devoted to enumeration, sieve methods (including the Principle of Inclusion-Exclusion), partially ordered sets, and rational generating functions. There are a large number of exercises, almost all with solutions, which greatly augment the text and provide entry into many areas not covered directly. The author stresses important connections with other areas of mathematics. This is a reissue of a book first published in 1986. The author has updated the references and included more problems. Graduate students and research mathematicians who wish to apply combinatorics to their work will find this an authoritative reference.
|Ch. 1||What Is Enumerative Combinatorics?||1|
|Ch. 2||Sieve Methods||64|
|Ch. 3||Partially Ordered Sets||96|
|Ch. 4||Rational Generating Functions||202|
|App||Graph Theory Terminology||293|
Posted June 15, 2001
It was only after having been prodded by Eratosthenes, then Librarian of Alexandria, that Archimedes was induced to write 'The Method.' Fermat is notorious for having written in the margins of Diophantus's 'Arithmetica', where there was never enough room for methods. Newton took an extra year writing 'Principia Mathematica' in order to conceal his methods. (Only after Leibniz began publishing did Newton talk openly about calculus.) Abel said about Gauss that 'he is like the fox, who effaces his tracks with his tail.' Fortunately, mathematicians of the first rank no longer deliberately hide their methods. Unfortunately, few of them seem both willing and able to write lucidly enough for nonspecialists to appreciate subtleties of approach. Richard Stanley is a refreshing exception. His two volume 'Enumerative Combinatorics' is already a classic, both for its depth and for its clarity. Reading these books, one achieves a sense, not only of 'what', but of 'why' and 'how'. Technique is generously illustrated, not only in the exposition, but in the explicit solutions of numerous well chosen exercises. These volumes comprise a masterpiece of mathematical writing.
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