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in 2019 (posthumously for
Flajolet). The award citation called the book "an authoritative and highly accessible compendium of its subject, which demonstrates the deep interface between combinatorial mathematics and classical analysis". Although the application of analytic methods in combinatorics
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Reviewer Toufik
Mansour calls it not only "a comprehensive theoretical treatment" but "an interesting read". Reviewer Christopher Hanusa writes that "the writing style is inviting, the subject material is contemporary and riveting", and he recommends the book to anyone "learning or working in
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is not primarily a textbook; for instance, it has no exercises. Nevertheless, it can be used as the textbook for an upper-level undergraduate elective, graduate course, or seminar, although reviewer Miklós Bóna writes that some selection is needed, as it "has enough material for three or more
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The final part investigates the behavior of random combinatorial structures, rather than the total number of structures, using the same toolbox. Beyond expected values for combinatorial quantities of interest, it also studies limit theorems and
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The five chapters of the second part of the book, roughly half of the text and "the heart of the book", concern the application of tools from complex analysis to the generating function, in order to understand the
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as part of the reinterpretation process. The chapters in this part divide the material into the enumeration of unlabeled objects, the enumeration of labeled objects, and multivariate generating functions.
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of the function can be used to derive accurate estimates of the resulting integrals. After an introductory chapter and a chapter giving examples of the possible behaviors of
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The main part of the book is organized into three parts. The first part, covering three chapters and roughly the first quarter of the book, concerns the
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for these quantities. Three appendices provide background on combinatorics and asymptotics, in complex analysis, and in probability theory.
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of the numbers of objects in a combinatorial class. In particular, for sufficiently well-behaved generating functions,
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are associated with formulas that describe their structures, and then those formulas are reinterpreted to produce the
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to understand the growth rates of the numbers of combinatorial objects. It was written by
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semesters". It can also be a reference for researchers in this subject.
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of the classes, in some cases using tools such as the
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35:is a book on the mathematics of
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472:Enumerative combinatorics
102:Cauchy's integral formula
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130:large deviations theory
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32:Analytic Combinatorics
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203:Leroy P. Steele Prize
118:meromorphic functions
61:Leroy P. Steele Prize
81:generating functions
59:in 2009. It won the
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396:"Review of
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212:G. H. Hardy
98:asymptotics
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230:References
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