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Analytic combinatorics in several variables. (English) Zbl 1297.05004

Cambridge Studies in Advanced Mathematics 140. Cambridge: Cambridge University Press (ISBN 978-1-107-03157-9/hbk; 978-1-139-38186-4/ebook). xiii, 380 p. (2013).
The book describes the recent progress of analytic combinatorics, which refers to “the use of complex anaytic methods to solve problems in combinatorial enumeration”, by means of the study of generating functions. The main progress made from the 1990s (when the theory was “in its infancy”) up to now are described, including mainly asymptotic results in one variable (Part I) and many variables (Part III).
The book also discusses the necessary mathematical background in detail, such as Fourier-Laplace integrals, Laurent series and Gröbner bases (Part II), integration on manifolds and Morse theory (Part IV).

MSC:

05-02 Research exposition (monographs, survey articles) pertaining to combinatorics
05A15 Exact enumeration problems, generating functions
05A16 Asymptotic enumeration
05C90 Applications of graph theory
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