Pemantle R., Wilson M.C. Analytic Combinatorics in Several Variables

Cambridge University Press, UK, 2013. — 396 p. — (Cambridge Studies in Advanced Mathematics 140) — ISBN: 1107031575This book is the first to treat the analytic aspects of combinatorial enumeration from a multivariate perspective. Analytic combinatorics is a branch of enumeration that uses analytic techniques to estimate combinatorial quantities: generating functions are defined and their coefficients are then estimated via complex contour integrals. The multivariate case involves techniques well known in other areas of mathematics but not in combinatorics. Aimed at graduate students and researchers in enumerative combinatorics, the book contains all the necessary background, including a review of the uses of generating functions in combinatorial enumeration as well as chapters devoted to saddle point analysis, Groebner bases, Laurent series and amoebas, and a smattering of differential and algebraic topology. All software along with other ancillary material can be located via the book's website, http://www.cs.auckland.ac.nz/~mcw/Research/mvGF/asymultseq/ACSVbook/.Combinatorial EnumerationGenerating Functions
Univariate Asymptotics
Mathematical Background
Fourier-Laplace Integrals in One Variable
Fourier-Laplace Integrals in More than One Variable
Techniques of Symbolic Computation via Gröbner Bases
Cones, Laurent Series, and Amoebas
Multivariate Enumeration
Overview of Analytic Methods for Multivariate Generating Functions
Smooth Point Asymptotics
Multiple Point Asymptotics
Cone Point Asymptotics
Worked Examples
Extensions
Appendixes
Integration on Manifolds
Morse Theory
Stratification and Stratified Morse Theory