Robin (University of Pennsylvania) Pemantle & Mark C. (University of Auckland) Wilson 
Analytic Combinatorics in Several Variables [PDF ebook] 

This 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/mv GF/asymultseq/ACSVbook/.