Analytic Combinatorics in Several Variables

Aimed at graduate students and researchers in enumerative combinatorics, this book is the first to treat the analytic aspects of combinatorial enumeration from a multivariate perspective.

Robin Pemantle (Author), Mark C. Wilson (Author)

9781107031579, Cambridge University Press

Hardback, published 31 May 2013

392 pages, 53 b/w illus. 2 tables 64 exercises
23.5 x 15.6 x 2.7 cm, 0.66 kg

'The organization of the book is exemplary. A thorough and well-designed introduction provides full context and is worth rereading as one works through the book … The treatment of analytic methods for multivariate generating functions in this book is breathtaking. A detailed overview is followed by thorough chapters on smooth point asymptotics, multiple point asymptotics, and cone point asymptotics, then four worked examples, and extensions. The end result, a combination of analytic, Morse-theoretic, algebraic, topological, and asymptotic methods, is surprisingly effective. Indeed, it is astonishing that the authors have found relevant ways to exploit such a broad spectrum of mathematical tools to address the problem at hand.' Robert Sedgewick, Bulletin of the AMS

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/mvGF/asymultseq/ACSVbook/.

Part I. Combinatorial Enumeration: 1. Introduction
2. Generating functions
3. Univariate asymptotics
Part II. Mathematical Background: 4. Saddle integrals in one variable
5. Saddle integrals in more than one variable
6. Techniques of symbolic computation via Grobner bases
7. Cones, Laurent series and amoebas
Part III. Multivariate Enumeration: 8. Overview of analytic methods for multivariate generating functions
9. Smooth point asymptotics
10. Multiple point asymptotics
11. Cone point asymptotics
12. Worked examples
13. Extensions
Part IV. Appendices: Appendix A. Manifolds
Appendix B. Morse theory
Appendix C. Stratification and stratified Morse theory.

Subject Areas: Combinatorics & graph theory [PBV], Discrete mathematics [PBD], Mathematics [PB]