Analytic Combinatorics in Several Variables (Cambridge Studies in Advanced Mathematics)
Cambridge University Press, 5/31/2013
EAN 9781107031579, ISBN10: 1107031575
Hardcover, 392 pages, 15.3 x 3 x 2.8 cm
Language: English
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.