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Analytic Combinatorics in Several Variables

2nd Edition

Part of Cambridge Studies in Advanced Mathematics

  • Date Published: February 2024
  • availability: Available
  • format: Hardback
  • isbn: 9781108836623

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About the Authors
  • Discrete structures model a vast array of objects ranging from DNA sequences to internet networks. The theory of generating functions provides an algebraic framework for discrete structures to be enumerated using mathematical tools. This book is the result of 25 years of work developing analytic machinery to recover asymptotics of multivariate sequences from their generating functions, using multivariate methods that rely on a combination of analytic, algebraic, and topological tools. The resulting theory of analytic combinatorics in several variables is put to use in diverse applications from mathematics, combinatorics, computer science, and the natural sciences. This new edition is even more accessible to graduate students, with many more exercises, computational examples with Sage worksheets to illustrate the main results, updated background material, additional illustrations, and a new chapter providing a conceptual overview.

    • Develops classical tools in topology and analysis from a computational point of view to motivate the study of abstract mathematical theories, and to show how pure mathematics can be applied to concrete applications in mathematics, computer science, and the natural sciences
    • Provides an exposition that weaves in the considerable background material in a way that a graduate student can successfully tackle
    • Allows readers to replicate computations and check their work in Sage worksheets
    • Comprehensively surveys forty years of development of an emerging field, and shows that combinatorial problems can draw together many fields of mathematics, including algebraic geometry, harmonic analysis, and singularity theory
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    Reviews & endorsements

    'A definitive treatment of a challenging but very useful subject. There is a wide variety of situations calling for the estimation of the coefficients of a multivariate generating function. The authors have done a superb job of classifying and elucidating the myriad of available techniques for achieving this aim.' Richard P. Stanley, University of Miami

    'This book is an invaluable resource that is certain to have dramatic impact on research and teaching in this rapidly developing area of mathematics. The first edition broke new ground; this edition prepares the field for others to harvest new knowledge with important applications in many scientific disciplines.' Robert Sedgewick, Princeton University

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    Product details

    • Edition: 2nd Edition
    • Date Published: February 2024
    • format: Hardback
    • isbn: 9781108836623
    • length: 592 pages
    • dimensions: 236 x 158 x 37 mm
    • weight: 1.02kg
    • availability: Available
  • Table of Contents

    Part I. Combinatorial Enumeration:
    1. Introduction
    2. Generating functions
    3. Univariate asymptotics
    Part II. Mathematical Background:
    4. Fourier–Laplace integrals in one variable
    5. Multivariate Fourier–Laplace integrals
    6. Laurent series, amoebas, and convex geometry
    Part III. Multivariate Enumeration:
    7. Overview of analytic methods for multivariate generating functions
    8. Effective computations and ACSV
    9. Smooth point asymptotics
    10. Multiple point asymptotics
    11. Cone point asymptotics
    12. Combinatorial applications
    13. Challenges and extensions
    Appendices: A. Integration on manifolds
    B. Algebraic topology
    C. Residue forms and classical Morse theory
    D. Stratification and stratified Morse theory
    References
    Author index
    Subject index.

  • Resources for

    Analytic Combinatorics in Several Variables

    Robin Pemantle, Mark C. Wilson, Stephen Melczer

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  • Authors

    Robin Pemantle, University of Pennsylvania
    Robin Pemantle is Merriam Term Professor of Mathematics at the University of Pennsylvania, working in the fields of probability theory and combinatorics. He received his bachelor's degree from Berkeley and his Ph.D. from MIT. He is a Fellow of the AMS and IMS and a winner of the Rollo Davidson Prize.

    Mark C. Wilson, University of Massachusetts, Amherst
    Mark C. Wilson is Senior Teaching Faculty at the College of Information and Computer Sciences at the University of Massachusetts, Amherst. He received his Ph.D. in mathematics from the University of Wisconsin–Madison. He is Editor-in-Chief of 'Notices of the American Mathematical Society' and life member of the Combinatorial Mathematics Society of Australasia.

    Stephen Melczer, University of Waterloo, Ontario
    Stephen Melczer is Assistant Professor in the Department of Combinatorics and Optimization at the University of Waterloo. He received doctorates from the École normale supérieure de Lyon and the University of Waterloo. He is a recipient of a Governor General Silver Academic Medal and previously published the textbook 'An Invitation to Analytic Combinatorics.'

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