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An Invitation to Analytic Combinatorics

From One to Several Variables

  • Book
  • © 2021

Overview

Authors:
  1. Stephen Melczer

    1. University of Waterloo, Waterloo, Canada

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  • Interactive computer algebra worksheets allow deeper understanding of the material
  • Many fully worked detailed examples help to illustrate advanced theoretical methods
  • Unique blend of theory, computation, and applications
  • Appeals to students and researchers in multiple areas of mathematics, computer science, and physics

Part of the book series: Texts & Monographs in Symbolic Computation (TEXTSMONOGR)

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Table of contents (11 chapters)

  1. Front Matter

    Pages i-xviii
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  2. Introduction

    • Stephen Melczer
    Pages 1-18

  3. Correction to: An Invitation to Analytic Combinatorics

    • Stephen Melczer
    Pages C1-C1

  4. Background and Motivation

    1. Front Matter

      Pages 19-19
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    2. Generating Functions and Analytic Combinatorics

      • Stephen Melczer
      Pages 21-91

    3. Multivariate Series and Diagonals

      • Stephen Melczer
      Pages 93-141

    4. Lattice Path Enumeration, The Kernel Method, and Diagonals

      • Stephen Melczer
      Pages 143-181

  5. Smooth ACSV and Applications

    1. Front Matter

      Pages 183-183
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    2. The Theory of ACSV for Smooth Points

      • Stephen Melczer
      Pages 185-246

    3. Application: LatticeWalks and Smooth ACSV

      • Stephen Melczer
      Pages 247-262

    4. Automated Analytic Combinatorics

      • Stephen Melczer
      Pages 263-304

  6. Non-Smooth ACSV

    1. Front Matter

      Pages 305-305
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    2. Beyond Smooth Points: Poles on a Hyperplane Arrangement

      • Stephen Melczer
      Pages 307-350

    3. Multiple Points and Beyond

      • Stephen Melczer
      Pages 351-385

    4. Application: Lattice Paths, Revisited

      • Stephen Melczer
      Pages 387-405

  7. Back Matter

    Pages 407-418
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Keywords

About this book

This book uses new mathematical tools to examine broad computability and complexity questions in enumerative combinatorics, with applications to other areas of mathematics, theoretical computer science, and physics. A focus on effective algorithms leads to the development of computer algebra software of use to researchers in these domains.

After a survey of current results and open problems on decidability in enumerative combinatorics, the text shows how the cutting edge of this research is the new domain of Analytic Combinatorics in Several Variables (ACSV). The remaining chapters of the text alternate between a pedagogical development of the theory, applications (including the resolution by this author of conjectures in lattice path enumeration which resisted several other approaches), and the development of algorithms. The final chapters in the text show, through examples and general theory, how results from stratified Morse theorycan help refine some of these computability questions.

Complementing the written presentation are over 50 worksheets for the SageMath and Maple computer algebra systems working through examples in the text.



Reviews

“Melczer has done a good job of mathematically encompassing the issues related to computation and complexity in this domain, adopting an algorithmic approach to explain the underlying computer algebra and its associated software. … I found the theory and applications to be quite lucidly explained … . The target readership includes graduate and advanced undergraduate students of mathematics and computer science, as well as researchers of these and allied areas.” (Soubhik Chakraborty, Computing Reviews, August 17, 2022)

“This book is grounded in computation, which is useful for the comprehension of the subject to the reader. This book provides an accessible introduction to the subject for researchers in combinatorics and broader areas.” (Andrés R. Vindas Meléndez, zbMATH 1468.05002, 2021)

 


Authors and Affiliations

  • University of Waterloo, Waterloo, Canada

    Stephen Melczer

About the author

Stephen Melczer is an Assistant Professor in the Department of Combinatorics and Optimization at the University of Waterloo. Prior to joining Waterloo, he was a CRM-ISM postdoctoral fellow at the Université du Québec à Montréal, a postdoctoral fellow at the University of Pennsylvania, and a visiting scholar at the University of Illinois Urbana-Champaign. His research adapts methods from algebraic and differential geometry, analysis, and topology to create effective tools for combinatorics, mathematics, and computer science. He received doctorates from the École normale supérieure de Lyon and the University of Waterloo in 2017, and is a recipient of a Governor General Silver Academic Medal.



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