Invitation to Analytic Combinatorics : From One to Several Variables

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 theory can 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.

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.