“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)
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.
