Analytic combinatorics is a powerful branch of modern mathematics with several applications to physics and computer science, not to speak of mathematics itself. 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.

After a helpful literature review of open problems on the topic, the author addresses analytic combinatorics in several variables (ACSV), both for smooth and non-smooth points. I found the theory and applications to be quite lucidly explained, with helpful worksheets for the SageMath and Maple computer algebra systems.

The book has three parts and an introductory chapter 1. Part 1, “Background and Information” (chapters 2 to 4) covers generating functions and analytic combinatorics, multivariate series and diagonals, and lattice path enumeration. Part 2, “Smooth ACSV and Aapplications” (again, three chapters) looks at the theory of ACSV for smooth points, lattice walks, and automated analytic combinatorics. Finally, Part 3, “Non-Smooth ACSV” (chapters 8 to 10) discusses topics such as poles on a hyperplane arrangement, multiple points, and lattice paths.

The target readership includes graduate and advanced undergraduate students of mathematics and computer science, as well as researchers of these and allied areas. Given that Pemantle and Wilson’s textbook Analytic combinatorics in several variables [1] is meant more for researchers, Melczer’s, in contrast, serves more pedagogical interests and is therefore a welcome addition to the literature. The prerequisite for understanding this useful book is sound knowledge of advanced calculus and mathematical analysis (including analysis of sequences and series).