# Analytic Combinatorics in Several Variables, First Edition

**Note: The second edition of this book, written by Robin Pemantle, Mark C. Wilson, and Stephen Melczer, will be published in 2022. If you are interested in reading a draft manuscript in early 2022, please contact one of the authors.**

This book by Robin Pemantle and Mark C. Wilson is aimed at graduate-level researchers in the field of analytic combinatorics, and more experienced researchers in fields in which enumerative questions arise that can be solved by the methods of analytic combinatorics. In contrast to the magnum opus of Flajolet and Sedgewick, this book deals almost exclusively with problems involving more than one variable.

This book forms a part of the more general Analytic Combinatorics in Several Variables project.

## How to get the book

- Published version available from mid-2013, for example from Amazon.
- First draft sent to Cambridge University Press 2012-11-08.

## Reviews and Citations

- Citations on Google Scholar
- Robert Sedgewick in Bulletin of the American Mathematical Society (July 2018): “The organization of the book is exemplary … The treatment of analytic methods for multivariate generating functions in this book is breathtaking … Whether or not one cares about applications, this book is an extremely well- written treatment of a relevant contemporary topic that many mathematicians will see as an opportunity to learn and appreciate new areas of mathematics and how they interact.”
- Bruce Richmond in Mathematical Reviews: “This book is a timely and well-written description of contemporary multivariate analytic combinatorics, and the authors deserve much credit for the progress since 1995. The book is written as a text with helpful exercises to illuminate the concepts; students will see several branches of modern mathematics and computer algebra interact. "
- Miklos Bona in ACM SIGACT News Vol 45 No 2, 2014: “It should be clear from all the above that there is no other book on the market that comes remotely close to this one.”
- Michael Albert in New Zealand Mathematical Society Newsletter, Dec 2013: “Since roughly the turn of this century the authors have been at the forefront of the development of more general, less ad hoc, techniques for such and related problems. This book develops, summarises and illustrates the results of their work both theoretically and in practice. As such, it fills a gap that was very much in need of filling.” … “I for one am very grateful to have it on my desk now.”
- D.V. Feldman in Choice 51.6 (Feb 2014), 1048: “Few books ever made a better case for the grand unity of mathematics.” … “It deserves a place on college library shelves even if nearly all undergraduates will find it too daunting, for it provides a nearly universal answer to the ‘what can I do with this stuff?’ question that students pose in so many basic courses. Recommended.”

## Errata in published edition

Owing to an error by the publisher, early copies of the book are missing the list of symbols (which is very important for understanding the book). It can be downloaded from the publisher’s website.

Thanks to all who have reported errors. They are listed below with the name of the finder and the date of notification. *Note: the second edition of the book corrects many additional errors from the first edition, and contains an overall simplified presentation.*

- p90, Thm 5.1.2: the assumption \(A(0) \not= 0\) is not necessary for the form of the expansion, only in order to give a nice formula for the leading term. Charles Burnette 2016-08-24.
- p192, Eq 9.5.9: second term on right side should have \(H_x^2\), so that the expression is symmetric in \(x\) and \(y\). Repeated on p160. Jim Tao 2013-07-20.
- p198, Eq 9.5.16: subexponential factor missing - correct version is on p160. Jim Tao 2013-07-23.
- p252: “…vertices of the Birkhoff” -> “lattice points in the dilated Birkhoff”. Brendan McKay 2012-11-28.
- p306, Eq 13.2.1: R-tilde should be F-tilde.
- p315, Exercise 13.1: based on a misconception, probably meaningless.
- In many places, “i.e.,” is found where it makes no sense, and should be “that is”. Michael Albert 2013-12.