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An Invitation to Analytic Combinatorics: From One to Several Variables

Stephen Melczer
Publication Date: 
Number of Pages: 
Texts & Monographs in Symbolic Computation
[Reviewed by
Miklós Bóna
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The subtitle is more descriptive than the title as the main focus of the book is indeed the study of multivariate generating functions that count combinatorial objects.   This is a difficult subject that needs sufficient preparation, and the author, to his credit, devotes four chapters out of ten to introduce and motivate the field. In this first part of the book, we get a very brief overview of analytic combinatorics in one variable, and of a very frequent way of connecting univariate and multivariate generating functions through the concept of diagonals. We see the first few examples of lattice path enumeration, which will be a major source of examples in later parts of the book, and we also get an overview of the Kernel method. 
The main reason that analytic combinatorics in several variables is much more difficult than analytic combinatorics in one variable is that the generating functions that occur can behave in many more different ways. This phenomenon serves as the distinguishing principle for the rest of the book. Part II focuses on the smooth case, that is, when the set of singularities of the generating function at hand form a manifold. The author develops a general theory, and shows how nice properties of structures turn into nice properties of their generating functions. Lattice paths are the most frequent examples studied in this part.
Part III examines the case when the singular variety is not a manifold. In this case, we will see that this variety will be a union of hyperplanes, which will allow for a deep analysis in this case as well. More generally, a geometric approach is developed. New methods are discussed for computability issues of differentiably finite power series.
There are exercises (called Problems) at the end of each chapter. Their solutions are not included. For a subject that is this difficult, including at least a few answers would have been helpful. On the other hand, the author has made a very large number of computational worksheets available on his website. These include computer algebra code for all the "worked examples" for the first seven chapters of the book (at the time of this review), both in Sage code, and in static html format.  
Miklós Bóna is a Professor and Distinguished Teaching Scholar at the University of Florida, and the author and editor of several books. His main research interest is enumerative combinatorics.