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Enumerative combinatorics. Volume 2. Paperback ed. (English) Zbl 0978.05002
Cambridge Studies in Advanced Mathematics. 62. Cambridge: Cambridge University Press. xii, 585 p. (2001).
This long awaited Volume 2 of Enumerative combinatorics by Richard Stanley was well worth the wait! At 581 pages it is almost twice the length of Volume 1. Much of the increase in size has to do with the inclusion of even more exercises and solutions than in Volume 1. These carefully selected and organized problems along with their solutions is one of the assets of this book as a graduate text.
Volume 2 has just three chapters. Chapters 5 (Trees and the composition of generating functions) and 6 (Algebraic, D-finite, and noncommutation generating functions) continue the development of generating functions initiated in the last chapter of Volume 1. Chapter 7 (Symmetric functions) includes many traditional topics (e.g. Pólya enumeration) but in this newly designed framework of symmetric functions. Seeing so many seemingly disjoint topics arranged and unified by one central theme makes reading Chapter 7 a joy. Since writing this sentence for the review of the hard cover, the reviewer has had the experience of leading a seminar based on Chapter 7. Only with such close reading does it become apparent just how skillfully the author has arranged the topics within this chapter.
This magnificant two-volume work is best described by a quote from Gian-Carlo Rota’s forward to Volume 2: I find it impossible to predict when Richard Stanleys two-volume exposition of combinatorics may be superseded. No one will try, let along be able, to match the thoroughness of coverage, the care for detail, the definitiveness of proof, the elegance of presentation.

05-02 Research exposition (monographs, survey articles) pertaining to combinatorics
05A15 Exact enumeration problems, generating functions
05A16 Asymptotic enumeration
06A07 Combinatorics of partially ordered sets
05E05 Symmetric functions and generalizations