Enumerative Combinatorics, Volume 1, Second Edition by Richard P. Stanley Cambridge University Press, 2012 626 pages, softcover Paperback $43.00, Hardcover $102.00

摘要

Chapter 1 develops a deep theory about counting those most basic of combinatorial objects: permutations. The geometrical or graphical representation of permutations is given a prominent place, and with that there is opportunity to explore combinatorial arguments for these objects. Chapter 1 begins, appropriately enough, with a section entitled "How to Count." It goes through some elementary counting problems, as one might see in an introductory course in discrete mathematics, and then introduces generating functions, the fundamental tool of enumerative combinatorics. Although generating functions are used throughout the chapter - and indeed throughout the book - they are not the raison d'etre of the text, unlike in the case of some other books on the topic such as Goulden and Jackson's Combinatorial Enumeration and Wilf's Generatingfunctionology. This introductory section also introduces the notion of a bijective or combinatorial proof and demonstrates the cleanness of the approach through examples that show the kind of intuitive understanding that a combinatorial proof can provide.