Enumerative Combinatorics: Volume 1

Publisher Description (unedited Publisher Data) This Second Volume Of A Two-volume Basic Introduction To Enumerative Combinatorics Covers The Composition Of Generating Functions, Trees, Algebraic Generating Functions, D-finite Generating Functions, Noncommutative Generating Functions, And Symmetric Functions. The Chapter On Symmetric Functions Provides The Only Available Treatment Of This Subject Suitable For An Introductory Graduate Course On Combinatorics, And Includes The Important Robinson-schensted-knuth Algorithm. Also Covered Are Connections Between Symmetric Functions And Representation Theory. An Appendix By Sergey Fomin Covers Some Deeper Aspects Of Symmetric Function Theory, Including Jeu De Taquin And The Littlewood-richardson Rule. As In Volume 1, The Exercises Play A Vital Role In Developing The Material. There Are Over 250 Exercises, All With Solutions Or References To Solutions, Many Of Which Concern Previously Unpublished Results. Graduate Students And Research Mathematicians Who Wish To Apply Combinatorics To Their Work Will Find This An Authoritative Reference. Library Of Congress Subject Headings For This Publication: Combinatorial Enumeration Problems. V. 1. What Is Enumerative Combinatorics? -- Sieve Methods -- Partially Ordered Sets -- Rational Generating Functions -- Graph Theory Terminology -- V. 2. Trees And The Composition Of Generating Functions -- Algebraic, D-finite, And Noncommutative Generating Functions -- Symmetric Functions -- Knuth Equivalence, Jeu De Taquin, And The Littlewood-richardson Rule -- The Characters Of Gl (n, C). Richard P. Stanley. Originally Published: Monterey, Calif. : Wadsworth & Brooks/cole Advanced Books & Software, C1986- . (the Wadsworth & Brooks/cole Mathematics Series). Includes Bibliographical References And Index. [2 pages per pdf page]Main subject category: • Enumerative CombinatoricsContents: • What is Enumerative Combinatorics • Sieve Methods • Partially Ordered Sets • Rational Generating Functions • Appendix Graph Theory Terminology • IndexThis book, the first of a two-volume basic introduction to enumerative combinatorics, concentrates on the theory and application of generating functions, a fundamental tool in enumerative combinatorics.Richard Stanley covers those parts of enumerative combinatorics with the greatest applications to other areas of mathematics. The four chapters are devoted to an accessible introduction to enumeration, sieve methods ‒ including the Principle of Inclusion-Exclusion, partially ordered sets, and rational generating functions.A large number of exercises, almost all with solutions, augment the text and provide entry into many areas not covered directly. Graduate students and research mathematicians who wish to apply combinatorics to their work will find this an authoritative reference. This is the second volume of a two-volume work on the subject of enumerative combinatorics, an area of mathematics with connections to many other topics within and outside of mathematics, such as computer science, spectroscopy, algebraic geometry, algebraic topology, and representation theory. Many topics covered (in particular, the theory of symmetric functions) are not available in any other textbook at this level, and the usefulness of the book is enhanced by over 250 exercises with solutions.Although primarily intended as a textbook for graduate students and a resource for professional mathematicians, some parts of the book will be accessible to mathematics undergraduates and even interested amateurs. Enumerative combinatorics deals with the basic problem of counting how many objects have a given property, a subject of great applicability. This book provides an introduction at a level suitable for graduate students. Extensive exercises with solutions show connections to other areas of mathematics. If F(x) and G(x) are formal power series with G(0) = 0, then we have seen (after Proposition 1.1.9) that the composition F(G(x) is a well-defined formal power series.