Algebraic Combinatorics

1. Front Matter

   Pages i-xii

   PDF

2. Walks in Graphs

   • Richard P. Stanley

   Pages 1-9

3. Cubes and the Radon Transform

   • Richard P. Stanley

   Pages 11-19

4. Random Walks

   • Richard P. Stanley

   Pages 21-30

5. The Sperner Property

   • Richard P. Stanley

   Pages 31-41

6. Group Actions on Boolean Algebras

   • Richard P. Stanley

   Pages 43-55

7. Young Diagrams and q-Binomial Coefficients

   • Richard P. Stanley

   Pages 57-73

8. Enumeration Under Group Action

   • Richard P. Stanley

   Pages 75-101

9. A Glimpse of Young Tableaux

   • Richard P. Stanley

   Pages 103-133

10. The Matrix-Tree Theorem

    • Richard P. Stanley

    Pages 135-150

11. Eulerian Digraphs and Oriented Trees

    • Richard P. Stanley

    Pages 151-161

12. Cycles, Bonds, and Electrical Networks

    • Richard P. Stanley

    Pages 163-185

13. Miscellaneous Gems of Algebraic Combinatorics

    • Richard P. Stanley

    Pages 187-207

14. Back Matter

    Pages 209-223

    PDF

Written by one of the foremost experts in the field, Algebraic Combinatorics is a unique undergraduate textbook that will prepare the next generation of pure and applied mathematicians. The combination of the author’s extensive knowledge of combinatorics and classical and practical tools from algebra will inspire motivated students to delve deeply into the fascinating interplay between algebra and combinatorics. Readers will be able to apply their newfound knowledge to mathematical, engineering, and business models.

The text is primarily intended for use in a one-semester advanced undergraduate course in algebraic combinatorics, enumerative combinatorics, or graph theory.  Prerequisites include a basic knowledge of linear algebra over a field, existence of finite fields, and group theory.  The topics in each chapter build on one another and include extensive problem sets as well as hints to selected exercises. Key topics include walks on graphs, cubes and the Radon transform, the Matrix–Tree Theorem, and the Sperner property. There are also three appendices on purely enumerative aspects of combinatorics related to the chapter material: the RSK algorithm, plane partitions, and the enumeration of labeled trees.

Richard Stanley is currently professor of Applied Mathematics at the Massachusetts Institute of Technology. Stanley has received several awards including the George Polya Prize in applied combinatorics, the Guggenheim Fellowship, and the Leroy P. Steele Prize for mathematical exposition. Also by the author: Combinatorics and Commutative Algebra, Second Edition, © Birkhauser.

• Matrix-Tree Theorem
• Radon transform
• Sperner property
• algebraic combinatorics
• combinatorics

“The chosen topics represent a sample of enumerative combinatorics suitable for the elementary algebra available to an undergraduate student. … This is a book that can be used to teach a topics course for senior undergraduates. It will help them to solidify their just-acquired abstract algebra and at the same it will introduce them to some beautiful topics in pure or applied combinatorics.” (Felipe Zaldivar, MAA Reviews, maa.org, December, 2015)

“This gentle book provides the perfect stepping-stone up. The various chapters treat diverse topics … . Stanley’s emphasis on ‘gems’ unites all this--he chooses his material to excite students and draw them into further study. … Summing Up: Highly recommended. Upper-division undergraduates and above.” (D. V. Feldman, Choice, Vol. 51 (8), April, 2014)

• (MIT), Dept. Mathematics, Massachusetts Institute of Technology, Cambridge, USA

  Richard P. Stanley

Professor Richard Stanley is one of the most well-known algebraic combinatorists in the world. He is currently professor of Applied Mathematics at the Massachusetts Institute of Technology. Amongst his several visiting professorships, Stanley has received numerous awards including the George Polya Prize in Applied Combinatorics, Guggenheim Fellowship, admission to both the American Academy and National Academies of Sciences, Leroy P. Steele Prize for Mathematical Exposition, Rolf Schock Prize in Mathematics, Senior Scholar at Clay Mathematics Institute, Aisenstadt Chair, Honorary Doctor of Mathematics from the University of Waterloo, and an honorary professorship at the Nankai University. Professor Stanley has had over 50 doctoral students and is well known for his excellent teaching skills. Stanley’s list of publications amount to over 155.

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