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Enumerative Combinatorics, Second Edition

Book Description

Richard Stanley's two-volume basic introduction to enumerative combinatorics has become the standard guide to the topic for students and experts alike. This thoroughly revised second edition of Volume 1 includes ten new sections and more than 300 new exercises, most with solutions, reflecting numerous new developments since the publication of the first edition in 1986. The author brings the coverage up to date and includes a wide variety of additional applications and examples, as well as updated and expanded chapter bibliographies. Many of the less difficult new exercises have no solutions so that they can more easily be assigned to students. The material on P-partitions has been rearranged and generalized; the treatment of permutation statistics has been greatly enlarged; and there are also new sections on q-analogues of permutations, hyperplane arrangements, the cd-index, promotion and evacuation and differential posets.

Table of Contents

  1. Cover
  2. Half Title
  3. Title Page
  4. Copyright
  5. Dedication
  6. Contents
  7. Preface
  8. Acknowledgments
  9. 1. What Is Enumerative Combinatorics?
    1. 1.1 How to Count
    2. 1.2 Sets and Multisets
    3. 1.3 Cycles and Inversions
    4. 1.4 Descents
    5. 1.5 Geometric Representations of Permutations
    6. 1.6 Alternating Permutations, Euler Numbers, and the cd-Index of S[sub(n)]
    7. 1.7 Permutations of Multisets
    8. 1.8 Partition Identities
    9. 1.9 The Twelvefold Way
    10. 1.10 Two q-Analogues of Permutations
    11. Notes
    12. Bibliography
    13. Exercises for Chapter 1
    14. Solutions to Exercises
  10. 2. Sieve Methods
    1. 2.1 Inclusion-Exclusion
    2. 2.2 Examples and Special Cases
    3. 2.3 Permutations with Restricted Position
    4. 2.4 Ferrers Boards
    5. 2.5 V-Partitions and Unimodal Sequences
    6. 2.6 Involutions
    7. 2.7 Determinants
    8. Notes
    9. Bibliography
    10. Exercises for Chapter 2
    11. Solutions to Exercises
  11. 3. Partially Ordered Sets
    1. 3.1 Basic Concepts
    2. 3.2 New Posets from Old
    3. 3.3 Lattices
    4. 3.4 Distributive Lattices
    5. 3.5 Chains in Distributive Lattices
    6. 3.6 Incidence Algebras
    7. 3.7 The Möbius Inversion Formula
    8. 3.8 Techniques for Computing Möbius Functions
    9. 3.9 Lattices and Their Möbius Functions
    10. 3.10 The Möbius Function of a Semimodular Lattice
    11. 3.11 Hyperplane Arrangements
    12. 3.12 Zeta Polynomials
    13. 3.13 Rank Selection
    14. 3.14 R-Labelings
    15. 3.15 (P,ω)-Partitions
    16. 3.16 Eulerian Posets
    17. 3.17 The cd-Index of an Eulerian Poset
    18. 3.18 Binomial Posets and Generating Functions
    19. 3.19 An Application to Permutation Enumeration
    20. 3.20 Promotion and Evacuation
    21. 3.21 Differential Posets
    22. Notes
    23. Bibliography
    24. Exercises for Chapter 3
    25. Solutions to Exercises
  12. 4. Rational Generating Functions
    1. 4.1 Rational Power Series in One Variable
    2. 4.2 Further Ramifications
    3. 4.3 Polynomials
    4. 4.4 Quasipolynomials
    5. 4.5 Linear Homogeneous Diophantine Equations
    6. 4.6 Applications
    7. 4.7 The Transfer-Matrix Method
    8. Notes
    9. Bibliography
    10. Exercises for Chapter 4
    11. Solutions to Exercises
  13. Appendix: Graph Theory Terminology
  14. First Edition Numbering
  15. List of Notation (Partial)
  16. Index