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 two covers the composition of generating functions, in particular the exponential formula and the Lagrange inversion formula, labelled and unlabelled trees, algebraic, D-finite, and 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 and focusing on combinatorics, especially the Robinson–Schensted–Knuth algorithm. An appendix by Sergey Fomin covers some deeper aspects of symmetric functions, including jeu de taquin and the Littlewood–Richardson rule. The exercises in the book play a vital role in developing the material, and this second edition features over 400 exercises, including 159 new exercises on symmetric functions, all with solutions or references to solutions.
Author(s): Richard P. Stanley
Series: Cambridge Studies in Advanced Mathematics 208
Edition: Second
Publisher: Cambridge University Press
Year: 2023
Language: English
Pages: 802
Cover
Half Title Page
Series Page
Endorsements
Title Page
Copyright Page
Dedication
Contents
Preface to Second Edition
Preface
5 Trees and the Composition of Generating Functions
5.1 The Exponential Formula
5.2 Applications of the Exponential Formula
5.3 Enumeration of trees
5.4 The Lagrange inversion formula
5.5 Exponential structures
5.6 Oriented trees and the Matrix-Tree Theorem
Notes
Exercises
Solutions
6 Algebraic, D-finite, and Noncommutative Generating Functions
6.1 Algebraic generating functions
6.2 Examples of algebraic series
6.3 Diagonals
6.4 D-finite generating functions
6.5 Noncommutative generating functions
6.6 Algebraic formal series
6.7 Noncommutative diagonals
Notes
Exercises
Solutions
7 Symmetric Functions
7.1 Symmetric functions in general
7.2 Partitions and their orderings
7.3 Monomial symmetric functions
7.4 Elementary symmetric functions
7.5 Complete homogeneous symmetric functions
7.6 An involution
7.7 Power sum symmetric functions
7.8 Specializations
7.9 A scalar product
7.10 The combinatorial definition of Schur functions
7.11 The RSK algorithm
7.12 Some consequences of the RSK algorithm
7.13 Symmetry of the RSK algorithm
7.14 The dual RSK algorithm
7.15 The classical definition of Schur functions
7.16 The Jacobi–Trudi identity
7.17 The Murnaghan–Nakayama rule
7.18 The characters of the symmetric group
7.19 Quasisymmetric functions
7.20 Plane partitions and the RSK algorithm
7.21 Plane partitions with bounded part size
7.22 Reverse plane partitions and the Hillman–Grassl correspondence
7.23 Applications to permutation enumeration
7.24 Enumeration under group action
Notes
Exercises
Solutions
Supplementary Problems
Supplementary Solutions
Appendix 1 Knuth Equivalence, Jeu de Taquin, and theLittlewood–Richardson Rule
A1.1 Knuth equivalence and Greene’s theorem
A1.2 Proofs of Theorems A1.1.1, A1.1.4 and A1.1.6
A1.3 Jeu de taquin
A1.4 The Sch¨utzenberger involution
A1.5 The Littlewood–Richardson Rule
A1.6 Variations of the Littlewood–Richardson rule
A1.7 Notes
Appendix 2 The Characters of GL(n,C)
A2.1 Basic definitions and results
A2.2 Two operations on symmetric functions
Bibliography
Index
