Author(s): Richard P. Stanley.
Series: Cambridge Studies in Advanced Mathematics 208
Edition: 2
Publisher: Cambridge University Press
Year: 2023
Language: English
Pages: 783
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 for Chapter 5
Solutions to Exercises for Chapter 5
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 for Chapter 6
Solutions to Exercises for Chapter 6
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–Grasslcorrespondence
7.23 Applications to permutation enumeration
7.24 Enumeration under group action
Notes
Exercises for Chapter 7
Solutions for Chapter 7
Supplementary Problems for Chapter 7
Supplementary Solutions for Chapter 7
Appendix 1 - Knuth Equivalence, Jeu de Taquin, and the Littlewood–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 Schützenberger 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
