Enumerative Combinatorics Volume 2 2nd Edition by by Richard Stanley 0521560691 9780521560696

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ISBN 10: 0521560691

ISBN 13: 9780521560696

Author: Richard P. Stanley

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 There are over 250 exercises, all with solutions or references to solutions, many of which concern previously unpublished results.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.

Enumerative Combinatorics Volume 2 2nd Table of contents:

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

References

Exercises

Solutions to Exercises

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

References

Exercises

Solutions to Exercises

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

References

A1 Knuth Equivalence, Jeu de Taquin, and the Littlewood-Richardson Rule

A1.1 Knuth Equivalence and Greene’s Theorem

A1.2 Jeu de Taquin

Al.3 The Littlewood-Richardson Rule

Notes

References

A2 The Characters of GL(n, C)

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