The $q,t$-Catalan Numbers and the Space of Diagonal Harmonics: With an Appendix on the Combinatorics of Macdonald Polynomials
About this Title
James Haglund, University of Pennsylvania, Philadelphia, Philadelphia, PA
Publication: University Lecture Series
Publication Year 2008: Volume 41
ISBNs: 978-0-8218-4411-3 (print); 978-1-4704-2185-4 (online)
MathSciNet review: MR2371044
MSC: Primary 05E05; Secondary 05A05, 05A30, 33D52
This book contains detailed descriptions of the many exciting recent developments in the combinatorics of the space of diagonal harmonics, a topic at the forefront of current research in algebraic combinatorics. These developments led in turn to some surprising discoveries in the combinatorics of Macdonald polynomials, which are described in Appendix A. The book is appropriate as a text for a topics course in algebraic combinatorics, a volume for self-study, or a reference text for researchers in any area which involves symmetric functions or lattice path combinatorics.
The book contains expository discussions of some topics in the theory of symmetric functions, such as the practical uses of plethystic substitutions, which are not treated in depth in other texts. Exercises are interspersed throughout the text in strategic locations, with full solutions given in Appendix C.
Graduate students and research mathematicians interested in combinatorics.
Table of Contents
- Chapter 1. Introduction to $q$-analogues and symmetric functions
- Chapter 2. Macdonald polynomials and the space of diagonal harmonics
- Chapter 3. The $q, t$-Catalan numbers
- Chapter 4. The $q, t$-Schröder polynomial
- Chapter 5. Parking functions and the Hilbert series
- Chapter 6. The shuffle conjecture
- Chapter 7. The proof of the $q, t$-Schröder theorem
- Appendix A. The combinatorics of Macdonald polynomials
- Appendix B. The Loehr-Warrington conjecture
- Appendix C. Solutions to exercises