Decorated Dyck Paths, Polyominoes, and the Delta Conjecture.

Cover
Title page
Introduction
Acknowledgments
Part 1. Definitions and results
Chapter 1. Background and definitions
1.1. Symmetric and quasisymmetric functions
1.2. Combinatorial definitions
Chapter 2. Conjectures
2.1. The Delta conjecture
2.2. The generalised Delta conjecture
2.3. Our conjecture with \pmaj
2.4. Our conjecture with polyominoes
2.5. Our square conjecture
Chapter 3. Our results
3.1. A decorated , -Schröder
3.2. A decorated , -Narayana
3.3. Links with the Delta conjecture
3.4. A symmetry result
3.5. A new , -square
3.6. Symmetric functions identities
3.7. A few open problems
Part 2. Proofs
Chapter 4. Symmetric functions
4.1. Basic identities
4.2. A summation formula
4.3. Three families of plethystic formulae
4.4. Another symmetric function identity
4.5. Two theorems and a corollary
4.6. Δ_{ } ( _{ }) at =1/
Chapter 5. Combinatorics of decorated Dyck paths
5.1. Haglund's (sweep) map
5.2. The map exchanging peaks and falls
5.3. Combinatorial recursions
Chapter 6. Combinatorics of polyominoes
6.1. Parallelogram polyominoes
6.2. Reduced polyominoes
6.3. Two car parking functions
6.4. Partially labelled Dyck paths
6.5. A new \dinv statistic on parallelogram polyominoes
6.6. A \bounce statistic on partially labelled Dyck paths
Chapter 7. Putting the pieces together
7.1. Combinatorial interpretations of plethystic formulae
7.2. Proof of the decorated , -Schröder
7.3. Proof of the decorated , -Narayana
Chapter 8. Square paths
8.1. A new , -square
8.2. Observations when =1/
Appendix A. Proof of the elementary lemmas
Bibliography
Back Cover.