04533nam 22005533 450 991097010200332120231110220624.097814704717051470471701(MiAaPQ)EBC29379001(Au-PeEL)EBL29379001(CKB)24267882600041(OCoLC)1336954399(EXLCZ)992426788260004120220721d2022 uy 0engurcnu||||||||txtrdacontentcrdamediacrrdacarrierDecorated Dyck Paths, Polyominoes, and the Delta Conjecture1st ed.Providence :American Mathematical Society,2022.©2022.1 online resource (138 pages)Memoirs of the American Mathematical Society ;v.278Print version: D'Adderio, Michele Decorated Dyck Paths, Polyominoes, and the Delta Conjecture Providence : American Mathematical Society,c2022 9781470471576 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."We discuss the combinatorics of decorated Dyck paths and decorated parallelogram polyominoes, extending to the decorated case the main results of both Haglund ("A proof of the Schroder conjecture", 2004) and Aval et al. ("Statistics on parallelogram polyominoes and a analogue of the Narayana numbers", 2014). This settles in particular the cases and of the Delta conjecture of Haglund, Remmel and Wilson ("The delta conjecture", 2018). Along the way, we introduce some new statistics, formulate some new conjectures, prove some new identities of symmetric functions, and answer a few open problems in the literature (e.g., from Aval, Bergeron and Garsia [2015], Haglund, Remmel and Wilson [2018], and Zabrocki [2019]). The main technical tool is a new identity in the theory of Macdonald polynomials that extends a theorem of Haglund in "A proof of the Schroder conjecture" (2004)"--Provided by publisher.Memoirs of the American Mathematical Society Combinatorial analysisSymmetric functionsCombinatorics -- Algebraic combinatorics -- Symmetric functions and generalizationsmscCombinatorial analysis.Symmetric functions.Combinatorics -- Algebraic combinatorics -- Symmetric functions and generalizations.511/.6511.605E05mscD'Adderio Michele1734842Iraci Alessandro1802306Wyngaerd Anna Vanden1802307MiAaPQMiAaPQMiAaPQBOOK9910970102003321Decorated Dyck Paths, Polyominoes, and the Delta Conjecture4347953UNINA