Decorated Dyck Paths, Polyominoes, and the Delta Conjecture

• Autore: D'Adderio Michele
• Pubblicazione: Providence : , : American Mathematical Society, , 2022; ©2022
• Edizione: 1st ed.
• Descrizione fisica: 1 online resource (138 pages)
• Disciplina: 511/.6; 511.6
• Soggetto topico: Combinatorial analysis; Symmetric functions; Combinatorics -- Algebraic combinatorics -- Symmetric functions and generalizations
• Classificazione: 05E05
• Altri autori: IraciAlessandro WyngaerdAnna Vanden
• Nota di contenuto: 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.
• Sommario/riassunto: "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)"--
• Titolo autorizzato: Decorated Dyck Paths, Polyominoes, and the Delta Conjecture
• ISBN: 9781470471705; 1470471701
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Lingua di pubblicazione: Inglese
• Record Nr.: 9910970102003321
• Lo trovi qui: Univ. Federico II
• Serie: Memoirs of the American Mathematical Society