E-raamat: Decorated Dyck Paths, Polyominoes, and the Delta Conjecture

"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)"--

D'Adderio, Iraci, and Wyngaerd discuss the combinatorics of decorated Dyck paths and decorated parallelogram polyominoes, extending to the decorated case in the main results of both Haglund (2001) and Aval et al. (2014). This settles in particular the cases of the Delta conjecture of Haglund, Remmel, and Wilson (2018), they say. They cover background and definitions, conjectures, and their results, then provide proofs for symmetric functions, combinatorics of decorated Dyck paths, combinatorics of polyominoes, putting the pieces together, and square paths. Annotation ©2022 Ringgold, Inc., Portland, OR (protoview.com)