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 2004] and [Aval et al. 2014]. This settles in particular the cases \(\langle\cdot,e_{n-d}h_d\rangle\) and \(\langle\cdot,h_{n-d}h_d\rangle\)...

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Published in:arXiv.org 2020-11
Main Authors: D'Adderio, Michele, Iraci, Alessandro, Anna Vanden Wyngaerd
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Combinatorial analysis
Decoration
Identities
Mathematical analysis
Polynomials
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description We discuss the combinatorics of decorated Dyck paths and decorated parallelogram polyominoes, extending to the decorated case the main results of both [Haglund 2004] and [Aval et al. 2014]. This settles in particular the cases \(\langle\cdot,e_{n-d}h_d\rangle\) and \(\langle\cdot,h_{n-d}h_d\rangle\) of the Delta conjecture of Haglund, Remmel and Wilson (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 [Haglund et al. 2018], [Zabrocki 2016], [Aval et al. 2015]). The main technical tool is a new identity in the theory of Macdonald polynomials that extends a theorem of Haglund in [Haglund 2004]. This is an edited merge of arXiv:1712.08787 and arXiv:1709.08736
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subjects Combinatorial analysis
Decoration
Identities
Mathematical analysis
Polynomials
title Decorated Dyck paths, polyominoes, and the Delta conjecture
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