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arXiv · 2011.09568

Decorated Dyck paths, polyominoes, and the Delta conjecture

Michele D'Adderio·Alessandro Iraci·Anna Vanden Wyngaerd
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Abstract

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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Michele D'Adderio, Alessandro Iraci, Anna Vanden Wyngaerd. 2020-11-18. Decorated Dyck paths, polyominoes, and the Delta conjecture. https://doi.org/10.1090/memo/1370

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