Decorated Dyck paths, polyominoes, and the Delta conjecture

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