  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
