Parallelogram polyominoes, partially labelled Dyck paths, and the Delta conjecture (FULL VERSION)

We introduce area, bounce and dinv statistics on decorated parallelogram polyominoes, and prove that some of their q,t-enumerators match $\langle \Delta_{h_m} e_{n+1},s_{k+1,1^{n-k}}\rangle$, extending in this way the work in (Aval et al. 2014). Also, we provide a bijective connection between decorated parallelogram polyominoes and decorated labelled Dyck paths, which allows us to prove the combinatorial interpretation of the coefficient $\langle \Delta_{e_{m+n-k-1}}'e_{m+n},h_m h_n\rangle$ predicted by the Delta conjecture in (Haglund et al. 2015). Finally, we define a statistic pmaj on partially labelled Dyck paths, which provides another conjectural combinatorial interpretation of ${\mathrm{\Delta }}_{h_{\mathrm{\ell }}}{\mathrm{\Delta }}_{e_{n-k-1}}^{\mathrm{\prime }}{e}_n$, cf. (Haglund et al. 2015). This is the full version of (D’Adderio, Iraci 2017) arXiv:1711.03923.