In 2015 Haglund, Remmel and Wilson conjectured a combinatorial formula for the symmetric function $\Delta_{h_m}\Delta_{e_{n-k-1}}'e_n$ in terms of partially labelled Dyck paths, which goes under the name of "generalized Delta conjecture". In the first part of the talk we will state this conjecture, discuss some motivating background, and survey what is known about this conjecture.

In the second part of the talk we will do the same for our recent (related) conjectured combinatorial formula for $[n-k]_t/[n]_t\Delta_{h_m}\Delta_{e_{n-k}}\omega(p_n)$ in terms of partially labelled square paths, which we call "generalized Delta square conjecture".

This is joint work with Alessandro Iraci and Anna Vanden Wyngaerd.