Theta operators at $t=1$, Macdonald cumulants, and LLT positivity

We study the Theta operators of D'Adderio-Iraci-Vanden Wyngaerd at $t=1$ and show that the power-sum-indexed operators $\Theta_{\mathsf p_k}|_{t=1}$ agree with a commuting family of derivations when restricted to symmetric functions of positive degree. Writing $\widetilde{h}_a$ for the modified Macdonald function indexed by the single row $(a)$ we establish that $\Theta_{\mathsf p_\mu}\widetilde{h}_a|_{t=1}$ is, up to a normalization, the single-row Macdonald cumulant of Dol\k{e}ga. We use these results to show that $\Theta_{\mathsf p_\mu}\widetilde{h}_a|_{t=1}$ and $\Theta_{\mathsf e_\lambda}\widetilde{h}_a|_{t=1}$ are both sums of vertical-strip LLT polynomials indexed by certain plane trees. The former further shows that single-row Macdonald cumulants are LLT-positive, thereby yielding a stronger form of the higher-order Macdonald positivity conjecture of Dol\k{e}ga when all shapes are single rows.