  We prove a combinatorial formula for LLT cumulants of melting lollipops as a
positive combination of LLT polynomials indexed by spanning trees. The result
gives an affirmative answer to a general positivity question for this class of
unicellular LLT cumulants, and gives an independent proof of their
Schur-positivity. In the special case of the complete graph, we also express
the formula in terms of parking functions.
