Macdonald cumulant positivity for arbitrary partition shapes

Prove that, for every finite tuple of partitions, the associated Macdonald cumulant is Schur positive, meaning that all of its Schur coefficients belong to \(\mathbb Z_{\ge 0}[q,t]\).

Background

The paper recalls a conjecture of Dołęga asserting positivity for Macdonald cumulants associated with arbitrary tuples of partitions. The paper proves a stronger positivity statement only in the special case in which every partition in the tuple is a single row, leaving the general-shape conjecture unresolved.

Macdonald cumulants generalize ordinary Macdonald positivity: for one partition, the conjecture reduces to Haiman’s theorem, while for multiple partitions it concerns multivariate q,tq,t-Kostka polynomials and their Schur expansions.

References

For all partitions \lambda1,\dots,\lambdar, the Macdonald cumulant \kappa(\lambda1,\dots,\lambdar) is Schur positive, i.e. its Schur coefficients lie in \mathbb Z_{\ge0}[q,t].

— Theta operators at $t=1$, Macdonald cumulants, and LLT positivity  (2609.29957 - Haglund et al., 24 Sep 2026) in Conjecture 1.6, Section 1