  Recently, Nekrasov discovered a new "genus" for Hilbert schemes of points on
$\mathbb{C}^4$. We conjecture a DT/PT correspondence for Nekrasov genera for
toric Calabi-Yau 4-folds. We verify our conjecture in several cases using a
vertex formalism. Taking a certain limit of the equivariant parameters, we
recover the cohomological DT/PT correspondence for toric Calabi-Yau 4-folds
recently conjectured by the first two authors. Another limit gives a
dimensional reduction to the $K$-theoretic DT/PT correspondence for toric
3-folds conjectured by Nekrasov-Okounkov. As an application of our techniques,
we find a conjectural formula for the generating series of $K$-theoretic stable
pair invariants of the local resolved conifold. Upon dimensional reduction to
the resolved conifold, we recover a formula which was recently proved by
Kononov-Okounkov-Osinenko.
