Integrals of characteristic classes of tautological sheaves on the Hilbert
scheme of points on a surface frequently arise in enumerative problems. We use
the K-theoretic Donaldson-Thomas theory of certain toric Calabi-Yau threefolds
to study K-theoretic variants of such expressions. We study limits of the
K-theoretic Donaldson-Thomas partition function of a toric Calabi-Yau threefold
under certain one-parameter subgroups called slopes, and formulate a condition
under which two such limits coincide. We then explicitly compute the limits of
components of the partition function under so-called preferred slopes,
obtaining explicit combinatorial expressions related to the refined topological
vertex of Iqbal, Kos\c{c}az and Vafa. Applying these results to specific
Calabi-Yau threefolds, we deduce dualities satisfied by a generating function
built from tautological bundles on the Hilbert scheme of points on
$\mathbb{C}^2$. We then use this duality to study holomorphic Euler
characteristics of exterior and symmetric powers of tautological bundles on the
Hilbert scheme of points on a general surface.