  Recently, Marian-Oprea-Pandharipande established (a generalization of) Lehn's
conjecture for Segre numbers associated to Hilbert schemes of points on
surfaces. Extending work of Johnson, they provided a conjectural correspondence
between Segre and Verlinde numbers. For surfaces with holomorphic 2-form, we
propose conjectural generalizations of their results to moduli spaces of stable
sheaves of any rank.
  Using Mochizuki's formula, we derive a universal function which expresses
virtual Segre and Verlinde numbers of surfaces with holomorphic 2-form in terms
of Seiberg-Witten invariants and intersection numbers on products of Hilbert
schemes of points. We prove that certain canonical virtual Segre and Verlinde
numbers of general type surfaces are topological invariants and we verify our
conjectures in examples.
  The power series in our conjectures are algebraic functions, for which we
find expressions in several cases and which are permuted under certain Galois
actions. Our conjectures imply an algebraic analog of the Mari\~{n}o-Moore
conjecture for higher rank Donaldson invariants. For ranks $3$ and $4$, we
obtain explicit expressions for Donaldson invariants in terms of Seiberg-Witten
invariants.
