  We interprete results of Markman on monodromy operators as a universality
statement for descendent integrals over moduli spaces of stable sheaves on $K3$
surfaces. This yields effective methods to reduce these descendent integrals to
integrals over the punctual Hilbert scheme of the $K3$ surface. As an
application we establish the higher rank Segre-Verlinde correspondence for $K3$
surfaces as conjectured by G\"ottsche and Kool.
