  We perform a study of the moduli space of framed torsion-free sheaves on
Hirzebruch surfaces by using localization techniques. We discuss some general
properties of this moduli space by studying it in the framework of
Huybrechts-Lehn theory of framed modules. We classify the fixed points under a
toric action on the moduli space, and use this to compute the Poincare
polynomial of the latter. This will imply that the moduli spaces we are
considering are irreducible. We also consider fractional first Chern classes,
which means that we are extending our computation to a stacky deformation of a
Hirzebruch surface. From the physical viewpoint, our results provide the
partition function of N=4 Vafa-Witten theory on total spaces of line bundles on
P1, which is relevant in black hole entropy counting problems according to a
conjecture due to Ooguri, Strominger and Vafa.
