  We show how the "finite Quot scheme method" applied to Le Potier's strange
duality on del Pezzo surfaces leads to conjectures (valid for all smooth
complex projective surfaces) relating two sets of universal power series on
Hilbert schemes of points on surfaces: those for top chern classes of
tautological sheaves, and those for Euler characteristics of line bundles. We
have verified these predictions computationally for low order. We then give an
analysis of these conjectures in small ranks. We also give a combinatorial
proof of a formula predicted by our conjectures: the top chern class of the
tautological sheaf on $S^{[n]}$ associated to the structure sheaf of a point is
equal to $(-1)^n$ times the $n$th Catalan number.
