Let $X$ be a smooth projective curve of postive genus or else a del Pezzo surface and let $Q(V,c)$ be the Grothendieck Quot scheme associated to a vector bundle $V$ and a Chern class $c$ (of the quotient) on $X$. For some choices of $V$ and $c$ this is a finite, reduced scheme that is therefore something we might enumerate. When $X$ is a curve this was done by Marian and Oprea, and the answer is given by the Verlinde formula. When $X$ is a del Pezzo surface, there seems to be a similar connection with Goettsche-type formulas that begs a deeper understanding. This is joint work with Thomas Goller and Drew Johnson.