We show that the Quot scheme $$\text{Quot}_{\mathbb{A}^3}(\mathcal{O}^r,n)$$ admits a symmetric obstruction theory, and we compute its virtual Euler characteristic. We extend the calculation to locally free sheaves on smooth $3$-folds, thus refining a special case of a recent Euler characteristic calculation of Gholampour–Kool. We then extend Toda’s higher rank DT/PT correspondence on Calabi–Yau $3$-folds to a local version centered at a fixed slope stable sheaf. This generalises (and refines) the local DT/PT correspondence around the cycle of a Cohen–Macaulay curve. Our approach clarifies the relation between Gholampour–Kool’s functional equation for Quot schemes, and Toda’s higher rank DT/PT correspondence.