**Abstract:** Given any diagram (a finite collection of boxes on a grid), one can define an associated symmetric function. In many cases, these symmetric functions contain interesting and nontrivial information related to the diagram: for Young diagrams one obtains Schur functions; for skew diagrams, skew Schur functions; for permutation diagrams, Stanley symmetric functions, which describe reduced words. Liu defined a collection of subvarieties of the Grassmannian indexed by diagrams, and conjectured that their cohomology classes are represented by the corresponding diagram symmetric functions. I will give a counterexample to Liu's conjecture, along with results limiting how badly it can fail in the case of permutation diagrams. I will also discuss a connection to rank varieties (a special case of Knutson-Lam-Speyer's positroid varieties), and some new results on their cohomology classes.