Inspired by Bar-Natan’s description of Khovanov homology, we discuss thin posets and their capacity to support homology and cohomology theories which categorify rank-statistic generating functions. Additionally, we present two main applications. The first, a categorification of certain generalized Vandermonde determinants gotten from the Bruhat order on the symmetric group by applying a special TQFT to smoothings of torus link diagrams. The second is a broken circuit model for chromatic homology, categorifying Whitney’s broken circuit theorem for the chromatic polynomial of graphs.