Counting cocircuits and convex two-colourings is #P-complete
  We prove that the problem of counting the number of colourings of the vertices of a graph with at most two colours, such that the colour classes induce connected subgraphs is #P-complete. We also show that the closely related problem of counting the number of cocircuits of a graph is #P-complete. 
Andrew J. Goodall, Steven D. Noble