  We relate the counting of honeycomb dimer configurations on the cylinder to
the counting of certain vertices in Kirillov-Reshetikhin crystal graphs. We
show that these dimer configurations yield the quantum Kostka numbers of the
small quantum cohomology ring of the Grassmannian, i.e. the expansion
coefficients when multiplying a Schubert class repeatedly with different Chern
classes. This allows one to derive sum rules for Gromov-Witten invariants.
