  Here we prove that counting maximum matchings in planar, bipartite graphs is
#P-complete. This is somewhat surprising in the light that the number of
perfect matchings in planar graphs can be computed in polynomial time. We also
prove that counting non-necessarily perfect matchings in planar graphs is
already #P-complete if the problem is restricted to bipartite graphs. So far
hardness was proved only for general, non-necessarily bipartite graphs.
