  It was recently shown by van den Broeck at al. that the symmetric weighted
first-order model counting problem (WFOMC) for sentences of two-variable logic
FO2 is in polynomial time, while it is Sharp-P_1 complete for some
FO3-sentences. We extend the result for FO2 in two independent directions: to
sentences of the form "phi and \forall\exists^=1 psi" with phi and psi in FO2,
and to sentences formulated in the uniform one-dimensional fragment of FO, a
recently introduced extension of two-variable logic with the capacity to deal
with relation symbols of all arities. Note that the former generalizes the
extension of FO2 with a functional relation symbol. We also identify a complete
classification of first-order prefix classes according to whether WFOMC is in
polynomial time or Sharp-P_1 complete.
