Weighted model counting beyond two-variable logic

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-P1 complete for some FO3-sentences. We extend the result for FO2 in two independent directions: to sentences of the form "phi and forallexists^=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-P1 complete.