A question to the #P-complete proof of the permanent from Ben-Dor/Halevi
This topic has been extensively investigated in recent years under the name of Holographic Algorithms by researchers such as Valiant, Cai, Lu, Xia, Lipton, and others. Essentially all tractable cases of #CSP (counting constraint satisfaction problems) have been identified in terms of dichotomy theorems (FP vs. #P-complete). In particular, Matchgate computations have been identified as the specific class of counting problems that become tractable on planar graphs. See e.g. this link for further references.