# Special cases of hard counting problems that are easy

We know that bipartite planar perfect matching count is easy, permanent mod $3^t$ is easy for orthogonal matrices, permanent mod $2$ is easy, bounded rank permanent is easy.

Outside of permanent which other hard counting problems have special cases that are easy? One example is counting points in polytope in fixed dimension.

• Your question looks a bit broad to me. First you talk about counting modulo, so are you interested in this kind of counting problems or exact counting? Which area? #SAT is easy on some structured instances. So is #VertexCover. How do you even define counting problem (like permanent is not really a counting problem even if we can define it this way)? Function in #P? But then every function in FP is easy... – holf Apr 5 '18 at 8:20
• I changed the wording. I am interested in both counting modulo and exact counting. – T.... Apr 5 '18 at 10:08