Number of "hamiltonian tours" from upper left to lower left corner of a grid graph?
I think a divide and conquer strategy is going to work here. The asymptotic runtime isn't great, but fortunately our graph is of fixed size. Pick some arbitrary node, $v$, and partition the rest of the nodes into two sets. The number of hamiltonian tours is the number of paths that start at the upper left, go through all nodes in the first side of the partition and end at $v$ multiplied by the number of paths that start at $v$, go through every node in the second half and end at the bottom left corner summed over every possible such partition. Recursively do this for each partition.