I was wondering if there's a formula or combinatorial method (perhaps by using induction or recursion) for counting the number of self-avoiding rook paths on a m × n grid for small values of m and n (for instance m, n < 7).
While studying smaller cases, I realized that you can simply prove by induction that for 2 × n grids the number of these paths is 2^(n - 1) and there also seems to be a recursive function for grids of 3 × n provided in here.