Given an $m\times n$ chess board, we place $p$ $2\times 1$ dominoes on the board so that they don't overlap. How many ways can we place them?


When each square of the board is covered by a domino this is the well-known tiling problem. Is there any research on the case where $p$ is small such that we have to leave some empty squares on the board?



An "incomplete" tiling?
If we sum over all $p$, then we obtain the number of monomer-dimer tilings of an $m\times n$ chessboard. This number is known to be #P-complete and hence very likely computationally intractable. For a reference, see citation [7] of the paper by Yong Kung [here](https://arxiv.org/abs/cond-mat/0610690).


