Generalization of a statistic on linear domino arrangements

In this paper, we generalize an earlier statistic on square-and-domino tilings by considering only those squares covering a multiple of $k$, where $k$ is a fixed positive integer. We consider the distribution of this statistic jointly with the one that records the number of dominos in a tiling. We derive both finite and infinite sum expressions for the corresponding joint distribution polynomials, the first of which reduces when $k=1$ to a prior result. The cases $q=0$ and $q=-1$ are noted for general $k$. Finally, the case $k=2$ is considered specifically, where further results may be given, including a combinatorial proof when $q=-1$.