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.