Generalizations of Two Statistics on Linear Tilings

In this paper, we study generalizations of two well-known statistics on linear square-and-domino tilings by considering only those dominos whose right half covers a multiple of k, where k is a fixed positive integer. Using the method of generating functions, we derive explicit expressions for the joint distribution polynomials of the two statistics with the statistic that records the number of squares in a tiling. In this way, we obtain two families of q-generalizations of the Fibonacci polynomials. When k = 1, our formulas reduce to known results concerning previous statistics. Special attention is payed to the case k = 2. As a byproduct of our analysis, several combinatorial identities are obtained.