$k$-tilings of the Aztec diamond
We study $k$-tilings ($k$-tuples of domino tilings) of the Aztec diamond of rank $m$. We assign a weight to each $k$-tiling, depending on the number of vertical dominos and also on the number of ``interactions" between the different tilings. We will compute the generating polynomials of the $k$-tilings by relating them to an integrable colored vertex model. We will then prove some combinatorial results about $k$-tilings in certain limits of the interaction strength.