Enumeration of Tilings of Diamonds and Hexagons with Defects

We show how to count tilings of Aztec diamonds and hexagons with defects using determinants. In several cases these determinants can be evaluated in closed form. In particular, we obtain solutions to open problems 1, 2, and 10 in James Propp's list of problems on enumeration of matchings [22]. 1. Introduction While studying dimer models, P. W. Kasteleyn [15] noticed that tilings of very simple figures by very simple tiles can be not only plausible physical models, but also starting points for some very interesting enumeration problems. Kasteleyn himself solved the problem of counting tilings of a rectangle by dominoes. He also found a general method (now known as Kasteleyn matrices) for computing the number of tilings of any bipartite planar graph in polynomial time. Kasteleyn's method has proven very useful for computational-experimental work, but it does not, of itself, provide proofs of closed formulas for specific enumeration problems. We shall see a few examples of problems for w..