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 problems 1, 2, and 10 in James Propp's list of problems on enumeration of matchings. 
Harald Helfgott and Ira M. Gessel