| The field of exact enumeration of tilings (equivalently, perfect matchings) dates back to the early 1900's when MacMahon proved his classical theorem on the number of plane partitions that fit in a given box. The field has developed vastly during the last two and a half decades. In this thesis, we consider new aspects of the enumeration of tilings by using various methods. We solve an open problem on quasi-hexagons due to James Propp, and prove Matt Blum's conjecture on hexagonal dungeons. The above open problem and conjecture both stood open for more than fourteen years in the field. We also consider several new variants of the Aztec diamond whose tilings are enumerated by perfect powers. In addition, we investigate a new family of regions on the triangular lattice, called quartered hexagons, and prove a simple product formula for their number of tilings. |
