An Illustrative Study of the Enumeration of Tilings: Conjecture Discovery and Proof Techniques

We have exact formulas for the number of tilings of only a small number of regions: Aztec Diamonds, Fortresses, and regions composed of Lozenge Tiles. However, these regions, particularly aztec diamonds and lozenge tilings are fundamental, and numerous other regions types can be reduced to weighted versions of these graphs using urban renewal techniques. This paper will illustrate the process by which you can generate conjectures about new region types and then prove those conjectures using various graph manipulation techniques and combinatorial arguments to reduce them to previously known results, in particular to the case of weighted aztec diamonds. The example used throughout is a conjecture involving tilings of the plane using triangles and squares, originally proposed in [5] and proved in [3]. We will follow the conjecture from the beginning, using it to illustrate the process of the choice of the boundary conditions determining a finite region, computer experimentation, conjecture formation, and proof, primarily by means of urban renewal and a related transformation. The proof which is developed here is by no means the shortest or smoothest version; it is rather presented in the natural structure of its discovery in an attempt to elucidate the process.