Tiling problems, automata, and tiling graphs

This paper continues the investigation of tiling problems via formal languages, which was begun in papers by Merlini, Sprugnoli, and Verri. Those authors showed that certain tiling problems could be encoded by regular languages, which lead automatically to generating functions and other combinatorial information on tilings. We introduce a method of simplifying the DFA’s recognizing these language, which leads to bijective proofs of certain tiling identities. We apply these ideas to some other tiling problems, including three-dimensional tilings and tilings with triangles and rhombi. We also study graph-theoretic variations of these tiling problems.