Enumerating Distinct Chessboard Tilings

Counting the number of distinct colorings of various discrete objects, via Burnside’s Lemma and Pólya Counting, is a traditional problem in combinatorics. Motivated by a method for proving upper bounds on the order of the minimal recurrence relation satisfied by a set of tiling instances, we address a related problem in a more general setting. Given an m× n chessboard and a fixed set of (possibly colored) tiles, how many distinct tilings exist, up to symmetry? More specifically, we are interested in the sequences formed by counting the number of distinct tilings of boards of size (m× 1), (m× 2), (m× 3) . . ., for a fixed set of tiles and some natural number m. We present explicit results and closed forms for several well known classes of tiling problems as well as a general result showing that all such sequences satisfy some linear, homogeneous, constant–coefficient recurrence relation. Additionally, we give a characterization of all 1 × n distinct tiling problems in terms of the generalized Fibonacci tilings.