Tilings of Rectangular Regions by Rectangular Tiles: Counts Derived from Transfer Matrices

Abstract

Step by step completion of a left-to-right tiling of a rectangular floor with tiles of a single shape starts from one edge of the floor, considers the possible ways of inserting a tile at the leftmost uncovered square, passes through a sequence of rugged shapes of the front line between covered and uncovered regions of the floor, and finishes with a straight front line at the opposite edge. We count the tilings by mapping the front shapes to nodes in a digraph, then counting closed walks on that digraph with the transfer matrix method. Generating functions are detailed for tiles of shape 1 x 3, 1 x 4 and 2 x 3 and modestly wide floors. Equivalent results are shown for the 3-dimensional analog of filling bricks of shape 1x 1 x 2, 1 x 1 x 3, 1 x 1 x 4, 1 x 2 x 2 or 1 x 2 x 3 into rectangular containers of small cross sections.