Tilings of Rectangular Regions by Rectangular Tiles: Counts Derived from Transfer Matrices
Abstract
Step by step completion of a lefttoright 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 3dimensional 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.
 Publication:

arXiv eprints
 Pub Date:
 June 2014
 arXiv:
 arXiv:1406.7788
 Bibcode:
 2014arXiv1406.7788M
 Keywords:

 Mathematics  Combinatorics;
 52C20;
 05B45 (Primary) 05A15 (Secondary)
 EPrint:
 21 pages, 21 figures