[Paper Review] Tilings of Rectangular Regions by Rectangular Tiles: Counts Derived from Transfer Matrices

Richard J. Mathar|arXiv (Cornell University)|Jun 30, 2014

Cellular Automata and Applications7 references4 citations

TL;DR

This paper develops a transfer matrix method to count tilings of rectangular regions and 3D rectangular rooms using 2D and 3D rectangular tiles, modeling tiling progress via state vectors that track the profile of covered regions. It derives explicit generating functions for $1\times3$, $1\times4$, $2\times3$, $1\times1\times2$, $1\times1\times3$, $1\times1\times4$, and $1\times2\times3$ tiles on modestly wide floors and rooms, enabling exact enumeration of tilings via closed walks on state digraphs.

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.

Motivation & Objective

• To develop a systematic method for counting the number of ways to tile rectangular regions and 3D rectangular containers using fixed rectangular tiles.
• To model the tiling process as a sequence of state transitions based on the profile of the front edge between covered and uncovered regions.
• To apply the transfer matrix method to convert tiling enumeration into counting closed walks on a directed graph of state vectors.
• To derive explicit generating functions for specific tile shapes and container dimensions, enabling exact computation of tiling counts.

Proposed method

• Represent each intermediate tiling state by a height vector $ (h_1 h_2 \ldots h_m) $, where $ h_i $ encodes the penetration of tiles into the uncovered region at column $ i $, with $ 0 \leq h_i \leq t_n $.
• Construct a directed graph (digraph) where nodes are state vectors and edges represent valid tile placements that advance the tiling from one state to another.
• Use the transfer matrix method to count closed walks of length $ N $ starting and ending at the all-zero state vector $ (00\ldots0) $, which corresponds to complete tilings.
• Derive ordinary generating functions $ T_{t_m \times t_n}(m,z) = \sum_N T(m,N) z^N $ by applying graph-theoretic rules: edge weights $ z $, serial and parallel path composition, and geometric series for loops.
• Apply pruning rules to eliminate disconnected subgraphs and simplify the generating function expressions.
• Use the generating function structure to extract exact tiling counts via coefficient extraction, with results validated against known integer sequences (e.g., OEIS A000079, A103143, A237357, A237358, A233291).

Experimental results

Research questions

• RQ1 How can the number of distinct tilings of a rectangular $ m \times n $ floor using $ t_m \times t_n $ rectangular tiles be computed efficiently, including both orientations?
• RQ2 What is the structure of the state space of partial tilings, and how can it be encoded as a finite digraph for enumeration?
• RQ3 Can generating functions be derived for tiling problems with $ 1\times3 $, $ 1\times4 $, $ 2\times3 $, and 3D bricks like $ 1\times1\times4 $ and $ 1\times2\times3 $, and what do they reveal about tiling counts?
• RQ4 How do the generating functions for 2D and 3D tiling problems relate to known integer sequences and combinatorial structures?
• RQ5 What is the role of symmetry and connectivity in the state digraph, and how can irrelevant components be removed to simplify the generating function?

Key findings

• For $ 1\times3 $ tiles on a $ 1\times n $ floor, the generating function is derived and shown to produce integer sequences matching known combinatorial counts.
• For $ 1\times4 $ tiles on a $ 2\times n $ floor, the generating function is $ T_{1\times4}(2,4,z) = \frac{1 - z^4 - z^8 + z^{12} - z^6}{(1 - z^2 - 2z^4 + z^8)(-z^{12} + z^4 + 1 - z^8 + z^6)} $, with coefficients matching squares of $ 1\times4 $ tiling counts.
• For $ 1\times1\times4 $ bricks in a $ 2\times4\times n $ room, the generating function yields coefficients $ T(2,4,N) = T_{1\times4}^2(4,N) $, confirming a structural relationship between 2D and 3D tiling counts.
• For $ 1\times2\times3 $ bricks in a $ 2\times2\times n $ room, the generating function is $ \frac{1}{1 - 2z^2} $, producing powers of 2, reflecting the forced return to the straight profile after each placement.
• For $ 3\times3\times n $ rooms with $ 1\times2\times3 $ bricks, the generating function is $ \frac{1 - z^3}{1 - 22z^6 - 7z^3 - 36z^9} $, yielding counts $ 1, 6, 64, 616, 5936, 57408, \ldots $, matching OEIS A237357.
• For $ 3\times4\times n $ rooms with $ 1\times2\times3 $ bricks, the generating function has a complex rational form with numerator and denominator polynomials, producing coefficients $ 1, 1, 11, 64, 296, 1716, 9123, \ldots $, matching OEIS A237358.