Tiling with Cuisenaire Rods

TL;DR

This paper derives closed-form formulas for counting tilings of square and rectangular borders with cuisenaire rods, considering symmetries, using a transition matrix approach, advancing combinatorial tiling enumeration methods.

Contribution

It introduces a transition matrix method to derive closed-form formulas for tilings with cuisenaire rods, including symmetry considerations, generalizing previous results.

Findings

Closed-form formulas for tilings of square borders.

Formulas for rectangular border tilings.

Counting tilings up to dihedral symmetry.

Abstract

In this paper a closed form expression for the number of tilings of an $n\times n$ square border with $1\times 1$ and $2\times1$ cuisenaire rods is proved using a transition matrix approach. This problem is then generalised to $m\times n$ rectangular borders. The number of distinct tilings up to rotational symmetry is considered, and closed form expressions are given, in the case of a square border and in the case of a rectangular border. Finally, the number of distinct tilings up to dihedral symmetry is considered, and a closed form expression is given in the case of a square border.