
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 square border with and cuisenaire rods is proved using a transition matrix approach. This problem is then generalised to 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.
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Taxonomy
TopicsQuasicrystal Structures and Properties · Advanced Materials and Mechanics · Supramolecular Self-Assembly in Materials
