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arXiv · 2106.14755

Counting Divisions of a $2\times n$ Rectangular Grid

Jacob Brown
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Abstract

Consider a $2\times n$ rectangular grid composed of $1\times 1$ squares. Cutting only along the edges between squares, how many ways are there to divide the board into $k$ pieces? Building off the work of Durham and Richmond, who found the closed-form solutions for the number of divisions into 2 and 3 pieces, we prove a recursive relationship that counts the number of divisions of the board into $k$ pieces. Using this recursion, we obtain closed-form solutions for the number of divisions for $k=4$ and $k=5$ using fitting techniques on data generated from the recursion. Furthermore, we show that the closed-form solution for any fixed $k$ must be a polynomial on $n$ with degree $2k-2$.

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Jacob Brown. 2021-06-28. Counting Divisions of a $2\times n$ Rectangular Grid. https://arxiv.org/abs/2106.14755

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