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

TL;DR

This paper develops recursive and closed-form formulas to count the ways to divide a 2-by-n grid into k pieces using edge cuts, extending previous results for k=2 and 3 to higher k values.

Contribution

It introduces a recursive relationship for counting grid divisions into k pieces and derives polynomial closed-form solutions for fixed k.

Findings

Recursive formula for counting divisions into k pieces.

Closed-form polynomial solutions for fixed k.

Extension of previous work for k=2 and 3 to higher k.

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 $2k-2$.