Jacob Brown '22, Mathematics and Physics, presented "Counting Divisions of a 2 by n Rectangular Grid" at the 2021 MAA Mathfest, held virtually August 4-7. Jacob's research was sponsored by Professors Emilie Wiesner and Dan Visscher from the Math Department.
The abstract for the presentation:
Consider a 2 by n rectangular grid composed of 1 by 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.