Explorations into Monomer-Dimer Tilings of Planar Regions

The properties of monomer-dimer tilings of planar regions has been a focused area of study in the mathematical community for many years. Applications include areas such as diatomic molecular bonding and ice-formation. As my research has gone forth, discoveries have been made regarding the number of monomer-dimer tilings in specific regions. We also were able to characterize these larger tilings with polynomials that have not been published by others. Using mathematical programming, I have also found the probability distribution of where monomers will land in a completely random tiling of these square regions. Thorough research has also been done on n by n, 2 by n, and 1 by n regions, and tilings of these regions using "bonding" between what begins as a region of only monomers and turns into one that has both monomers and dimers. Patterns have been confirmed regarding how many steps it takes for these regions to converge or "freeze" and how many dimers are expected to exist after converging. This has been confirmed both through simulations and mathematical analysis.