On the complexity of Jensen's algorithm for counting fixed polyominoes

AbstractRecently I. Jensen published a novel transfer-matrix algorithm for computing the number of polyominoes in a rectangular lattice. However, his estimation of the computational complexity of the algorithm (O((2)n), where n is the size of the polyominoes), was based only on empirical evidence. In contrast, our research provides some solid proof. Our result is based primarily on an analysis of the number of some class of strings that plays a significant role in the algorithm. It turns out that this number is closely related to Motzkin numbers. We provide a rigorous computation that roughly confirms Jensen's estimation. We obtain the bound O(n5/2(3)n) on the running time of the algorithm, while the actual number of polyominoes is about C4.06n/n, for some constant C>0

On the enumeration of permutominoes

Although the exact counting and enumeration of polyominoes remain challenging open problems, several positive results were achieved for special classes of polyominoes. We give an algorithm for direct enumeration of permutominoes by size, or, equivalently, for the enumeration of grid orthogonal polygons. We show how the construction technique allows us to derive a simple characterization of the class of convex permutominoes, which has been extensively investigated. The approach extends to other classes, such as the row convex and the directed convex permutominoes.Fondo Europeo de Desarrollo RegionalFundação para a Ciência e a Tecnologi