Two Types of Hexagonal Polyominoes Counted by Area and Number of Columns
A column-convex polyomino is a polyomino in which every column is free of gaps. A level-one column-subconvex polyomino is a polyomino in which every column either has no gaps or has exactly one gap that consists of exactly one cell. In this Demonstration examples of both types are constructed using hexagonal cells.
A level- column-subconvex polyomino is a polyomino in which each column has at most one gap and no gap is of length greater than . Unfortunately, it becomes very difficult to compute the generating functions for these as increases. For more information, see S. Feretić and A. J. Guttmann, "The Generalizations of Column-Convex Polygons," Journal of Physics A: Mathematical and Theoretical, 42(48), 2009. doi: 10.1088/1751-8113/42/48/2009.