Two generalizations of column-convex polygons

Abstract

Column-convex polygons were first counted by area several decades ago, and the result was found to be a simple, rational, generating function. In this work we generalize that result. Let a p-column polyomino be a polyomino whose columns can have 1, 2, ..., p connected components. Then column-convex polygons are equivalent to 1-convex polyominoes. The area generating function of even the simplest generalization, namely 2-column polyominoes, is unlikely to be solvable. We therefore define two classes of polyominoes which interpolate between column-convex polygons and 2-column polyominoes. We derive the area generating functions of those two classes, using extensions of existing algorithms. The..