Two generalizations of column-convex polygons

Svjetlan Feretić, Anthony J Guttmann
2009 Journal of Physics A: Mathematical and Theoretical  
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 columnconvex 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
more » ... refore 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 growth constants of both classes are greater than the growth constant of column-convex polyominoes. Rather tight lower bounds on the growth constants complement a comprehensive asymptotic analysis.
doi:10.1088/1751-8113/42/48/485003 fatcat:kjmzkqfvvrf2phxyalsgycn6py