The area generating function for simple-2-column polyominoes with hexagonal cells

TL;DR

This paper extends the enumeration of column-convex polygons to a new class of 2-column polyominoes with hexagonal cells, deriving their area generating function and analyzing their growth constants.

Contribution

It introduces a new class of polyominoes interpolating between column-convex and 2-column polyominoes, and derives their area generating function using an extended algorithm.

Findings

The growth constant of the new class exceeds that of column-convex polygons.

A tight lower bound on the growth constant is established.

The derived generating function provides insights into the combinatorial complexity.

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 chapter 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 to 2-column polyominoes, is unlikely to be solvable. We therefore define a class of polyominoes which interpolates between column-convex polygons and 2-column polyominoes. We derive the area generating function of that class, using an extension of an existing algorithm. The growth constant of the new class is greater than the growth constant of column-convex polygons. A rather tight lower bound on the growth constant complements a compelling numerical analysis.