Polyominoes with nearly convex columns: An undirected model

TL;DR

This paper introduces a new class of polyominoes called level m column-subconvex polyominoes, analyzes their generating functions for hexagonal cells, and compares their growth constants to related models.

Contribution

It defines a novel super-set of column-convex polyominoes, computes their area generating functions for specific levels, and provides growth constants for these new models.

Findings

Generated area functions are complex q-series.

Growth constants increase with level, approaching other models.

New model extends understanding of polyomino growth behaviors.

Abstract

Column-convex polyominoes were introduced in 1950's by Temperley, a mathematical physicist working on "lattice gases". By now, column-convex polyominoes are a popular and well-understood model. There exist several generalizations of column-convex polyominoes; an example is a model called multi-directed animals. In this paper, we introduce a new sequence of supersets of column-convex polyominoes. Our model (we call it level m column-subconvex polyominoes) is defined in a simple way. We focus on the case when cells are hexagons and we compute the area generating functions for the levels one and two. Both of those generating functions are complicated q-series, whereas the area generating function of column-convex polyominoes is a rational function. The growth constants of level one and level two column-subconvex polyominoes are 4.319139 and 4.509480, respectively. For comparison, the growth constants of column-convex polyominoes, multi-directed animals and all polyominoes are 3.863131, 4.587894 and 5.183148, respectively.