Polyominoes with nearly convex columns: A model with semidirected blocks

TL;DR

This paper introduces a broader class of polyominoes with nearly convex columns, expanding solvable models by relaxing semidirectedness constraints, and analyzes their enumeration and growth properties.

Contribution

It extends previous models by loosening semidirectedness restrictions, resulting in larger solvable subclasses with increased growth constants.

Findings

Larger solvable subclasses of polyominoes with nearly convex columns.

Invariance under vertical reflection for the new classes.

Increased growth constants compared to previous models.

Abstract

In most of today's exactly solved classes of polyominoes, either all members are convex (in some way), or all members are directed, or both. If the class is neither convex nor directed, the exact solution uses to be elusive. This paper is focused on polyominoes with hexagonal cells. Concretely, we deal with polyominoes whose columns can have either one or two connected components. Those polyominoes (unlike the well-explored column-convex polyominoes) cannot be exactly enumerated by any of the now existing methods. It is therefore appropriate to introduce additional restrictions, thus obtaining solvable subclasses. In our recent paper, published in this same journal, the restrictions just mentioned were semidirectedness and an upper bound on the size of the gap within a column. In this paper, the semidirectedness requirement is made looser. The result is that now the exactly solved subclasses are larger and have greater growth constants. These new polyomino families also have the advantage of being invariant under the reflection about the vertical axis.