[Paper Review] Polyominoes with nearly convex columns: A semidirected model

Svjetlan Feretić|arXiv (Cornell University)|Oct 23, 2009

Point processes and geometric inequalities12 references6 citations

TL;DR

This paper introduces level-$m$ cheesy polyominoes on the hexagonal lattice as a generalization of column-convex polyominoes, allowing columns with up to two connected components separated by gaps of size at most $m$. Using a semi-directed constraint and the Temperley method, the authors derive rational area generating functions and show that growth constants increase with $m$, reaching 4.2886 for level 3, exceeding column-convex polyominoes but falling short of multi-directed animals.

ABSTRACT

Column-convex polyominoes are by now a well-explored model. So far, however, no attention has been given to polyominoes whose columns can have either one or two connected components. This little known kind of polyominoes seems not to be manageable as a whole. To obtain solvable models, one needs to introduce some restrictions. This paper is focused on polyominoes with hexagonal cells. The restrictions just mentioned are semidirectedness and an upper bound on the size of the gap within a column. The solvable models so obtained have rational area generating functions, as column-convex polyominoes do. However, the growth constants of the new models are 4.114908 and more, whereas the growth constant of column-convex polyominoes is 3.863131.

Motivation & Objective

• To generalize column-convex polyominoes by allowing columns with up to two connected components and bounded gaps.
• To develop a solvable model with rational area generating functions, avoiding the intractability of unrestricted nearly convex column polyominoes.
• To explore whether higher growth constants can be achieved compared to known solvable models like column-convex or directed polyominoes.
• To lay groundwork for future models such as polyominoes with cheesy blocks and column-subconvex polyominoes.
• To compare the asymptotic growth of these models with known classes, including multi-directed animals and all polyominoes.

Proposed method

• Define level-$m$ cheesy polyominoes as rightward-semi-directed polyominoes where each column has at most two connected components and the gap between components is at most $m$ cells.
• Apply the 'turbo' version of the Temperley method, developed by Bousquet-Mélou and Svrtan, to derive area generating functions.
• Use recursive decomposition based on column structure and semi-directed constraints to model the growth of polyominoes step-by-step.
• Implement symbolic computation to derive rational generating functions for levels $m = 1, 2, 3$, using exact enumeration of small cases.
• Validate results by comparing computed counts with known values for small $n$, ensuring consistency with column-convex and other polyomino classes.
• Analyze asymptotic behavior by extracting growth constants from the dominant singularity of the rational generating functions.

Experimental results

Research questions

• RQ1 Can a generalization of column-convex polyominoes that allows bounded gaps in columns be made exactly solvable with rational area generating functions?
• RQ2 What is the growth rate of such generalized polyominoes, and how does it compare to known models like column-convex or multi-directed animals?
• RQ3 Does increasing the gap size bound $m$ lead to a monotonic increase in the growth constant, and what is the limiting behavior as $m \to \infty$?
• RQ4 Is the semi-directed constraint essential for achieving rational generating functions, or can the model be generalized further without losing solvability?
• RQ5 Can the perimeter generating function of level-1 cheesy polyominoes be computed, and does it have a positive radius of convergence?

Key findings

• The area generating function for level-1 cheesy polyominoes is rational, with a growth constant of approximately 4.1149.
• For level-2 cheesy polyominoes, the growth constant increases to approximately 4.2318, and for level-3, it reaches 4.2886.
• The asymptotic number of $n$-celled level-$m$ cheesy polyominoes grows as $c_m \times \lambda_m^n$, where $\lambda_m$ increases with $m$, with $c_1 \approx 0.1441$, $c_2 \approx 0.1210$, and $c_3 \approx 0.1082$.
• The growth constant of column-convex polyominoes (3.8631) is surpassed by all levels of cheesy polyominoes, indicating a faster growth rate.
• The growth constants of cheesy polyominoes appear to approach a limit of approximately 4.346 as $m \to \infty$, based on observed first differences.
• The model is not expected to be enumeratable by perimeter, as its perimeter generating function is conjectured to have zero radius of convergence.