A Partition-Based Generating Function for Row-Convex Polyominoes

TL;DR

This paper introduces a novel generating function approach for enumerating row-convex polyominoes using integer partitions, providing exact formulas and asymptotic analysis.

Contribution

It establishes a new connection between integer partitions and polyomino enumeration, enabling exact and asymptotic analysis of convex polyominoes.

Findings

Derived an exact generating function for convex polyominoes.

Established the asymptotic growth rate involving cosine oscillations.

Provided numerical examples for small areas.

Abstract

An alternative generating function is proposed to enumerate row-convex polyominoes without internal holes on a discrete grid. The approach is based on integer partitions of the total area, where each partition corresponds to a sequence of row lengths, and the product of all permutations of the parts accounts for all possible horizontal alignments of consecutive rows. Summing over the products yields a formula for the total number of convex polyominoes of a given size. Numerical examples are provided for small areas, and the exact generating function is derived via a transfer series argument, establishing the asymptotic growth S(N) as A2^(N) cos(N*theta) + phi) with theta = arctan(sqrt(7)/3). The method establishes a direct connection between integer partitions and polyomino enumeration, offering a simple yet effective framework for both exact and asymptotic combinatorial analysis. Potential applications include shape priors in discrete image analysis, grid-based modeling, and combinatorial generation of convex structures.