A Partition-Based Generating Function for Row-Convex Polyominoes
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.
Keywords
Cite
@article{arxiv.2605.03203,
title = {A Partition-Based Generating Function for Row-Convex Polyominoes},
author = {Vincenzo M. Scarrica},
journal= {arXiv preprint arXiv:2605.03203},
year = {2026}
}