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| Format: | Preprint |
| Veröffentlicht: |
2026
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| Online-Zugang: | https://arxiv.org/abs/2605.03203 |
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| _version_ | 1866915979067392000 |
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| author | Scarrica, Vincenzo M. |
| author_facet | Scarrica, Vincenzo M. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_03203 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A Partition-Based Generating Function for Row-Convex Polyominoes Scarrica, Vincenzo M. Combinatorics Computer Vision and Pattern Recognition 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. |
| title | A Partition-Based Generating Function for Row-Convex Polyominoes |
| topic | Combinatorics Computer Vision and Pattern Recognition |
| url | https://arxiv.org/abs/2605.03203 |