Latin squares with three disjoint subsquares of the same order
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866912620674547712 |
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| author | Kemp, Tara Lefevre, James G. |
| author_facet | Kemp, Tara Lefevre, James G. |
| contents | Given an integer partition $P = (h_1h_2\dots h_k)$ of $n$, a realization of $P$ is a latin square with disjoint subsquares of orders $h_1,h_2,\dots,h_k$. Most known results restrict either $k$ or the number of different integers in $P$. There is little known for partitions with arbitrary $k$ and subsquares of at least three orders. It has been conjectured that if $h_1=h_2=h_3\geq h_4\geq\dots\geq h_k$ then a realization of $P$ always exists. We prove this conjecture, and thus show the existence of realizations for many general partitions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_00364 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Latin squares with three disjoint subsquares of the same order Kemp, Tara Lefevre, James G. Combinatorics Given an integer partition $P = (h_1h_2\dots h_k)$ of $n$, a realization of $P$ is a latin square with disjoint subsquares of orders $h_1,h_2,\dots,h_k$. Most known results restrict either $k$ or the number of different integers in $P$. There is little known for partitions with arbitrary $k$ and subsquares of at least three orders. It has been conjectured that if $h_1=h_2=h_3\geq h_4\geq\dots\geq h_k$ then a realization of $P$ always exists. We prove this conjecture, and thus show the existence of realizations for many general partitions. |
| title | Latin squares with three disjoint subsquares of the same order |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2510.00364 |