An update on the existence of integer Heffter arrays
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866911200243089408 |
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| author | Morini, Fiorenza Pellegrini, Marco Antonio |
| author_facet | Morini, Fiorenza Pellegrini, Marco Antonio |
| contents | An integer Heffter array $H(m,n;s;k)$ is an $m\times n$ partially filled array whose entries are the elements of a subset $Ω\subset \mathbb{Z}$ such that $\{Ω,-Ω\}$ is a partition of the set $\{1,2,\ldots,2nk\}$ and such that the following conditions are satisfied: each row contains $s$ filled cells, each column contains $k$ filled cells, the elements in every row and column add up to $0$. It was conjectured by Dan Archdeacon that an integer $\H(m,n;s;k)$ exists if and only if $ms=nk$, $3\leq s \leq n$, $3\leq k\leq m$ and $nk\equiv 0,3\pmod 4$. In this paper, we provide new constructions of these objects that allow us to prove the validity of Archdeacon's conjecture in each admissible case, except when $k=3,5$ and $s\not \equiv 0\pmod 4$ is such that $\gcd(s,k)=1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_08302 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | An update on the existence of integer Heffter arrays Morini, Fiorenza Pellegrini, Marco Antonio Combinatorics 05B20, 05B30 An integer Heffter array $H(m,n;s;k)$ is an $m\times n$ partially filled array whose entries are the elements of a subset $Ω\subset \mathbb{Z}$ such that $\{Ω,-Ω\}$ is a partition of the set $\{1,2,\ldots,2nk\}$ and such that the following conditions are satisfied: each row contains $s$ filled cells, each column contains $k$ filled cells, the elements in every row and column add up to $0$. It was conjectured by Dan Archdeacon that an integer $\H(m,n;s;k)$ exists if and only if $ms=nk$, $3\leq s \leq n$, $3\leq k\leq m$ and $nk\equiv 0,3\pmod 4$. In this paper, we provide new constructions of these objects that allow us to prove the validity of Archdeacon's conjecture in each admissible case, except when $k=3,5$ and $s\not \equiv 0\pmod 4$ is such that $\gcd(s,k)=1$. |
| title | An update on the existence of integer Heffter arrays |
| topic | Combinatorics 05B20, 05B30 |
| url | https://arxiv.org/abs/2510.08302 |