Signed magic arrays: existence and constructions
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866914965644902400 |
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| author | Morini, Fiorenza Pellegrini, Marco Antonio |
| author_facet | Morini, Fiorenza Pellegrini, Marco Antonio |
| contents | Let $m,n,s,k$ be four integers such that $1\leqslant s \leqslant n$, $1\leqslant k\leqslant m$ and $ms=nk$. A signed magic array $SMA(m,n; s,k)$ is an $m\times n$ partially filled array whose entries belong to the subset $Ω\subset \mathbb{Z}$, where $Ω=\{0,\pm 1, \pm 2,\ldots, \pm (nk-1)/2\}$ if $nk$ is odd and $Ω=\{\pm 1, \pm 2, \ldots, \pm nk/2\}$ if $nk$ is even, satisfying the following requirements: $(a)$ every $ω\in Ω$ appears once in the array; $(b)$ each row contains exactly $s$ filled cells and each column contains exactly $k$ filled cells; $(c)$ the sum of the elements in each row and in each column is $0$. In this paper we construct these arrays when $n$ is even and $s,k\geqslant 5$ are odd coprime integers. This allows us to give necessary and sufficient conditions for the existence of an $SMA(m,n; s,k)$ for all admissible values of $m,n,s,k$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_04101 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Signed magic arrays: existence and constructions Morini, Fiorenza Pellegrini, Marco Antonio Combinatorics 05B15, 05C78, 05B30 Let $m,n,s,k$ be four integers such that $1\leqslant s \leqslant n$, $1\leqslant k\leqslant m$ and $ms=nk$. A signed magic array $SMA(m,n; s,k)$ is an $m\times n$ partially filled array whose entries belong to the subset $Ω\subset \mathbb{Z}$, where $Ω=\{0,\pm 1, \pm 2,\ldots, \pm (nk-1)/2\}$ if $nk$ is odd and $Ω=\{\pm 1, \pm 2, \ldots, \pm nk/2\}$ if $nk$ is even, satisfying the following requirements: $(a)$ every $ω\in Ω$ appears once in the array; $(b)$ each row contains exactly $s$ filled cells and each column contains exactly $k$ filled cells; $(c)$ the sum of the elements in each row and in each column is $0$. In this paper we construct these arrays when $n$ is even and $s,k\geqslant 5$ are odd coprime integers. This allows us to give necessary and sufficient conditions for the existence of an $SMA(m,n; s,k)$ for all admissible values of $m,n,s,k$. |
| title | Signed magic arrays: existence and constructions |
| topic | Combinatorics 05B15, 05C78, 05B30 |
| url | https://arxiv.org/abs/2410.04101 |