论文标题
魔术部分填充了亚伯群的阵列
Magic partially filled arrays on abelian groups
论文作者
论文摘要
在本文中，我们介绍了一类特殊的部分填充阵列。一个魔术填充的阵列$ \ mathrm {mpf}_Ω（m，n; s，k）$在亚abelian group $（γ，+）$的子集$ω$上是一个部分$ $ m \ times n $，带有$ω$的条目，$ω$（I）$（i）$（i）$（i）$（i）$（i）$ ch y rane omω$ earray earray; $（ii）$每行包含$ s $填充的单元格，每列包含$ k $填充的单元格； $（iii）$存在（不一定是不同的）元素$ x，y inγ$，使每行的元素的总和为$ x $，每列中的元素的总和为$ y $。特别是，如果$ x = y =0_γ$，我们有一个零和魔法部分填充数组$ {}^0 \ mathrm {mpf}_Ω（m，n; s，k）$。这些对象的示例是魔术矩形，$γ$ - 魔术矩形，签名的魔术阵列，（整数或非整数）Heffter数组。在这里，我们提供了具有空单元格的魔术矩形，即$ \ mathrm {mpf}_Ω（m，n; s，k）$中的魔术矩形的必要条件。当$ω$是Abelian Group $γ$或其非零元素的集合时，我们还会构建零和魔法部分填充的数组。
In this paper we introduce a special class of partially filled arrays. A magic partially filled array $\mathrm{MPF}_Ω(m,n; s,k)$ on a subset $Ω$ of an abelian group $(Γ,+)$ is a partially filled array of size $m\times n$ with entries in $Ω$ such that $(i)$ every $ω\in Ω$ appears once in the array; $(ii)$ each row contains $s$ filled cells and each column contains $k$ filled cells; $(iii)$ there exist (not necessarily distinct) elements $x,y\in Γ$ such that the sum of the elements in each row is $x$ and the sum of the elements in each column is $y$. In particular, if $x=y=0_Γ$, we have a zero-sum magic partially filled array ${}^0\mathrm{MPF}_Ω(m,n; s,k)$. Examples of these objects are magic rectangles, $Γ$-magic rectangles, signed magic arrays, (integer or non integer) Heffter arrays. Here, we give necessary and sufficient conditions for the existence of a magic rectangle with empty cells, i.e., of an $\mathrm{MPF}_Ω(m,n;s,k)$ where $Ω=\{1,2,\ldots,nk\}\subset\mathbb{Z}$. We also construct zero-sum magic partially filled arrays when $Ω$ is the abelian group $Γ$ or the set of its nonzero elements.
