Magic partially filled arrays on abelian groups

Abstract

In this paper we introduce a special class of partially filled arrays. A magic partially filled array ${\text{MPF}}_{\Omega }(m,n;s,k)$ on a subset $\Omega$ of an abelian group $(\Gamma ,+)$ is a partially filled array of size $m\times n$ with entries in $\Omega$ such that (i) every $\omega \in \Omega$ 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 $x,y\in \Gamma$ 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 $x=y=0_{\Gamma }$, we have a zero-sum magic partially filled array ${}^0\text{MPF}_{\Omega }(m,n;s,k)$. Examples of these objects are magic rectangles, $\Gamma$-magic rectangles, signed magic arrays, (integer or noninteger) Heffter arrays. Here, we give necessary and sufficient conditions for the existence of a magic rectangle with empty cells, that is, of an ${\text{MPF}}_{\Omega }(m,n;s,k)$ where $\Omega =\{1,2,\text{…},nk\}\subset \mathbb{Z}$. We also construct zero-sum magic partially filled arrays when $\Omega$ is the abelian group $\Gamma$ or the set of its nonzero elements.