Magic partially filled arrays on abelian groups

TL;DR

This paper introduces a new class of partially filled arrays called magic partially filled arrays on abelian groups, providing existence conditions and constructions for these arrays, including zero-sum variants and examples like magic rectangles and Heffter arrays.

Contribution

It defines the concept of magic partially filled arrays on abelian groups, establishes necessary and sufficient conditions for their existence, and constructs specific examples such as zero-sum arrays.

Findings

Necessary and sufficient conditions for existence of magic rectangles with empty cells.

Construction methods for zero-sum magic arrays over abelian groups.

Examples include magic rectangles, $ ext{Gamma}$-magic rectangles, and Heffter arrays.

Abstract

In this paper we introduce a special class of partially filled arrays. A magic partially filled array $\mathrm{MPF}_\Omega(m,n; s,k)$ 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 ${}^0\mathrm{MPF}_\Omega(m,n; s,k)$. Examples of these objects are magic rectangles, $\Gamma$-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 $\mathrm{MPF}_\Omega(m,n;s,k)$ where $\Omega=\{1,2,\ldots,nk\}\subset\mathbb{Z}$. We also construct zero-sum magic partially filled arrays when $\Omega$ is the abelian group $\Gamma$ or the set of its nonzero elements.