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 MPFΩ(m,n;s,k) ${\text{MPF}}_{{\rm{\Omega }}}(m,n;s,k)$ on a subset normalΩ ${\rm{\Omega }}$ of an abelian group (normalΓ,+) $({\rm{\Gamma }},+)$ is a partially filled array of size m×n $m\times n$ with entries in normalΩ ${\rm{\Omega }}$ such that (i) every ω∈normalΩ $\omega \in {\rm{\Omega }}$ appears once in the array; (ii) each row contains s $s$ filled cells and each column contains k $k$ filled cells; (iii) there exist (not necessarily distinct) elements x,y∈normalΓ $x,y\in {\rm{\Gamma }}$ such that the sum of the elements in each row is x $x$ and the sum of the elements in each column is y $y$. In particular, if x=y=0Γ $x=y={0}_{{\rm{\Gamma }}}$, we have a zero‐sum magic partially filled array MPFΩ0(m,n;s,k) ${}^{0}\text{MPF}_{{\rm{\Omega }}}(m,n;s,k)$. Examples of these objects are magic rectangles, normalΓ ${\rm{\Gamma }}$‐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 MPFΩ(m,n;s,k) ${\text{MPF}}_{{\rm{\Omega }}}(m,n;s,k)$ where normalΩ={1,2,…,nk}⊂normalℤ ${\rm{\Omega }}=\{1,2,\ldots ,nk\}\subset {\rm{{\mathbb{Z}}}}$. We also construct zero‐sum magic partially filled arrays when normalΩ ${\rm{\Omega }}$ is the abelian group normalΓ ${\rm{\Gamma }}$ or the set of its nonzero elements.