Magic partially filled arrays on abelian groups

In this paper we introduce a special class of partially filled arrays. A magic partially filled array MPF_Ω( m , n ; s , k ) on a subset Ω of an abelian group (Γ, +) is a partially filled array of size m × n with entries in Ω such that (i) every ω ∈ 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 ∈ 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 0MPF_Ω ( m , n ; s , k ) . Examples of these objects are magic rectangles, Γ ‐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 MPF_Ω( m , n ; s , k ) where Ω = {1, 2, ..., nk } ⊂ Z .We also construct zero‐sum magic partially filled arrays when Ω is the abelian group Γ or the set of its nonzero elements.