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Magic partially filled arrays on abelian groups.
- Source :
-
Journal of Combinatorial Designs . Aug2023, Vol. 31 Issue 8, p347-367. 21p. - Publication Year :
- 2023
-
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 Ω ${\rm{\Omega }}$ of an abelian group (Γ,+) $({\rm{\Gamma }},+)$ is a partially filled array of size m×n $m\times n$ with entries in Ω ${\rm{\Omega }}$ such that (i) every ω∈Ω $\omega \in {\rm{\Omega }}$ 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∈Γ $x,y\in {\rm{\Gamma }}$ 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, Γ ${\rm{\Gamma }}$‐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 Ω={1,2,...,nk}⊂ℤ ${\rm{\Omega }}=\{1,2,\ldots ,nk\}\subset {\rm{{\mathbb{Z}}}}$. We also construct zero‐sum magic partially filled arrays when Ω ${\rm{\Omega }}$ is the abelian group Γ ${\rm{\Gamma }}$ or the set of its nonzero elements. [ABSTRACT FROM AUTHOR]
- Subjects :
- *ABELIAN groups
*MAGIC
*RECTANGLES
Subjects
Details
- Language :
- English
- ISSN :
- 10638539
- Volume :
- 31
- Issue :
- 8
- Database :
- Academic Search Index
- Journal :
- Journal of Combinatorial Designs
- Publication Type :
- Academic Journal
- Accession number :
- 164136459
- Full Text :
- https://doi.org/10.1002/jcd.21886