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.

Morini, F., Pellegrini, M. A., Magic partially filled arrays on abelian groups, <<JOURNAL OF COMBINATORIAL DESIGNS>>, 2023; (31): 347-367. [doi:10.1002/jcd.21886] [https://hdl.handle.net/10807/238234]

Magic partially filled arrays on abelian groups

Pellegrini, Marco Antonio
Secondo
2023

Abstract

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.
2023
Inglese
Morini, F., Pellegrini, M. A., Magic partially filled arrays on abelian groups, <<JOURNAL OF COMBINATORIAL DESIGNS>>, 2023; (31): 347-367. [doi:10.1002/jcd.21886] [https://hdl.handle.net/10807/238234]
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/10807/238234
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