2024/10/05 by Morini, Fiorenza, Pellegrini, Marco Antonio
#05B15 #05B30 #05C78 #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2410.04101
Let m,n,s,k be four integers such that 1\leqslant s \leqslant n, 1\leqslant k\leqslant m and ms=nk. A signed magic array SMA(m,n; s,k) is an m× n partially filled array whose entries belong to the subset Ω⊂ ℤ, where Ω=\0,± 1, ± 2,…, ± (nk-1)/2\ if nk is odd and Ω=\± 1, ± 2, …, ± nk/2\ if nk is even, satisfying the following requirements: (a) every ω∈ Ω appears once in the array; (b) each row contains exactly s filled cells and each column contains exactly k filled cells; (c) the sum of the elements in each row and in each column is 0. In this paper we construct these arrays when n is even and s,k\geqslant 5 are odd coprime integers. This allows us to give necessary and sufficient conditions for the existence of an SMA(m,n; s,k) for all admissible values of m,n,s,k.