Magic rectangles, signed magic arrays and integer λ-fold relative Heffter arrays

2020/10/22 by Morini, Fiorenza, Pellegrini, Marco Antonio
#05B20 #05B30 #Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.2010.12333

Abstract

Let m,n,s,k be integers such that 4≤ s≤ n, 4≤ k ≤ m and ms=nk. Let λ be a divisor of 2ms and let t be a divisor of \frac2msλ. In this paper we construct magic rectangles MR(m,n;s,k), signed magic arrays SMA(m,n;s,k) and integer λ-fold relative Heffter arrays ^λH_t(m,n;s,k) where s,k are even integers. In particular, we prove that there exists an SMA(m,n;s,k) for all m,n,s,k satisfying the previous hypotheses. Furthermore, we prove that there exist an MR(m,n;s,k) and an integer ^λH_t(m,n;s,k) in each of the following cases: (i) s,k ≡ 0 \pmod 4; (ii) s≡ 2\pmod 4 and k≡ 0 \pmod 4; (iii) s≡ 0\pmod 4 and k≡ 2 \pmod 4; (iv) s,k≡ 2 \pmod 4 and m,n both even.

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