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 2ms/λ . 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 λHt(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 λHt(m, n; s, k) in each of the following cases: (i) s, k ≡ 0 (mod 4); (ii) s ≡ 2 (mod 4) and k ≡ 0 (mod 4); (iii) s ≡ 0 (mod 4) and k ≡ 2 (mod 4); (iv) s, k ≡ 2 (mod 4) and m, n both even.

Magic rectangles, signed magic arrays and integer λ-fold relative Heffter arrays / Morini, F.; Pellegrini, M. A.. - In: THE AUSTRALASIAN JOURNAL OF COMBINATORICS. - ISSN 2202-3518. - 80:2(2021), pp. 249-280.

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

Morini F.;
2021

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 2ms/λ . 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 λHt(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 λHt(m, n; s, k) in each of the following cases: (i) s, k ≡ 0 (mod 4); (ii) s ≡ 2 (mod 4) and k ≡ 0 (mod 4); (iii) s ≡ 0 (mod 4) and k ≡ 2 (mod 4); (iv) s, k ≡ 2 (mod 4) and m, n both even.
Magic rectangles, signed magic arrays and integer λ-fold relative Heffter arrays / Morini, F.; Pellegrini, M. A.. - In: THE AUSTRALASIAN JOURNAL OF COMBINATORICS. - ISSN 2202-3518. - 80:2(2021), pp. 249-280.
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11381/2894902
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