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 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 (mod4); (ii) s≡2 (mod4) and k≡0 (mod4); (iii) s≡0 (mod4) and k≡2 (mod 4); (iv)s, k≡2 (mod4) and m, n both even.

Morini, F., Pellegrini, M. A., Magic rectangles, signed magic arrays and integer λ-fold relative Heffter arrays, <<THE AUSTRALASIAN JOURNAL OF COMBINATORICS>>, 2021; (80): 249-280 [http://hdl.handle.net/10807/181461]

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

Fiorenza; Pellegrini
Secondo
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 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 (mod4); (ii) s≡2 (mod4) and k≡0 (mod4); (iii) s≡0 (mod4) and k≡2 (mod 4); (iv)s, k≡2 (mod4) and m, n both even.
Inglese
Morini, F., Pellegrini, M. A., Magic rectangles, signed magic arrays and integer λ-fold relative Heffter arrays, <<THE AUSTRALASIAN JOURNAL OF COMBINATORICS>>, 2021; (80): 249-280 [http://hdl.handle.net/10807/181461]
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/10807/181461
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