Rectangular Heffter arrays: a reduction theorem. (English) Zbl 1497.05025
Summary: Let \(m\), \(n\), \(s\), \(k\) be four integers such that \(3 \leq s \leq n\), \(3 \leq k \leq m\) and \(m s = n k\). Set \(d = \gcd(s, k)\). In this paper we show how one can construct a Heffter array \(\mathrm{H}(m, n; s, k)\) starting from a square Heffter array \(\mathrm{H}(n k / d; d)\) whose elements belong to \(d\) consecutive diagonals. As an example of application of this method, we prove that there exists an integer \(\mathrm{H}(m, n; s, k)\) in each of the following cases: (i) \(d \equiv 0\pmod 4\); (ii) \(5 \leq d \equiv 1\pmod 4\) and \(n k \equiv 3\pmod 4\); (iii) \(d \equiv 2\pmod 4\) and \(n k \equiv 0\pmod 4\); (iv) \(d \equiv 3\pmod 4\) and \(n k \equiv 0, 3\pmod 4\). The same method can be applied also for signed magic arrays \(\mathrm{SMA}(m, n; s, k)\) and for magic rectangles \(\mathrm{MR}(m, n; s, k)\). In fact, we prove that there exists an \(\mathrm{SMA}(m, n; s, k)\) when \(d \geq 2\), and there exists an \(\mathrm{MR}(m, n; s, k)\) when either \(d \geq 2\) is even or \(d \geq 3\) and \(nk\) are odd. We also provide constructions of integer Heffter arrays and signed magic arrays when \(k\) is odd and \(s \equiv 0\pmod 4\).
MSC:
| 05B30 | Other designs, configurations |
| 05B15 | Orthogonal arrays, Latin squares, Room squares |
| 05B10 | Combinatorial aspects of difference sets (number-theoretic, group-theoretic, etc.) |
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