Rectangular Heffter arrays: a reduction theorem

Let m,n,s,k be four integers such that 3≤s≤n, 3≤k≤m and ms=nk. Set d=gcd⁡(s,k). In this paper we show how one can construct a Heffter array H(m,n;s,k) starting from a square Heffter array H(nk/d;d) whose elements belong to d consecutive diagonals. As an example of application of this method, we prove that there exists an integer H(m,n;s,k) in each of the following cases: (i) d≡0(mod4); (ii) 5≤d≡1(mod4) and nk≡3(mod4); (iii) d≡2(mod4) and nk≡0(mod4); (iv) d≡3(mod4) and nk≡0,3(mod4). The same method can be applied also for signed magic arrays SMA(m,n;s,k) and for magic rectangles MR(m,n;s,k). In fact, we prove that there exists an SMA(m,n;s,k) when d≥2, and there exists an MR(m,n;s,k) when either d≥2 is even or d≥3 and nk are odd. We also provide constructions of integer Heffter arrays and signed magic arrays when k is odd and s≡0(mod4).