Rectangular Heffter arrays: a reduction theorem

Let $m,n,s,k$ be four integers such that $3\le s\le n$, $3\le k\le m$ 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 $H(nk\mathrm{/}d;d)$ 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: $(i)$ $d\equiv 0(\mathrm{m}\mathrm{o}\mathrm{d}4)$; $(ii)$ $5\leq d\equiv 1 \pmod 4$and$n k\equiv 3\pmod 4$;$(iii)$ $d\equiv 2(\mathrm{m}\mathrm{o}\mathrm{d}4)$ and $nk\equiv 0(\mathrm{m}\mathrm{o}\mathrm{d}4)$; $(iv)$ $d\equiv 3(\mathrm{m}\mathrm{o}\mathrm{d}4)$ and $nk\equiv 0,3(\mathrm{m}\mathrm{o}\mathrm{d}4)$. 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 $d\ge 2$, and there exists an $MR(m,n;s,k)$ when either $d\ge 2$ is even or $d\ge 3$ and $nk$ are odd. We also provide constructions of integer Heffter arrays and signed magic arrays when $k$ is odd and $s\equiv 0 \pmod 4$.