Rectangular Heffter arrays: a reduction theorem

Abstract

Let $m,n,s,k$ be four integers such that $3\leq s \leq n$, $3\leq k\leq m$ 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: $(i)$ $d\equiv 0 \pmod 4$; $(ii)$ $5\leq d\equiv 1 \pmod 4$ and $n k\equiv 3\pmod 4$; $(iii)$ $d\equiv 2 \pmod 4$ and $nk\equiv 0 \pmod 4$; $(iv)$ $d\equiv 3 \pmod 4$ and $n k\equiv 0,3\pmod 4$. The same method can be applied also for signed magic arrays $SMA(m,n;s,k)$ and for magic rectangles $MR(m,n;s,k)$. In fact, we prove that there exists an $SMA(m,n;s,k)$ when $d\geq 2$, and there exists an $MR(m,n;s,k)$ 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 $s\equiv 0 \pmod 4$.