Magic rectangles, signed magic arrays and integer $\lambda$-fold relative Heffter arrays

Abstract

Let $m,n,s,k$ be integers such that $4\le s\le n$, $4\le k\le m$ and $ms=nk$. Let $\lambda$ be a divisor of $2ms$ and let $t$ be a divisor of $\frac{2ms}{\lambda }$. In this paper we construct magic rectangles $MR(m,n;s,k)$, signed magic arrays $SMA(m,n;s,k)$ and integer $\lambda$-fold relative Heffter arrays ${}^{\lambda }{H}_t(m,n;s,k)$ 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 ${}^{\lambda }{H}_t(m,n;s,k)$ in each of the following cases: $(i)$ $s,k\equiv 0(\mathrm{mod}4)$; $(ii)$ $s\equiv 2(\mathrm{mod}4)$ and $k\equiv 0(\mathrm{mod}4)$; $(iii)$ $s\equiv 0(\mathrm{mod}4)$ and $k\equiv 2(\mathrm{mod}4)$; $(iv)$ $s,k\equiv 2(\mathrm{mod}4)$ and $m,n$ both even.