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

Abstract

Let $m,n,s,k$ be integers such that $4\leq s\leq n$, $4\leq k \leq m$ 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: $(i)$ $s,k \equiv 0 \pmod 4$; $(ii)$ $s\equiv 2\pmod 4$ and $k\equiv 0 \pmod 4$; $(iii)$ $s\equiv 0\pmod 4$ and $k\equiv 2 \pmod 4$; $(iv)$ $s,k\equiv 2 \pmod 4$ and $m,n$ both even.