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

TL;DR

This paper constructs various combinatorial arrays, including magic rectangles and signed magic arrays, under specific divisibility and parity conditions, expanding the existence results for these arrays.

Contribution

It establishes the existence of signed magic arrays for all parameters satisfying certain conditions and constructs magic rectangles and Heffter arrays in multiple cases based on divisibility and parity.

Findings

Existence of signed magic arrays for all parameters meeting the hypotheses.

Construction of magic rectangles under specific divisibility and parity conditions.

Development of integer λ-fold relative Heffter arrays in various cases.

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.