Related papers: Magic rectangles, signed magic arrays and integer …
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…
Let $v=2ms+t$ be a positive integer, where $t$ divides $2ms$, and let $J$ be the subgroup of order $t$ of the cyclic group $\mathbb{Z}_v$. An integer Heffter array $H_t(m,n;s,k)$ over $\mathbb{Z}_v$ relative to $J$ is an $m\times n$…
Let $m,n,s,k$ be four integers such that $1\leqslant s \leqslant n$, $1\leqslant k\leqslant m$ and $ms=nk$. A signed magic array $SMA(m,n; s,k)$ is an $m\times n$ partially filled array whose entries belong to the subset $\Omega\subset…
A {\em signed magic rectangle} $SMR(m,n;k, s)$ is an $m \times n$ array with entries from $X$, where $X=\{0,\pm1,\pm2,\ldots, $ $\pm (mk-1)/2\}$ if $mk$ is odd and $X = \{\pm1,\pm2,\ldots,\pm mk/2\}$ if $mk$ is even, such that precisely $k$…
A signed magic rectangle $SMR(m,n;r, s)$ is an $m \times n$ array with entries from $X$, where $X=\{0,\pm1,\pm2,\ldots, $ $\pm (ms-1)/2\}$ if $mr$ is odd and $X = \{\pm1,\pm2,\ldots,\pm mr/2\}$ if $mr$ is even, such that precisely $r$ cells…
In this paper, we make significant progress on a conjecture proposed by Dan Archdeacon on the existence of integer Heffter arrays $H(m,n;s,k)$ whenever the necessary conditions hold, that is, $3\leqslant s \leqslant n$, $3\leqslant…
An integer Heffter array $H(m,n;s;k)$ is an $m\times n$ partially filled array whose entries are the elements of a subset $\Omega\subset \mathbb{Z}$ such that $\{\Omega,-\Omega\}$ is a partition of the set $\{1,2,\ldots,2nk\}$ and such that…
A magic rectangle of order $m\times n$ with precisely $r$ filled cells in each row and precisely $s$ filled cells in each column, denoted $MR(m,n;r,s)$, is an arrangement of the numbers from 0 to $mr-1$ in an $m\times n$ array such that…
A signed magic array, $SMA(m, n;s,t)$, is an $m \times n$ array with the same number of filled cells $s$ in each row and the same number of filled cells $t$ in each column, filled with a certain set of numbers that is symmetric about the…
Let $\Gamma$ be a group of order $mnk$ and $MRS_{\Gamma}(m,n;k)=(a_{i,j}^s)_{m\times n}$ be a collection of $k$ arrays $m\times n$ whose entries are all distinct elements of $\Gamma$. If there exist elements $\rho,\sigma\in\Gamma$ such that…
Relative Heffter arrays, denoted by $\mathrm{H}_t(m,n; s,k)$, have been introduced as a generalization of the classical concept of Heffter array. A $\mathrm{H}_t(m,n; s,k)$ is an $m\times n$ partially filled array with elements in…
A $\Gamma$-magic rectangle set $MRS_{\Gamma}(a, b; c)$ of order $abc$ is a collection of $c$ arrays $(a\times b)$ whose entries are elements of group $\Gamma$, each appearing once, with all row sums in every rectangle equal to a constant…
A tight Heffter array H(m,n) is an m x n matrix with nonzero entries from Z_{2mn+1} such that i) the sum of the elements in each row and each column is 0, and ii) no element from {x,-x\ appears twice. We prove that H(m,n) exist if and only…
For integers $n,k,s$, we give a formula for the number $T(n,k,s)$ of order $k$ subsets of the ring $\mathbb{Z}/n\mathbb{Z}$ whose sum of elements is $s$ modulo $n$. To do so, we describe explicitly a sequence of matrices $M(k)$, for…
A Heffter array $H(n;k)$ is an $n\times n$ matrix such that each row and column contains $k$ filled cells, each row and column sum is divisible by $2nk+1$ and either $x$ or $-x$ appears in the array for each integer $1\leq x\leq nk$.…
Let s,t,m,n be positive integers such that sm=tn. Let M(m,s;n,t) be the number of m x n matrices over {0,1,2,...} with each row summing to s and each column summing to t. Equivalently, M(m,s;n,t) counts 2-way contingency tables of order m x…
Graph labeling is a well-known and intensively investigated problem in graph theory. Sparse anti-magic squares are useful in constructing vertex-magic labeling for graphs. For positive integers $n,d$ and $d<n$, an $n\times n$ array $A$…
Heffter arrays are combinatorial structures used to construct orthogonal cyclic cycle decompositions and biembeddings of complete graphs onto surfaces. A Heffter array $H(m,n;h,k)$ is an $m \times n$ partially filled array with distinct…
We consider the notion of a signed magic array, which is an $m \times n$ rectangular array with the same number of filled cells $s$ in each row and the same number of filled cells $t$ in each column, filled with a certain set of numbers…
The shiftable Heffter arrays are naturally generalized to the shiftable Heffter spaces. We present a recursive construction which starting from a single shiftable Heffter space leads to infinitely many other shiftable Heffter spaces of the…