Square Heffter arrays are $n\times n$ arrays such that each row and each 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 $1\leq x\leq nk$. Archdeacon noted that a Heffter array, satisfying two additional conditions, yields a face $2$-colourable embedding of the complete graph $K_{2nk+1}$ on an orientable surface, where for each color, the faces give a $k$-cycle system. Moreover, a cyclic permutation on the vertices acts as an automorphism of the embedding. These necessary conditions pertain to cyclic orderings of the entries in each row and each column of the Heffter array and are: (1) for each row and each column the sequential partial sums determined by the cyclic ordering must be distinct modulo $2nk+1$; (2) the composition of the cyclic orderings of the rows and columns is equivalent to a single cycle permutation on the entries in the array. We show that this construction is in fact reversible, giving an equivalence between these combinatorial objects. We construct Heffter arrays which satisfy conditions (1) and (2) whenever $n\equiv 1\mod 4$, $k\equiv 3\mod 4$ and $n$ is prime. We also show there are at least $[(p-2)!/e]^2$ non equivalent such Heffter arrays, where $p=(k-3)/4$.

中文翻译：

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使用整数 Heffter 数组的循环系统的双嵌入

Square Heffter 数组是 $n\times n$ 个数组，这样每行和每列包含 $k$ 个填充单元，每行和每列总和可以被 $2nk+1$ 整除，并且 $x$ 或 $-x$ 出现在每个整数 $1\leq x\leq nk$ 的数组。Archdeacon 指出，满足两个附加条件的 Heffter 数组会在可定向表面上产生一个面 $2$-可着色的完整图 $K_{2nk+1}$ 嵌入，其中对于每种颜色，面给出 $k$-循环系统。此外，顶点上的循环置换充当嵌入的自同构。这些必要条件与 Heffter 数组的每一行和每一列中条目的循环排序有关，并且是： (1) 对于每一行和每列，由循环排序确定的连续部分和必须是不同的模 $2nk+1$ ; (2) 行和列的循环排序的组合相当于数组中条目的单个循环排列。我们表明这种构造实际上是可逆的，从而在这些组合对象之间给出了等价性。我们构造满足条件 (1) 和 (2) 的 Heffter 数组，只要 $n\equiv 1\mod 4$, $k\equiv 3\mod 4$ 和 $n$ 是素数。我们还表明至少存在 $[(p-2)!/e]^2$ 非等价的 Heffter 数组，其中 $p=(k-3)/4$。