We construct some large families of permutations of ${\mathbb{Z}}_n$ such that the sum of any two permutations from a family is again a permutation of ${\mathbb{Z}}_n$ (not necessarily in the same family). Our families are significantly larger than the largest such families known so far in the literature (depending on the value of $n$, the improvement can be as large as exponential). We also show that our families are maximal in the sense that no permutations can be added to them while maintaining the required property.

中文翻译：

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Z n的大置换族，其成对和是置换

我们构造了一些大的排列族 ${\mathbb{ž}}_{ñ}$ 这样一个家庭的任何两个排列之和又是 ${\mathbb{ž}}_{ñ}$（不一定是同一个家庭）。我们的家庭远大于文献中迄今为止已知的最大家庭（取决于家庭的价值）。$ñ$，则改进幅度可以达到指数级）。我们还表明，在保持所需属性的同时，不能向其添加任何排列的意义上说，我们的家庭是最大的。