AI 中文总结
研究加法运算下模\(n\)整数具有不同部分和的排列数，提出改进算法计算模\(n\)整数的这些排列数，并通过双射将其与已知序列关联以计算新项。
AI 中文摘要
阶乘的概念可扩展到在群运算和群元素的任何排列下的阿贝尔群。值得注意的是，对于某些有限群，可以对群中的元素进行排序，使得没有两个“阶乘”是相同的。在本文中，我们计算了在加法运算下对模20和模22的整数进行排序以实现此结果的方法数量，并引入了一种改进算法来计算模\(n\)整数的这些排列数。我们通过本文证明的一个双射将这些数与一个已知序列中的值相关联。这个证明和其他引理使我们能够计算该序列的新项。
英文摘要
The notion of the factorial extends to abelian groups under the group operation and any permutation of the elements of the group. Notably, for some finite groups, it is possible to order elements in these groups such that no two ``factorials'' are the same. In this paper we count the number of ways to order elements to achieve this result for the integers modulo 20 and 22 under addition, as well as introduce an improved algorithm to count these permutations for the integers modulo $n$. We relate these numbers to the values in an already known sequence via a bijection which we prove in this paper. This proof and other lemmas allow us to calculate new terms for the sequence.
