We write $\text{fin}(\mathfrak{m})$ and $S_{\text{fin}}(\mathfrak{m})$ for the cardinalities of the set of finite subsets and the set of permutations with finitely many non-fixed points, respectively, of a set which is of cardinality $\mathfrak{m}$. In this paper, we investigate relationships between $\text{fin}(\mathfrak{m})$ and $S_{\text{fin}}(\mathfrak{m})$ for an infinite cardinal $\mathfrak{m}$ in the absence of the Axiom of Choice. We give conditions that make $\text{fin}(\mathfrak{m})$ and $S_{\text{fin}}(\mathfrak{m})$ comparable as well as give related consistency results.

中文翻译：

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有限子集和具有有限多个非不动点的集合的排列

我们写 $\text{鳍}(\mathfrak{米})$ 和 ${秒}_{\text{鳍}}(\mathfrak{米})$ 对于具有基数的集合的有限子集集合和具有有限多个非不动点的排列集合的基数，分别 $\mathfrak{米}$. 在本文中，我们研究了$\text{鳍}(\mathfrak{米})$ 和 ${秒}_{\text{鳍}}(\mathfrak{米})$ 对于无限的基数 $\mathfrak{米}$在没有选择公理的情况下。我们给出条件，使$\text{鳍}(\mathfrak{米})$ 和 ${秒}_{\text{鳍}}(\mathfrak{米})$ 并给出相关的一致性结果。