The partitions whose members are finite and the permutations with at most $n$ non-fixed points of a set

Abstract

We write $S_{\leq n}(A)$ and $\Part_{\fin}(A)$ for the set of permutations with at most $n$ non-fixed points, where $n$ is a natural number, and the set of partitions whose members are finite, respectively, of a set $A$. Among our results, we show, in the Zermelo-Fraenkel set theory, that $|\Part_{\fin}(A)| \nleq |S_{\leq n}(A)|$ for any infinite set $A$ and if $A$ can be linearly ordered, then $|S_{\leq n}(A)| < |\Part_{\fin}(A)|$ while the statement ``$|S_{\leq n}(A)|\leq|\Part_{\fin}(A)|$ for all infinite sets $A$" is not provable for $n\geq 3$.