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

We write $S_{\le n}(A)$ and ${\text{Part}}_{\text{fin}}(A)$ 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 $\mathrm{\mid }{\text{Part}}_{\text{fin}}(A)\mathrm{\mid }\nleqq \mathrm{\mid }{S}_{\le n}(A)\mathrm{\mid }$ for any infinite set $A$ and if $A$ can be linearly ordered, then $\mathrm{\mid }{S}_{\le n}(A)\mathrm{\mid }<\mathrm{\mid }{\text{Part}}_{\text{fin}}(A)\mathrm{\mid }$ while the statement ``$\mathrm{\mid }{S}_{\le n}(A)\mathrm{\mid }\le \mathrm{\mid }{\text{Part}}_{\text{fin}}(A)\mathrm{\mid }$ for all infinite sets $A$" is not provable for $n\ge 3$.