The permutations with n non-fixed points and the subsets with n elements of a set

Abstract

We write $S_n\left(a\right)$ and ${\left[a\right]}^n$ for the cardinalities of the set of permutations with n non-fixed points and the set of subsets with n elements, respectively, of a set which is of cardinality $a$, where n is a natural number greater than 1. With the Axiom of Choice, $S_n\left(a\right)$ and ${\left[a\right]}^n$ are equal for all infinite cardinals $a$. We show, in ZF, that if ${\text{AC}}_{\le n}$ is assumed, then ${\left[a\right]}^n\le S_n\left(a\right)\le {\left[a\right]}^{n+1}$ for any infinite cardinal $a$. Moreover, the assumption ${\text{AC}}_{\le n}$ cannot be removed for $n>2$ and the superscript $n+1$ cannot be replaced by n. We also show under ${\text{AC}}_{\le n}$ that for any infinite cardinal $a$, $S_n\left(a\right)\le {\left[a\right]}^n$ implies $a$ is Dedekind-infinite.