  A function is boundedly finite-to-one if there is a natural number $k$ such
that each point has at most $k$ inverse images. In this paper, we prove in
$\mathsf{ZF}$ (without the axiom of choice) several results concerning this
notion, among which are the following:
  (1) For each infinite set $A$ and natural number $n$, there is no boundedly
finite-to-one function from $\mathcal{S}(A)$ to $\mathcal{S}_{\leq n}(A)$,
where $\mathcal{S}(A)$ is the set of all permutations of $A$ and
$\mathcal{S}_{\leq n}(A)$ is the set of all permutations of $A$ moving at most
$n$ points.
  (2) For each infinite set $A$, there is no boundedly finite-to-one function
from $\mathcal{B}(A)$ to $\mathrm{fin}(A)$, where $\mathcal{B}(A)$ is the set
of all partitions of $A$ whose blocks are finite and $\mathrm{fin}(A)$ is the
set of all finite subsets of $A$.
