Cantor's theorem may fail for finitary partitions

A partition is finitary if all its members are finite. For a set $A$, $\mathcal{B}(A)$ denotes the set of all finitary partitions of $A$. It is shown consistent with $\mathsf{Z}\mathsf{F}$ (without the axiom of choice) that there exist an infinite set $A$ and a surjection from $A$ onto $\mathcal{B}(A)$. On the other hand, we prove in $\mathsf{Z}\mathsf{F}$ some theorems concerning $\mathcal{B}(A)$ for infinite sets $A$, among which are the following: (1) If there is a finitary partition of $A$ without singleton blocks, then there are no surjections from $A$ onto $\mathcal{B}(A)$ and no finite-to-one functions from $\mathcal{B}(A)$ to $A$. (2) For all $n\in \omega$, $\mathrm{\mid }{A}^n\mathrm{\mid }<\mathrm{\mid }\mathcal{B}(A)\mathrm{\mid }$. (3) $\mathrm{\mid }\mathcal{B}(A)\mathrm{\mid }\mathrm{\ne }\mathrm{\mid }\mathrm{s}\mathrm{e}\mathrm{q}(A)\mathrm{\mid }$, where $\mathrm{s}\mathrm{e}\mathrm{q}(A)$ is the set of all finite sequences of elements of $A$.