The finitary partitions with $n$ non-singleton blocks of a set

A partition is finitary if all its blocks are finite. For a cardinal $\mathfrak{a}$ and a natural number $n$, let $\mathrm{f}\mathrm{i}\mathrm{n}(\mathfrak{a})$ and ${\mathcal{B}}_n(\mathfrak{a})$ be the cardinalities of the set of finite subsets and the set of finitary partitions with exactly $n$ non-singleton blocks of a set which is of cardinality $\mathfrak{a}$, respectively. In this paper, we prove in $\mathsf{Z}\mathsf{F}$ (without the axiom of choice) that for all infinite cardinals $\mathfrak{a}$ and all non-zero natural numbers $n$, $$(2^{{\mathcal{B}}_n(\mathfrak{a})}{)}^{{\mathrm{\aleph }}_0}=2^{{\mathcal{B}}_n(\mathfrak{a})}$$ and $$2^{\mathrm{f}\mathrm{i}\mathrm{n}(\mathfrak{a}{)}^n}=2^{{\mathcal{B}}_{2^n-1}(\mathfrak{a})}\mathrm{.}$$ It is also proved consistent with $\mathsf{Z}\mathsf{F}$ that there exists an infinite cardinal $\mathfrak{a}$ such that $$2^{{\mathcal{B}}_1(\mathfrak{a})}<2^{{\mathcal{B}}_2(\mathfrak{a})}<2^{{\mathcal{B}}_3(\mathfrak{a})}<\cdots <2^{\mathrm{f}\mathrm{i}\mathrm{n}(\mathrm{f}\mathrm{i}\mathrm{n}(\mathfrak{a}))}\mathrm{.}$$