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

Abstract: A partition is finitary if all its blocks are finite. For a cardinal $\mathfrak{a}$ and a natural number $n$, let $\mathrm{fin}(\mathfrak{a})$ and $\mathscr{B}{n}(\mathfrak{a})$ 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 $\mathsf{ZF}$ (without the axiom of choice) that for all infinite cardinals $\mathfrak{a}$ and all non-zero natural numbers $n$, [ (2^{\mathscr{B}{n}(\mathfrak{a})})^{\aleph_0}=2^{\mathscr{B}_{n}(\mathfrak{a})} ] and [ 2^{\mathrm{fin}(\mathfrak{a})^n}=2^{\mathscr{B}_{2^n-1}(\mathfrak{a})}. ] It is also proved consistent with $\mathsf{ZF}$ that there exists an infinite cardinal $\mathfrak{a}$ such that [ 2^{\mathscr{B}{1}(\mathfrak{a})}<2^{\mathscr{B}{2}(\mathfrak{a})}<2^{\mathscr{B}_{3}(\mathfrak{a})}<\cdots<2^{\mathrm{fin}(\mathrm{fin}(\mathfrak{a}))}. ]