Given $a_{n}$ is the number of ways to partition elements of $[n]$ into non-empty sets, then further partition those sets into non-empty sets.
I've already determined the egf for non-empty sets of $[n]$ is $e^{x}-1$. But how would I take that one step further and determine the egf for partitions of a partition.
Can someone direct me on how to proceed with this?
The sequence $$(a_{i} : i \in \mathbb{N}) = (1, 3, 12, 60, 343, \ldots)$$ does not appear to currently be in the On-Line Encyclopedia of Integer Sequences, which suggests that there may not currently be any known closed form for the exponential generating function for this integer sequence. However, it is not difficult to construct explicit combinatorial formulas for this sequence, and it is plausible that such formulas may be used to evaluate the e.g.f. of $(a_{i})_{i \in \mathbb{N}}$.
As above, for $n \in \mathbb{N}$, let $a_{n}$ denote the number of ways to partition $\{ 1, 2, \ldots, n \}$ into nonempty sets and then partition these sets into nonempty sets. To clarify: $a_{n}$ is not the total number of set partitions resulting from this process, but rather the number of ways of partitioning sets through this process. For example, one may partition the set $\{ 1, 2, 3 \}$ as $\{ \{ 1 \}, \{ 2, 3 \} \}$ and then partition the elements in this partition as $S_{1} = \{ \{ 1 \}, \{ 2 \}, \{ 3 \} \}$, and one may partition the set $\{ 1, 2, 3 \}$ as $\{ \{ 2 \}, \{ 1, 3 \} \}$ and then partition the elements in this partition as $S_{2} = \{ \{ 1 \}, \{ 2 \}, \{ 3 \} \}$, but the 'way of obtaining' $S_{1}$ is counted separately with respect to the 'way of obtaining' $S_{2}$.
Let $B_{n}$ denote the $n^{\text{th}}$ Bell number, which is the number of ways of partitioning a set. Since the partitions of $\{ 1, 2, 3 \}$ are $\{ \{1, 2, 3\} \}$, $\{ \{2, 3\}, \{1\} \}$, $ \{ \{1, 3\}, \{2\} \}$, $ \{ \{1, 2\}, \{3\} \}$, and $ \{ \{1\}, \{2\}, \{3\} \}$, we have that:
$$a_{3} = B_{3} + 3 \cdot B_{2} \cdot B_{1} + B_{1}^3 = 12.$$
Similarly, we have that:
$$a_{4} = B_{4} + 4 \cdot B_{3} \cdot B_{1} + 3 \cdot B_{2} \cdot B_{2} + 6 \cdot B_{2} \cdot B_{1} \cdot B_{1} + B_{1} \cdot B_{1} \cdot B_{1} \cdot B_{1} = 60$$
More generally, $$a_{n} = \sum_{\lambda \vdash n} c_{\lambda} B_{\lambda_{1}} \cdot B_{\lambda_{2}} \cdot \cdots \cdot B_{\lambda_{\ell(\lambda)}} $$ where the coefficient $c_{\lambda}$ is equal to the number of partitions of $\{ 1, 2, \ldots, n \}$ into a set of size $\lambda_{1}$, a set of size $\lambda_{2}$, etc.