Let $\mathcal{A}=\{A_1, A_2, \ldots, A_m\}$ be a uniformly random set partition of $[n]$.

What can we say about $||\mathcal{A}||_2 = \sqrt{\sum_{i=1}^m |A_i|^2}$? It is clearly upper bounded by $n$, but since the ``typical'' size of these $A_i$ is more like $\log(n) - \log\log(n)$, it seems reasonable to expect that $||\mathcal{A}||_2$ is actually more like $O\left(\sqrt{n\log(n)}\right)$, or something like that. What I really need is two things: a) whether $||\mathcal{A}||_2$ concentrates around its expectation, and b) upper bounds on $\mathbb{E}\left[||\mathcal{A}||_2\right]$ better than the trivial $n$ bound.

In my search for papers on this topic, I've found the following result: If $Z_i = |\{j: |A_j| = i\}|$ is the number of parts of size $i$, then the vector $(Z_1, Z_2, \ldots, Z_n)$ is distributed as $Z_i \sim \mathrm{Po}\left(\frac{r^i}{i}\right)$ (where $r=r(n)$ is defined by $re^r = n$), independent other than for conditioning on $\sum_i iZ_i = n$. So my question could equivalently be phrased as asking about $$\mathbb{E}\left[\sqrt{\sum_i i^2 Z_i}\, \middle|\, \sum_i i Z_i = n \right]$$ but it's not clear to me if that is any easier than the original question.