I'm working on a problem where I need to calculate the expected maximum pseudotree size in a randomly-generated pseudoforest with $n$ nodes.
Expected maximum value is of course:
$$ E(x) = \sum_{i=1}^{n}i\cdot p(i) $$
Where $i$ is the size of the largest pseudotree in that forest and $p(i)$ is the probability of a tree of size $i$ being the largest tree in the forest.
I'm trying to calculate $p(i)$ based on the following formula, and I'm pretty sure this is where the problem is:
$$ p(i) = \frac{\text{graphs}(i)*\text{partitions}(n, i)}{(n-1)^n} $$
Where $\text{graphs}(i)$ is the number of pseudotrees with $i$ nodes (see here) and $\text{partitions}(n,i)$ is the number of partitions of $n$ elements where the largest cell has exactly $i$ values and the smallest has at least 2 (see here). Unfortunately for me, this seems to be under counting the numerator for $n>5$. My working theory is that there's a potential division of the elements emerging for larger values of $n$ that is not being counted. Any ideas?
Edit
I think what may be happening is that I'm forgetting about the different tree configurations for the rest of the partition. I'm investigating that now.
Edit
Current attempt at counting the remaining configurations:
$$ \sum_{j=1}^{n-i}graphs(j)\cdot partitions(n-i,j) $$
