Given a positive integer $n$, I want to know the expected value for the number of parts of a random partition of $n$.
I am aware that a similar question has been asked already: Expected number of parts of a uniformly selected partition of $n$ ; however, I do not want to select the partitions uniformly. I want to collect the partitions of $n$ and have the partition $\lambda$ appear $dim(\lambda)$ times. For example, for $n=3$ we have
$\{3\}$ (once)
$\{2,1\}$ (twice)
$\{1,1,1\}$ (once)
So the expected value will be $\frac{1+2+2+3}{4}=2.$
I wrote some code to compute this and it looks like the function is logarithmic in $n$, which is the case in the linked question, too. How is my question related to uniform case? Are there any known results for this?
Thank you