SOLVED: This is the Gumbel distribution
Let $\pi^n_i$ be set containing the terms in the $i$-th integer partition of the natural number $n$, according to whatever enumeration. For example, for $n = 5$ we'll have some partition $j$ with $\pi^5_j = \{3,1,1\}$, and so $|\pi^5_j| = 3$.
I'm interested in knowing more about what distribution arises for the $|\pi^n_j|$ for a given $n$ and any useful property related to it. Knowing its interquartile range or estimating its peak and width for large $n$ would be ideal.
Here's an example of this distribution for $n = 60$, where we can see the majority of partitions have around 10-25 terms. This distribution $f_n(k)$ can be computed using the recurrence relation
$$ f_n(k) = f_{n-k}(k) + f_{n-1}(k-1)$$
for $f_0(0) = 1$, and $f_n(k) = 0$ if $n \le 0$ or $k \le 0$, and $n,k$ not simultaneously zero. I am unsure if it is possible to get the information I'm looking for from this relation, as I'm unfamiliar with the appropriate techniques.
This problem boils down to the distribution of rows/columns in Young diagrams for a given $n$, if I'm not mistaken. I've been trying to find out more about this but I think I don't know the right search terms.
I tried fitting a few distributions and the Log-Normal distribution seems incredibly accurate, but I cannot make sense of why it would appear here or what its parameters should be.
Any pointer in the right direction would be helpful.
This is the Gumbel distribution. A proof was given by Erdos and Lehner.