Let $P(n,k)$ denote the number of partitions of $n$ into $k$ parts. 
I would like to know for given $n$, which $k$ does maximize $P(n,k)$?

Additionally, information on the maximum of $P(n,k)$, for fixed $n$, would also be of interest.

I'm interested in varied information but please note I'm still not out of high school! 

I've done some research on the topic earlier and found out that for $P(100, k)$, it is $7$ that yields the greatest value, even superseding $k=100(1/2)=50$. 

I am also aware of an asymptotic formula [from an MO question](https://mathoverflow.net/questions/72418/what-are-the-best-known-bounds-on-the-number-of-partitions-of-n-into-exactly-k):

$$P(n, k)\sim\frac{1}{2\pi n} \left( \frac{e^2 n}{k^2} \right)^k$$

Also, I know about that triangular-looking grid for $P(n,k)$. But then, what is derived from it?

Would [this PDF](http://www.cs.uwaterloo.ca/journals/JIS/VOL8/Doslic/doslic33.pdf) be of any help?