I don't have a strong CS background so apologies if the question is trivially simple:
So I am working on an algorithm, say $A(n)$ , which runs over all integer partitions of $n$. Now the algorithm calls a sub-routine $S(\pi)$ for each of the partition $\pi$ (and does nothing else). The time-complexity of running the subroutine is $O(n^c)$ for some constant $c$.
Now the number of integer partitions is $\frac{1}{4n\sqrt 3}\exp\Big(\pi \sqrt \frac{2n}{3}\Big)$. And hence, the time complexity of my algorithm should be $O(e^{\pi\sqrt \frac{2n}{3}}n^{c'})$ or maybe just $O(e^{\pi\sqrt \frac{2n}{3}})$ or just $O(e^{\sqrt n})$ ?
Also, in my understanding I can call this SubEXP in the sense of $\textbf{III}$ category presented here.
I would basically like to understand how best to describe this complexity in an academically accurate manner. Thanks alot !