We start with a fixed partition of $[n]$, namely $(1,1...,1)=(1^{(n)})$.
We define an operation, say F, that takes two numbers within the partition and adds them up. So the first step leads to the partition $(2,1^{(n-2)})$.
Repeating it another time, we get two possibilities $(3,2,1^{(n-5)})$ and $(2,2,1^{(n-4)})$
It is clear that this process will end after $n-1$ steps when all the numbers are added up, leaving us with the partition $(n)$.
What I'm interested in finding is the total number of such partitions for a particular $n$.
I initially tried variants of Bell numbers, but to no avail. I don't think there's a direct link to them, but I would be happy if anyone could prove me wrong on that. Writing the sequence out by hand, I obtain $$1,2,3,5,8,11,15,22....$$I couldn't find an obvious pattern and it seems too unwieldy to manipulate by hand.
If anyone can point me towards the right direction, that would really be great! Thank you!
The answer is right there on Wikipedia! Duh-oh! I'm not certain how to close this question, so I will just write a short answer and accept it! Do let me know if I should be doing something else instead.
So, accordingly, there's no known closed-form expression for the partition function, but there exists generating functions and recurrence relations for it. And that's the answer I was looking for!
I was trying to find the number of possible partitions obtained from applying $F$ to $(1^{(n)})$ $k$ times, but this direction doesn't seem that fruitful.
Thank you to all who looked at and answered my query!