Um, well, I think the title pretty much says it all.
Nevertheless, allow me to explain.
I am aware of a certain partition function $Q(n, k)$ that is supposed to remove duplication of parts and $k$ is the number of terms it contains.
I have a hunch that like $P(n, k)$ where $k$ limits the number of parts is equivalent to $k$ being the largest part, the same would apply to $Q(n, k)$. But then, I think probably not.
I'd like to know if any generating functions exist for what I have described (both removal of duplicate/repeated parts and largest part $k$). And one last question if you can't (or can, whatever) answer that question, is $Q(n, k)$ always lesser than n? (I think so...)
Well, thanks for stopping to read this and to anyone who bothered to help! PS Note that $q(n, k)$ and $Q(n, k)$ are different! Head over to Wolfram|Alpha for more detail.
Edit 1
I'm sorry, but I'm just a high school student (Class IX) and don't know what that is supposed to mean. Hope you can help me with it, thanks again.
Question 1: Where did this new variable $x$ come in from?
Question 2: One more thing, does this generating function produce a result always lesser than $m$?
The last question: If I'm right, I can interpret the core of your statement (excl. the coefficient part) as the product $(1 + x^k)$ as $k$ varies from $1$ to $k$... where's $m$ in all this? And for what am I going to find the coefficient of $x^m$, given there is no $x^m$ in the equation? (I'm pretty sure that ain't an algebraic coefficient)
Edit 2
Heck Mr. Garry, I just forgot that! Well, if $x$ is a placeholder of sorts, but what is it actually doing there if it's not supposedto be there? That said, is it a reference to any kind of big pi notation or such?
Edit 3
Fine, I've understood it now. I'm telling this here as writing it as an answer seems to look like me answering myself. Anyway, now that I think about it, is it further possible to restrict $Q(n, k)$ so as to include only, say, $t$ parts, such that it becomes a function where n is partitioned into $t$ parts - no less, no more - and the largest part is $k$? Is that possible?
Oh my. I seriously ask a lot of questions.