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.
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)
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?
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.