Let me put this into an example.
We have an array of the first $r$ (example $r=7$) natural numbers;
$1, 2, 3, 4, 5, 6, 7$
and given $k=3$, so you get to select any three integers from the pool above. Taken $1$, $6$ and $7$, we have their sum $14$. We get to partition this number ($14$) such that its $k$ parts all appear in the array above, like $14=2+5+7=3+4+7$ and so on
In this manner, which $k$ numbers' sum do you think can be partitioned into the most number of $k$ integers, in which any part can be at most $r$ from that array? Let me guess, this should be the sum of the last $k$ integers in the pool, right?
BTW I've used the terms pool and array, and integer and natural number in their same contexts. Sorry for that.
Yes, I guess it involves partitioning of the numbers' sum into distinct $k$ parts of which the largest part $r$ is the largest number in the array. Therefore, reinstating my previous example, we have $Q(14, 3:7)$ where 7 is the largest number in the array, consequently no part among the $k(=3)$ parts can exceed the number 7.
By the way, I remember having seen an asymptotic formula for $Q(n, k:r)$. I don't remember where that was. I'll let everyone know as soon as I find out.
To put this question in another way, which $k$ numbers' sum from the array (of the first $r$ natural numbers, in ascending order) will have the greatest number of $k$ distinct partitions with each part equal to or lesser than $r$?