How many collections of six positive, odd integers have a sum of $18$? Note that $1 + 1 + 1 + 3 + 3 + 9$ and $9 + 1 + 3 + 1 + 3 + 1$ are considered to be the same collection.
Let the six odd positive integers be $2x_1+1$, $2x_2+1$, $\cdots$, $2x_6+1$ where $x_i$ are non-negative integers. The fact that their sum is $18$ implies $$x_1+x_2+x_3+x_4+x_5+x_6=6$$
However, because sequence of $x_i$ does not matter per the problem's description, this is equivalent to putting $6$ indistiguishable balls into $6$ indistguishable boxes, empty box is permitted. As the number is small, we can manually list all the possible cases (in fact, we can also do the same without first simplifying):
Hence, the final answer is $\boxed{11}$.