Consider the symmetric group $S_{10}$. List all possible orders of the elements of this group and provide examples of elements of each order.
Attempt: the order of an element in the symmetric group is the lcm of the cycle lengths in writing that elements as disjoint cycles. So I believe we need to enumerate the ways of writing $10$ as a sum of natural numbers. For instance, we have $10=3+2+5$, so as the lcm is $30$ we could have an element of order $30$, an example being $(1\ 2\ 3)(4\ 5)(6\ 7\ 8\ 9\ 10)$. Is this right? Even still, I'm not sure how to answer this question 'nicely' without a great deal of calculation and mess on the page. I want my solution to be neat.
Yes, from a standpoint of what has to be done, you simply need the set of all $\rm{lcm}'$s of partitions of $10$, in this case. So there's in a sense a nice neat solution. Tedious is the particulars. I'm not going to endeavor to do that here.
Ramanujan was into partitions. I've heard the foremost authority now is George Andrews at Penn State. There's a function that counts the number of partitions of $n$, but it's I think difficult.
According to a comment solving this problem in general would solve the Riemann hypothesis. So watch out!
For getting started: $10=2+2+2+2+2$ gives us an element of order $\rm{lcm}(2,2,2,2,2)=2$. Or, $10=2+3+5$ gives an element of order $\rm{lcm}(2,3,5)=30$. $10=5+5$ gives $\rm {lcm}(5,5)=5$. Etc... So there's apparently $37$ more to check (according to the comments).
And it's easy also to provide elements of those orders: $$(12)(34)(56)(78)(9\,10),(12)(345)(6789\,10),(12345)(6789\,10)$$, respectively.