To my knowledge there doesn't exist a $O(n)$ worst-case algorithm that solves the following problem:
Given a sequence of length $n$ consisting of finite integers, find the permutation where every element is less than or equal to its successor.
But is there a proof that it doesn't exist, in the transdichotomous model of computation?
Note that I'm not limiting the range of the integers. I'm not limiting solutions to comparison sorts either.
Integers can be stably sorted in $O(n)$ time with $O(1)$ additional space. More precisely, if you have $n$ integers in the range $[1, n^c]$, the can be sorted in O(n) time.
This was only shown a couple of years ago by a team including the late Mihai Pătrașcu (which should surprise nobody who is familiar with his work). It's a remarkable result which I'm surprised more people don't know about, because it means that the problem of sorting integers is (theoretically) solved.
There is a practical algorithm (given in the paper above) if you're allowed to modify keys. Basically, you can compress sorted integers more than you can compress unsorted integers, and the extra space that you gain is precisely equal to the extra memory needed to do the radix sort. They also give an impractical algorithm which supports read-only keys.
