Is the anecdote about the impetus for quicksort really necessary? It seems to be only sourced by an interview.
Reword partition part.
Remove the part about the boss's bet. If this is a description of what actually happened, it's not of encyclopedic value; if it's an expression, it shouldn't be here at all.
Remove external links.
For the whole section, stricter sourcing should be implemented to get rid of this cruft.
The last paragraph seems to have some important information, but it also might need to be stripped down.
Due to its recursive nature, quicksort (like the partition routine) has to be formulated so as to be callable for a range within a larger array, even if the ultimate goal is to sort a complete array. This is untrue, as the sort is not necessarily in-place.
The caption on the image here is far too long.
Here we mention two specific partition methods. The author's we is not necessary here.
Try to find a review article or similar that discusses this scheme; primary sources shouldn't really be used here as they don't establish notability.
Sourcing needs to be vastly improved. There's only one citation in the entire first paragraph, which is quite chunky, and this is just one example.
Parentheticals should be refactored, they don't add much structurally and are far too long.
This section desperately needs to be refactored. The lead paragraph doesn't explain the actual implementation but describes edge cases that require a deeper knowledge of the implementation. I should probably come back to this after doing more research into quicksort.
Subsequent recursions (expansion on previous paragraph)