The incoming list is potentially 10^6 integers long, correct? That's potentially 1_000_000 integers.

This can be solved, it may involve packing your input into a huge string using pack, appending each int's packed representation onto the same string. Then manually implementing a sort that works in-place within the string, again using pack and unpack.

Messy! I don't think I want to invest the time in it. :)

Actually, 1 million integers in an array shouldn't be a deal-breaker, unless they're severely limiting your memory. My while-loop solution ought to be the most memory friendly. The map solutions generate a list of 1 .. n before sorting the inputs, so you're holding 2M integers rather than 1M in memory.