Stable Sorting, ie, Minimally-Disruptive Sorting

Question

Suppose I have a list of things (numbers, to keep things simple here) and I have a function I want to use to sort them by, using SortBy. For example, the following sorts a list

Answer 1

There is a variant of SortBy which breaks ties by using additional ordering functions:

SortBy[list,{f1, f2, ...}]

By counting ties you can thus obtain a stable sorting:

Module[{tie = 0}, 
 SortBy[{19, 301, 201, 502, 501, 101, 300}, {Mod[#, 10] &, (tie++) &}]]

yields

{300, 301, 201, 501, 101, 502, 19}