### Abstract

Counting sort is a well-known algorithm that sorts objects of any kind
mapped to integer keys, or else to keys in one-to-one correspondence
with some subset of the integers (e.g. alphabet letters). However, it
is suitable for direct use, viz. not just as a subroutine of another
sorting algorithm (e.g. radix sort), only if the key range is not
significantly larger than the number of the objects to be sorted.
This paper describes a tail-recursive generalization of counting sort
making use of a bounded number of counters, suitable for direct use in
case of a large, or even infinite key range of any kind, subject to
the only constraint of being a subset of an arbitrary linear order.
After performing a pen-and-paper analysis of how such algorithm has to
be designed to maximize its efficiency, this paper formalizes the
resulting generalized counting sort (GCsort) algorithm and then
formally proves its correctness properties, namely that (a) the
counters' number is maximized never exceeding the fixed upper
bound, (b) objects are conserved, (c) objects get sorted, and (d) the
algorithm is stable.