A regex to find if a letter occurs exactly $n times.

Is this what you were looking for as a first step?

#!/usr/bin/perl

# http://perlmonks.org/?node_id=1201500

use strict;
use warnings;

my @words = glob '{i,x}' x 6;

print "@words\n";

my $n = 1; # number of times a letter occurs in a word

my @oneletter = grep
  /^(?|
  (?=.*?(.)(?!.*\1))
  (?:
  (?:(?!\1).)*
  \1
  ){$n}
  (?:(?!\1).)*
  |
  (?=.*(.))
  (?:
  (?:(?!\1).)*
  \1
  ){$n}
  (?:(?!\1).)*
  )$
  /x, @words;

print "\n@oneletter\n\n", scalar @oneletter, "\n";
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Update: fails for some test cases