One solution:

use strict;

my @chars = ("a", "a", "b", "c");

my %non_repeat;
foreach (@chars) {
    if (exists($non_repeat{$_})) {
        $non_repeat{$_} = 2;
    } else {
        $non_repeat{$_} = 1;
    }
}

my @combinations;
my ($first, $second);
my @keys = keys %non_repeat;
my ($index1, $index2);
for ($index1 = 0; $index1 <= $#keys; $index1 ++) {
    if ($non_repeat{$keys[$index1]} == 2) {
        push @combinations, $keys[$index1] x 2;
    }
    for ($index2 = $index1 + 1; $index2 <= $#keys; $index2 ++) {
        push @combinations, $keys[$index1] . $keys[$index2];
        push @combinations, $keys[$index2] . $keys[$index1];
    }
}

print join(",", @combinations);
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