NextPermute() and NextPermuteNum() modify the input array in place, and (to traverse all permutations) require that the input starts off sorted. So minimally adapting your previous code to use Algorithm::Loops would look (untested) something like this:

#!/usr/bin/env perl

use strict;
use warnings;
use Algorithm::Loops qw(NextPermuteNum);
use Data::Dump;
use feature    qw(say);

my @n = sort { $a <=> $b } ( 1, 1, 1, 2, 3, 3, 3, 4, 5, 5, 5, 6, 7, 8,
+ 11, 12, 13, 14, 14, 15 );

my $m3 = qr/(.)\1\1/;
my $m2 = qr/(.)\1/;

do {
    my $s = pack( "(A*)*", @n );
    if ( $s =~ /($m3)/ ) {
        say $1;
        sleep 1;
        goto PERMUTE;
    }
    if ( $s =~ /($m2)/ ) {
        goto PERMUTE;
        say $1;
        sleep 1;
    }
    say join " ", @n;
  PERMUTE:
    ; # empty statement to attach the label to
} while (NextPermuteNum( @n ));

__END__
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As for the 15!/8! figure: imagine each of the 1s is a different colour, then there are 15! distinguishable ways to arrange the numbers. Take any one of those (valid solution or not) and it has 8 coloured 1s in some order; and each of the 8! ways to order those colours will appear once in what is otherwise the same arrangement of numbers.