Hm, not sure to understand everything that you are doing, but this is my proposed solution: 
use strict;
use warnings;
use Data::Dumper;


my $MTX = [
[4,-1,-2,-2, 0,-1],
[-1,5, 0, 2,-3, 1],
[-2,0, 6, 1,-3, 5],
[-2,2, 1, 7,-3, 0],
[0,-3,-3,-3, 8,-3],
[-1,1, 5, 0,-3, 9],
];

my (@diagonal, @result);
my $k = 3;
for my $i (0..5) {
    push @diagonal, $MTX->[$i][$i];
}
for my $i (0..5) {
    for my $j (0..5) {
        for my $k (0..5) {
            next if $i == $j or $i == $k or $j == $k;
            my $sc;
            $sc += $diagonal[$_] for ($i, $j, $k);
            next unless defined $sc;
            next if $sc < 17;
            push @{$result[$sc]}, [$i, $j, $k];
        }
    }
}
print Dumper \@result;

[download]
The result is as follows: 
$VAR18 = [
           [
             0,
             1,
             4
           ],
           [
             0,
             2,
             3
           ],
           [
             0,
             3,
             2
           ],
           [
             0,
             4,
             1
           ],
           [
             1,
             0,
             4
           ],
           [
             1,
             4,
             0
           ],
           [
             2,
             0,
             3
           ],
           [
             2,
             3,
             0
           ],
           [
             3,
             0,
             2
           ],
           [
             3,
             2,
             0
           ],
           [
             4,
             0,
             1
           ],
           [
             4,
             1,
             0
           ]
         ];
$VAR19 = [
           [
             0,
             1,
             5
           ],
           [
             0,
             2,
             4
           ],
           [
             0,
             4,
             2
           ],
           [
             0,
             5,
             1
           ],
           [
             1,
             0,
             5
           ],
           [
             1,
             2,
             3
           ],
           [
             1,
             3,
             2
           ],
           [
             1,
             5,
             0
           ],
           [
             2,
             0,
             4
           ],
           [
             2,
             1,
             3
           ],
           [
             2,
             3,
             1
           ],
           [
             2,
             4,
             0
           ],
           [
             3,
             1,
             2
           ],
           [
             3,
             2,
             1
           ],
           [
             4,
             0,
             2
           ],
           [
             4,
             2,
             0
           ],
           [
             5,
             0,
             1
           ],
           [
             5,
             1,
             0
           ]
         ];
$VAR20 = [
           [
             0,
             2,
             5
           ],
           [
             0,
             3,
             4
           ],
           [
             0,
             4,
             3
           ],
           [
             0,
             5,
             2
           ],
           [
             1,
             2,
             4
           ],
           [
             1,
             4,
             2
           ],
           [
             2,
             0,
             5
           ],
           [
             2,
             1,
             4
           ],
           [
             2,
             4,
             1
           ],
           [
             2,
             5,
             0
           ],
           [
             3,
             0,
             4
           ],
           [
             3,
             4,
             0
           ],
           [
             4,
             0,
             3
           ],
           [
             4,
             1,
             2
           ],
           [
             4,
             2,
             1
           ],
           [
             4,
             3,
             0
           ],
           [
             5,
             0,
             2
           ],
           [
             5,
             2,
             0
           ]
         ];
$VAR21 = [
           [
             0,
             3,
             5
           ],
           [
             0,
             5,
             3
           ],
           [
             1,
             2,
             5
           ],
           [
             1,
             3,
             4
           ],
           [
             1,
             4,
             3
           ],
           [
             1,
             5,
             2
           ],
           [
             2,
             1,
             5
           ],
           [
             2,
             5,
             1
           ],
           [
             3,
             0,
             5
           ],
           [
             3,
             1,
             4
           ],
           [
             3,
             4,
             1
           ],
           [
             3,
             5,
             0
           ],
           [
             4,
             1,
             3
           ],
           [
             4,
             3,
             1
           ],
           [
             5,
             0,
             3
           ],
           [
             5,
             1,
             2
           ],
           [
             5,
             2,
             1
           ],
           [
             5,
             3,
             0
           ]
         ];
$VAR22 = [
           [
             0,
             4,
             5
           ],
           [
             0,
             5,
             4
           ],
           [
             1,
             3,
             5
           ],
           [
             1,
             5,
             3
           ],
           [
             2,
             3,
             4
           ],
           [
             2,
             4,
             3
           ],
           [
             3,
             1,
             5
           ],
           [
             3,
             2,
             4
           ],
           [
             3,
             4,
             2
           ],
           [
             3,
             5,
             1
           ],
           [
             4,
             0,
             5
           ],
           [
             4,
             2,
             3
           ],
           [
             4,
             3,
             2
           ],
           [
             4,
             5,
             0
           ],
           [
             5,
             0,
             4
           ],
           [
             5,
             1,
             3
           ],
           [
             5,
             3,
             1
           ],
           [
             5,
             4,
             0
           ]
         ];
$VAR23 = [
           [
             1,
             4,
             5
           ],
           [
             1,
             5,
             4
           ],
           [
             2,
             3,
             5
           ],
           [
             2,
             5,
             3
           ],
           [
             3,
             2,
             5
           ],
           [
             3,
             5,
             2
           ],
           [
             4,
             1,
             5
           ],
           [
             4,
             5,
             1
           ],
           [
             5,
             1,
             4
           ],
           [
             5,
             2,
             3
           ],
           [
             5,
             3,
             2
           ],
           [
             5,
             4,
             1
           ]
         ];
$VAR24 = [
           [
             2,
             4,
             5
           ],
           [
             2,
             5,
             4
           ],
           [
             4,
             2,
             5
           ],
           [
             4,
             5,
             2
           ],
           [
             5,
             2,
             4
           ],
           [
             5,
             4,
             2
           ]
         ];
$VAR25 = [
           [
             3,
             4,
             5
           ],
           [
             3,
             5,
             4
           ],
           [
             4,
             3,
             5
           ],
           [
             4,
             5,
             3
           ],
           [
             5,
             3,
             4
           ],
           [
             5,
             4,
             3
           ]
         ];

[download]
I leave it up to you to check the results.

Je suis Charlie.

In reply to Re: Combinatorial problem by Laurent_R
in thread Combinatorial problem by baxy77bax

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