00111
01011
01101
01110
10011
10101
10110
11001
11010
11100

##</code><code>##

use 5.026;
use warnings;

use List::Util qw{ sum };
use Data::Dumper;

my @tests = (
   5,
   [ 1 ],
   [ 1, 2 ],
   [ 1, 2, 3 ],
   [ 1, 2, 3, 4 ],
   [ 1, 2, 3, 4, 5 ],
   [ 4, 17, 9, 23, 1, 14 ],
   );

foreach my $test ( @tests )
{
    my $raRes = allSums( $test ) or do {
        warn qq{$test: Not an array ref.\n};
        say q{-} x 25;
        next;
        };
    print Data::Dumper
       ->new( [ $test, $raRes ], [ qw{ test raRes }] )
       ->Indent( 1 )
       ->Dumpxs();
    say q{-} x 25;
}

sub allSums
{
    my $raNumbers = shift;
    return 0 unless ref( $raNumbers ) eq q{ARRAY};

    my $raSums = [ [], [], ];
    my $nElems = scalar @{ $raNumbers };

    if ( $nElems < 3 )
    {
        return $raSums;
    }

    foreach my $sumsOf ( 2 .. $nElems - 1 )
    {
        my $nZeros = $nElems - $sumsOf;
        my $rcNext = permutary( $nZeros, $sumsOf );

        while ( my $str = $rcNext->() )
        {
            my @posns;
            push @posns, pos $str while $str =~ m{(?=1)}g;
            unshift @{ $raSums->[ $sumsOf ] },
               { 
                  join( q{+}, @{ $raNumbers }[ @posns ] ),
                  sum @{ $raNumbers }[ @posns ]
               };
        }

    }
    return $raSums;
}

sub permutary
{
    no warnings qw{ portable };

    my ( $numZeros, $numOnes ) = @_;

    my $format = q{%0} . ( $numZeros + $numOnes ) . q{b};
    my $start  = oct( q{0b} . q{1} x $numOnes );
    my $limit  = oct( q{0b} . q{1} x $numOnes . q{0} x $numZeros );

    return sub
       {
           return undef if $start > $limit;
 
           my $binStr = sprintf $format, $start;
           die qq{Error: $binStr not $numOnes ones\n}
              unless $numOnes == $binStr =~ tr{1}{};

           my $jump = 0;
           if ( $binStr =~ m{(1+)$} )
           {
               $jump = 2 ** ( length($1) - 1 );
           }
           elsif ( $binStr =~ m{(1+)(0+)$} )
           {
               $jump  = 2 ** ( length($1) - 1 ) + 1;
               $jump += 2 ** $_ for 1 .. length( $2 ) - 1;
           }
           else
           {
               die qq{Error: $binStr seems malformed\n};
           }
           $start += $jump;
           return $binStr;
       };
}

##</code><code>##

5: Not an array ref.
-------------------------
$test = [
  1
];
$raRes = [
  [],
  []
];
-------------------------
$test = [
  1,
  2
];
$raRes = [
  [],
  []
];
-------------------------
$test = [
  1,
  2,
  3
];
$raRes = [
  [],
  [],
  [
    {
      '1+2' => 3
    },
    {
      '1+3' => 4
    },
    {
      '2+3' => 5
    }
  ]
];
-------------------------
$test = [
  1,
  2,
  3,
  4
];
$raRes = [
  [],
  [],
  [
    {
      '1+2' => 3
    },
    {
      '1+3' => 4
    },
    {
      '1+4' => 5
    },
    {
      '2+3' => 5
    },
    {
      '2+4' => 6
    },
    {
      '3+4' => 7
    }
  ],
  [
    {
      '1+2+3' => 6
    },
    {
      '1+2+4' => 7
    },
    {
      '1+3+4' => 8
    },
    {
      '2+3+4' => 9
    }
  ]
];
-------------------------
$test = [
  1,
  2,
  3,
  4,
  5
];
$raRes = [
  [],
  [],
  [
    {
      '1+2' => 3
    },
    {
      '1+3' => 4
    },
    {
      '1+4' => 5
    },
    {
      '1+5' => 6
    },
    {
      '2+3' => 5
    },
    {
      '2+4' => 6
    },
    {
      '2+5' => 7
    },
    {
      '3+4' => 7
    },
    {
      '3+5' => 8
    },
    {
      '4+5' => 9
    }
  ],
  [
    {
      '1+2+3' => 6
    },
    {
      '1+2+4' => 7
    },
    {
      '1+2+5' => 8
    },
    {
      '1+3+4' => 8
    },
    {
      '1+3+5' => 9
    },
    {
      '1+4+5' => 10
    },
    {
      '2+3+4' => 9
    },
    {
      '2+3+5' => 10
    },
    {
      '2+4+5' => 11
    },
    {
      '3+4+5' => 12
    }
  ],
  [
    {
      '1+2+3+4' => 10
    },
    {
      '1+2+3+5' => 11
    },
    {
      '1+2+4+5' => 12
    },
    {
      '1+3+4+5' => 13
    },
    {
      '2+3+4+5' => 14
    }
  ]
];
-------------------------
$test = [
  4,
  17,
  9,
  23,
  1,
  14
];
$raRes = [
  [],
  [],
  [
    {
      '4+17' => 21
    },
    {
      '4+9' => 13
    },
    {
      '4+23' => 27
    },
    {
      '4+1' => 5
    },
    {
      '4+14' => 18
    },
    {
      '17+9' => 26
    },
    {
      '17+23' => 40
    },
    {
      '17+1' => 18
    },
    {
      '17+14' => 31
    },
    {
      '9+23' => 32
    },
    {
      '9+1' => 10
    },
    {
      '9+14' => 23
    },
    {
      '23+1' => 24
    },
    {
      '23+14' => 37
    },
    {
      '1+14' => 15
    }
  ],
  [
    {
      '4+17+9' => 30
    },
    {
      '4+17+23' => 44
    },
    {
      '4+17+1' => 22
    },
    {
      '4+17+14' => 35
    },
    {
      '4+9+23' => 36
    },
    {
      '4+9+1' => 14
    },
    {
      '4+9+14' => 27
    },
    {
      '4+23+1' => 28
    },
    {
      '4+23+14' => 41
    },
    {
      '4+1+14' => 19
    },
    {
      '17+9+23' => 49
    },
    {
      '17+9+1' => 27
    },
    {
      '17+9+14' => 40
    },
    {
      '17+23+1' => 41
    },
    {
      '17+23+14' => 54
    },
    {
      '17+1+14' => 32
    },
    {
      '9+23+1' => 33
    },
    {
      '9+23+14' => 46
    },
    {
      '9+1+14' => 24
    },
    {
      '23+1+14' => 38
    }
  ],
  [
    {
      '4+17+9+23' => 53
    },
    {
      '4+17+9+1' => 31
    },
    {
      '4+17+9+14' => 44
    },
    {
      '4+17+23+1' => 45
    },
    {
      '4+17+23+14' => 58
    },
    {
      '4+17+1+14' => 36
    },
    {
      '4+9+23+1' => 37
    },
    {
      '4+9+23+14' => 50
    },
    {
      '4+9+1+14' => 28
    },
    {
      '4+23+1+14' => 42
    },
    {
      '17+9+23+1' => 50
    },
    {
      '17+9+23+14' => 63
    },
    {
      '17+9+1+14' => 41
    },
    {
      '17+23+1+14' => 55
    },
    {
      '9+23+1+14' => 47
    }
  ],
  [
    {
      '4+17+9+23+1' => 54
    },
    {
      '4+17+9+23+14' => 67
    },
    {
      '4+17+9+1+14' => 45
    },
    {
      '4+17+23+1+14' => 59
    },
    {
      '4+9+23+1+14' => 51
    },
    {
      '17+9+23+1+14' => 64
    }
  ]
];
-------------------------