I'm proud of it, so give a nonexpert a break. That said, any constructive feedback would be most welcome.

For a non-expert it's pretty good. There are a few inconsistencies but I'm sure you can spot those and polish them out in due course. The only bit which really grates is this:

# GENERATE THE BOARD

my @board;
my $inversions;
GEN:
@board = (0, shuffle(1..15));
# @board = (1,0,2..15); # for testing
$inversions = 0;
for my $a(1..15) {
  for my $b(1..15) {
    ++$inversions if($a<$b and $board[$a]>$board[$b])
  }
}
goto GEN if $inversions % 2;
[download]

There's really no need for a goto in here. You are clearly aware of conditional loops in Perl as you've used them elsewhere in this script. Let's re-write this to avoid the goto, avoid the special variables $a and $b and make it marginally more efficient by only checking the triangle rather than the square.

my @board;
my $inversions = 1;
while ($inversions % 2) {
    @board = (0, shuffle(1..15));
    $inversions = 0;
    for my $x (1 .. 15) {
        for my $y ($x + 1 .. 15) {
            $inversions++ if $board[$x] > $board[$y];
        }
    }
}
[download]

Hopefully I have not altered the logic of your board construction at all, just tweaked the code to do the same thing but in a slightly more Perlish way. For completeness I would probably put this in its own subroutine and just call my @board = setup_board() in the main script as the setup is entirely independent of the rest of the script.