I started with the assumption that the partition of 1 was [ [ 1 ] ] and that the partion of each larger one was each sub partion with 1 added to the end, and with one added to each element. Hmm that was more confusing than in my head. Well here, the partition of 2 is [ [ 1 + 1] , [1 , 2]][ [ 1 + 1] , [1 , 1]] and the partition of 3 is [ [ 2 + 1], [1 + 1, 2], [1, 2 + 1], [1 , 2 , 1]]. Well if that hasn't made any sense here is the code.

use strict;
use warnings;

use Data::Dumper;

sub partition {
    my $num = shift || 1;
    my $temp = [ [1] ];
    # updated 0 to 2 in the following, 
    # since we start with one, the first add_one gets us to two.
    $temp = add_one($temp) for (2 .. $num);  
    return $temp;
}

sub add_one {
  my $combinations = shift;
  my $temp;
  foreach my $combination (@$combinations) {
    foreach my $element (@$combination) {
        $element++;
        push @$temp, [ @$combination ];
        $element--;
    }
    push @$temp, [ @$combination, 1 ];
  }
  my $hash;
  foreach my $combination (@$temp) {
      $hash->{ join("-", sort @$combination) }++;
  }
  return [ map { [ split "-", $_ ] } keys %$hash ];  
}

print Dumper(partition(10) );
[download]

Oh and I added in some code to stop it from accumulating duplicates.

Updates: Thanks blockhead for noticeing some errors there