I've looked at permutations and combinations, and have decided they're not what I'm looking for. I have a game at home called Six Cubes. It's a dice game -- you roll six dice, and if specific combinations come up, you can store them away for points, and decide to roll the remaining dice or stop. The dice are your standard d6 (that's a gaming term).

I'm trying to calculate the number of rolls possible. Everything from "1 1 1 1 1 1" to "6 6 6 6 6 6". However, the roll "1 1 1 1 1 2" is no different from "1 1 1 2 1 1". I'm trying to determine the mathematical approach to calculating the number of rolls possible. I can produce them (there are 462), but I can't figure out a generic mathematical formula to get that number.

From a regex standpoint, it's /^(?=.{6}$)1*2*3*4*5*6*/. (This is extraneous, but it defines the strings that match. "111112" is valid, "121111" is not,, because it's "identical".)

Here's my Perl code. It looks eerily familiar, like I've seen this somewhere before, but I don't know where offhand:

sub unordered_sets {
  my ($in, $out, $left, $idx, $curr) = @_;
  push @$out, [@$curr] and return if $left == 0;
  $idx ||= 0;
  for ($idx .. $#$in) {
    push @$curr, $in->[$_];
    unordered_sets($in, $out, $left-1, $_, $curr);
    pop @$curr;
  }
}

my @sets;
unordered_sets [1..6], \@sets, 6;
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