Here's a version of the same algorithm working on array elements instead of characters. (Messier, isn't it?)

With 10 elements (states) it takes 1.5 seconds on my machine, but I think most of the time is taken by the printing.

How many neighbors can you have? (for code testing purposes.)

#!/usr/bin/perl -l

# http://perlmonks.org/?node_id=1198509

use strict;
use warnings;

my @elements = (0, 2, 3);
@elements = 0..9;

my ($first, $last) = @elements[0, -1];
my %next;
@next{@elements} = @elements[1 .. $#elements];

for my $count (1,2,3,4,7,8)
  {
  print "count=$count";

  my @set = ($first) x $count;
  local $, = ',';

  while(1)
    {
    print @set;
    my $i = $#set;
    $i-- while $i >= 0 and $set[$i] eq $last;
    $i < 0 and last;
    @set[$i .. $#set] = ($next{$set[$i]}) x ( @set - $i);
    }
  }
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