Update (again: previous code that appeared here was broken): If you modify the Fisher-Yates algorithm to force every element to swap with a higher element (instead of possibly "swapping" with itself), then almost all derangements are possible; all derangements become possible (thought not equally likely) if you use a random position in the list as the first location to swap from, and allow some swaps to be skipped if the candidates are already deranged.

use strict;
use warnings;

sub derange {
  my @list = @_;
  # Swap every element with something higher
  my $start = int rand @list;
  for (0..($#list-1)) {
    my $this = ($start + $_) % @_;
    next if $list[$this] ne $_[$this] and $_ < $#list-1 and rand > .52
+;
    my $other = $_ + 1 + int rand($#list - $_);
    $other = ($start + $other) % @_;
    @list[$this,$other] = @list[$other,$this];
  }
  "@list";
}

my @ar = ('a'..'d');
print "@ar\n";
my %countem;
$countem{derange @ar}++ for 1..5000;
print "$_: $countem{$_}\n" for (sort keys %countem);
[download]