Update: Like the saying goes: problems worthy of attack prove their worth by biting back. Immediately after I posted the scheme below I realized that it too fails to sample all the derangements. The decomposition of a derangement I give in the description of the algorithms is incorrect. For example, it does not describe the derangement 1,2,3,0 of 0,1,2,3. The moral of the story: do the math before doing the coding :) . I am beginning to see why my (admittedly very superficial) online searches for a derangement sampling algorithm turned up nothing cleverer than a rejection method.

OK, below I give a first pass at a function to generate a random derangement. I think it samples all derangements fairly, but I have not verified this:

my @cols = 0..11;
randomly_derange( \@cols );
@nums = map [ @{$_}[@cols] ], @nums;
sub randomly_derange {
my $arr = shift();
my @i = 0..$#$arr;
while ( @i ) {
my @swap = map splice( @i, rand( @i ), 1 ), 1, 2;
@{ $arr }[ @swap ] = @{ $arr }[ @swap[ 1, 0 ] ];
last if @i == 3;
}
if ( @i ) {
my @j = rand() < 0.5 ? @i[ 1, 2, 0 ] : @i[ 2, 0, 1 ];
@{ $arr }[ @i ] = @{ $arr }[ @j ];
}
}

Even if it is correct, I am sure that there is plenty of room for optimizing randomly_derange.

The idea behind it is this. Every derangement of a list having an even number of elements can be represented as a series of pairwise swaps. When the list has an odd number of elements, every derangement can be represented as a series of pairwise swaps, plus a 3-way derangement (consisting of a 1-rotation, left or right, of some 3-sublist of the original list). randomly_derange picks random swaps uniformly. When the list has an even number of elements, that's all there is to it. When the list has an odd number of elements, then at the end it performs one of two possible 3-way rotation of the three remaining unshuffled elements.