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Here is a recent article describing a simpler algorithm for sampling derangements. I also found slides for the presentation. Since the paper is so recent, I guess this means that a small modification of Fisher-Yates is unlikely to generate derangements, since someone would have already come up with it by now. Still, their algorithm is in-place and has better expected running time than retrying Fisher-Yates until you get a derangement.

Here is a Perl implementation I whipped up. It is slightly odd because I followed their lead and used array indexing from 1.

```sub rand_derangement {
my \$n = shift;
return if \$n == 1; ## no derangements of size 1

## precompute \$D[n] == number of derangements of size n
my @D = (1,0);
push @D, \$#D * (\$D[-1] + \$D[-2]) while \$#D < \$n;

my @A = (undef, 1 .. \$n);
my @mark = (1, (0) x \$n);

my (\$i, \$u) = (\$n, \$n);

while (\$u > 1) {
if (! \$mark[\$i]) {
my \$j = 0;
\$j = 1 + int rand(\$i-2) while \$mark[\$j];

@A[\$i,\$j] = @A[\$j,\$i];

if ( rand(1) < (\$u-1) * \$D[\$u-2] / \$D[\$u] ) {
\$mark[\$j] = 1;
\$u--;
}
\$u--;
}
\$i--;
}

return @A[1..\$n];
}

in thread Random Derangement Of An Array by Limbic~Region

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