What's the expected loop length of a randomly chosen mapping from 1..N to 1..N?

monte-carl.pl

use v6;

for 1..* -> $N {
    my $sum   = 0;
    my $count =  5 * $N**2;
    for ^$count {
        my @f := (^$N).roll($N).list;
        my %seen;
        my $current = 0;
        while !%seen{$current} {
            %seen{$current} = True;
            $current = @f[$current];
            $sum++;
        };
    }
    my $experiment = $sum / $count;
    my $theory     = theory($N);
    FIRST say "  N    THEORY     EXPERIMENT (ERR)";
    printf "%3d    %3f   %3f   (%2d%%)\n", $N, $theory, $experiment, abs($theory - $experiment) / $theory * 100;
}

sub postfix:<!>(Int $n) { [*] 1..$n }
sub theory($N) {
    ($N - 1)! * [+] (1..$N).map: {
         $_ * $_ / $N ** $_ / ($N - $_)!
    }
}

output.txt

  N    THEORY     EXPERIMENT (ERR)
  1    1.000000   1.000000   ( 0%)
  2    1.500000   1.550000   ( 3%)
  3    1.888889   1.888889   ( 0%)
  4    2.218750   2.137500   ( 3%)
  5    2.510400   2.472000   ( 1%)
  6    2.774691   2.733333   ( 1%)
  7    3.018139   3.008163   ( 0%)
  8    3.245018   3.356250   ( 3%)
  9    3.458316   3.456790   ( 0%)
 10    3.660216   3.628000   ( 0%)
 11    3.852372   3.804959   ( 1%)
 12    4.036074   3.962500   ( 1%)
 13    4.212348   4.280473   ( 1%)
 14    4.382029   4.355102   ( 0%)
 15    4.545807   4.548444   ( 0%)
 16    4.704258   4.683594   ( 0%)
 17    4.857871   4.898962   ( 0%)
 18    5.007063   5.041975   ( 0%)
 19    5.152196   5.211080   ( 1%)
 20    5.293585   5.314500   ( 0%)
 21    5.431504   5.488435   ( 1%)