Average loop length

Let f be a randomly chosen mapping from the numbers 1..N to the numbers 1..N. At some point, the sequence 1, f(1), f(f(1))... will contain a repetition, a number that occurring for the second time in the sequence.

Write a program or a script that estimates, for each N, the average length until the first such repetition.

Also calculate this expected length using an analytical formula, and optionally compare the simulated result with the theoretical one.

This problem comes from the end of Donald Knuth's Christmas tree lecture 2011.

Example of expected output:

 N    average    analytical    (error)
===  =========  ============  =========
  1     1.0000        1.0000  (  0.00%)
  2     1.4992        1.5000  (  0.05%)
  3     1.8784        1.8889  (  0.56%)
  4     2.2316        2.2188  (  0.58%)
  5     2.4982        2.5104  (  0.49%)
  6     2.7897        2.7747  (  0.54%)
  7     3.0153        3.0181  (  0.09%)
  8     3.2429        3.2450  (  0.07%)
  9     3.4536        3.4583  (  0.14%)
 10     3.6649        3.6602  (  0.13%)
 11     3.8091        3.8524  (  1.12%)
 12     3.9986        4.0361  (  0.93%)
 13     4.2074        4.2123  (  0.12%)
 14     4.3711        4.3820  (  0.25%)
 15     4.5275        4.5458  (  0.40%)
 16     4.6755        4.7043  (  0.61%)
 17     4.8877        4.8579  (  0.61%)
 18     4.9951        5.0071  (  0.24%)
 19     5.1312        5.1522  (  0.41%)
 20     5.2699        5.2936  (  0.45%)

Perl 6

Runs on Rakudo Warszawa (2012.12).

<lang perl6>my $MAX_N = 20; my $TRIALS = 10_000;

for 1 .. $MAX_N -> $N {

   my @lengths = loop-length(random-mapping($N)) xx $TRIALS;
   my $average = ([+] @lengths) / @lengths;
   my $analytical = analytical($N);
   my $percent_error = abs($analytical - $average) / $analytical * 100;

   FIRST say " N    average    analytical    (error)";
   FIRST say "===  =========  ============  =========";

   my @columns =
       $N.fmt('%3d'),
       $average.fmt('%9.4f'),
       $analytical.fmt('%12.4f'),
       $percent_error.fmt('    (%2d%%)'),
   ;
   say join '  ', @columns;

}

sub random-mapping($size) {

   # Not just a permutation, but a mapping from all 1..N to any 1..N
   # In other words, .roll, not .pick

   return hash map { $_ => (1..$size).roll }, 1..$size;

}

sub loop-length(%mapping) {

   my %seen;
   my $current = 1;
   my $steps = 0;
   while !%seen{$current}++ {
       $current = %mapping{$current};
       $steps++;
   }
   return $steps;

}

sub analytical($N) {

   sub postfix:<!>($n) { [*] 1..$n }
   return [+] (1..$N).map: -> $k { $N! * $k * $k / $N ** ($k + 1) / ($N - $k)! };

}</lang>

Example:

$ perl6 loop-lengths
 N    average    analytical    (error)
===  =========  ============  =========
  1     1.0000        1.0000  (  0.00%)
  2     1.4992        1.5000  (  0.05%)
  3     1.8784        1.8889  (  0.56%)
  4     2.2316        2.2188  (  0.58%)
  5     2.4982        2.5104  (  0.49%)
  6     2.7897        2.7747  (  0.54%)
  7     3.0153        3.0181  (  0.09%)
  8     3.2429        3.2450  (  0.07%)
  9     3.4536        3.4583  (  0.14%)
 10     3.6649        3.6602  (  0.13%)
 11     3.8091        3.8524  (  1.12%)
 12     3.9986        4.0361  (  0.93%)
 13     4.2074        4.2123  (  0.12%)
 14     4.3711        4.3820  (  0.25%)
 15     4.5275        4.5458  (  0.40%)
 16     4.6755        4.7043  (  0.61%)
 17     4.8877        4.8579  (  0.61%)
 18     4.9951        5.0071  (  0.24%)
 19     5.1312        5.1522  (  0.41%)
 20     5.2699        5.2936  (  0.45%)