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

loop-lengths.pl

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)! };
}

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%)