Let f be a uniformly-randomly chosen mapping from the numbers 1..N to the numbers 1..N (note: not necessarily a permutation of 1..N; the mapping could produce a number in more than one way or not at all). 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.



Task

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%)


(ns cyclelengths
  (:gen-class))
 
(defn factorial [n]
  " n! "
  (apply *' (range 1 (inc n))))             ; Use *' (vs. *) to allow arbitrary length arithmetic
 
(defn pow [n i]
  " n^i"
  (apply *' (repeat i n)))
 
(defn analytical [n]
  " Analytical Computation "
  (->>(range 1 (inc n))
      (map #(/ (factorial n) (pow n %) (factorial (- n %)))) ;calc n %))
      (reduce + 0)))
 
;; Number of random times to test each n
(def TIMES 1000000)
 
(defn single-test-cycle-length [n]
  " Single random test of cycle length "
  (loop [count 0
         bits 0
         x 1]
    (if (zero? (bit-and x bits))
      (recur (inc count) (bit-or bits x) (bit-shift-left 1 (rand-int n)))
        count)))
 
(defn avg-cycle-length [n times]
  " Average results of single tests of cycle lengths "
  (/
   (reduce +
           (for [i (range times)]
             (single-test-cycle-length n)))
  times))
 
;; Show Results
(println "\tAvg\t\tExp\t\tDiff")
(doseq [q (range 1 21)
        :let [anal (double (analytical q))
              avg (double (avg-cycle-length q TIMES))
              diff (Math/abs (* 100 (- 1 (/ avg anal))))]]
  (println (format "%3d\t%.4f\t%.4f\t%.2f%%" q avg anal diff)))
 