Problem Statement

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.

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:

Let's start with the solution:

F ffactorial(n)
   V result = 1.0
   L(i) 2..n
      result *= i
   R result

V MAX_N = 20
V TIMES = 1000000

F analytical(n)
   R sum((1..n).map(i -> ffactorial(@n) / pow(Float(@n), Float(i)) / ffactorial(@n - i)))

F test(n, times)
   V count = 0
   L(i) 0 .< times
      V (x, bits) = (1, 0)
      L (bits [&] x) == 0
         count++
         bits [|]= x
         x = 1 << random:(n)
   R Float(count) / times

print(" n      avg     exp.    diff\n-------------------------------")
L(n) 1 .. MAX_N
   V avg = test(n, TIMES)
   V theory = analytical(n)
   V diff = (avg / theory - 1) * 100
   print(‘#2 #3.4 #3.4 #2.3%’.format(n, avg, theory, diff))