Counting And Listing All Permutations

The applet below displays all the permutations of the set Nn = {1, ..., n}, where n changes from 0 to 7. N0 is of course empty. There is no special reason for selecting 7 as the upper size of the allowed sets - the algorithms described below are quite general. The problem is that of speed. I did not have enough patience to wait for the 7! = 5040 permutations of N7 to get listed on my 100 MHz machine. (In the development environment, N8 is permuted faster than N7 in a browser. P.S. This has been written in 1997!) Thus beware.

In the applet, every permutation is also presented as a product of cycles. For n = 3, we have a list of 6 elements of the symmetric group S3. Observing permutations as products of cycles it becomes obvious that elements of S3 represent rigid motions of an equilateral triangle. Indeed, envisage such a triangle and mark its vertices with numbers 1,2,3. Then (1 2 3) denotes the rotation of the vertices wherewith 1 goes to 2, 2 to 3, and 3 to 1. (1 3 2) is a rotation in the opposite direction. (1 2)(3) is the symmetry around the median from vertex 3, and so on.


This applet requires Sun's Java VM 2 which your browser may perceive as a popup. Which it is not. If you want to see the applet work, visit Sun's website at https://www.java.com/en/download/index.jsp, download and install Java VM and enjoy the applet.


What if applet does not run?

The applet offers three algorithms that generate the list of all the permutations: Recursive, Lexicographic and an algorithm due to B. Heap. I'll describe each in turn. In all the algorithms, N denotes the number of items to be permuted.

  1. Recursive algorithm

    The recursive algorithm is short and mysterious. It's executed with a call visit(0). Global variable level is initialized to -1 whereas every entry of the array Value is initialized to 0.

      void visit(int k)
      {
        level = level+1; Value[k] = level;    // = is assignment
        if (level == N)  // == is comparison
          AddItem();     // to the list box
        else
          for (int i = 0; i < N; i++)
            if (Value[i] == 0)
              visit(i);
        
        level = level-1; Value[k] = 0;
      }
    
      void AddItem()
      {
      // Form a string from Value[0], Value[1], ... Value[N-1].
      // At this point, this array contains one complete permutation.
      // The string is added to the list box for display.
      // The function, as such, is inessential to the algorithms.
      }
    

    Try this algorithm by hand to make sure you undestand how it works.

  2. Lexicographic order and finding the next permutation

    Permutation f precedes a permutation g in the lexicographic (alphabetic) order iff for the minimum value of k such that f(k)≠ g(k), we have f(k) < g(k). Starting with the identical permutation f(i) = i for all i, the second algorithm generates sequentially permutaions in the lexicographic order. The algorithm is described in [Dijkstra, p. 71].

      void getNext()
      {
        int i = N - 1;
        while (Value[i-1] >= Value[i]) 
          i = i-1;
    
        int j = N;
        while (Value[j-1] <= Value[i-1]) 
          j = j-1;
      
        swap(i-1, j-1);    // swap values at positions (i-1) and (j-1)
    
        i++; j = N;
        while (i < j)
        {
          swap(i-1, j-1);
          i++;
          j--;
        }
      }
    
  3. B. Heap's algorithm

    Heap's short and elegant algorithm is implemented as a recursive method HeapPermute [Levitin, p. 179]. It is invoked with HeapPermute(N).

    void HeapPermute(int n)
    {
      if (n == 1) 
        AddItem();
      else {
        for (int i = 0; i < n; i++) 
          HeapPermute(n-1);
          if (n % 2 == 1)  // if n is odd
            swap(0, n-1);
          else            // if n is even
            swap(i, n-1);
    }
    

Reference

  1. A. Levitin, Introduction to The Design & Analysis of Algorithms, Addison Wesley, 2003
  2. E. W. Dijkstra, A Discipline of Programming, Prentice-Hall, 1997 [FACSIMILE] (Paperback)
  3. R. Sedgewick, Algorithms in C, Addison-Wesley, 3rd edition (August 31, 2001)

Permutations

  • Transpositions
  • Groups of Permutations
  • Sliders
  • Puzzles on graphs
  • Equation in Permutations

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