Counting And Listing All Permutations

The applet below displays all the permutations of the set N_n = {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.

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.

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--; } }

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

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

Permutations

Transpositions

Groups of Permutations

Sliders

Puzzles on graphs