A random permutation is a random ordering of a set of objects, that is, a permutation-valued random variable. The use of random permutations is often fundamental to fields that use randomized algorithms such as coding theory, cryptography, and simulation. A good example of a random permutation is the shuffling of a deck of cards: this is ideally a random permutation of the 52 cards.
One method of generating a random permutation of a set of size n uniformly at random (i.e., each of the n! permutations is equally likely to appear) is to generate a sequence by taking a random number between 1 and n sequentially, ensuring that there is no repetition, and interpreting this sequence (x1, ..., xn) as the permutation
\begin{pmatrix} 1&2&3& … &n\\ x1&x2&x3& … &xn\\ \end{pmatrix},
shown here in two-line notation.
This brute-force method will require occasional retries whenever the random number picked is a repeat of a number already selected. This can be avoided if, on the ith step (when x1, ..., xi - 1 have already been chosen), one chooses a number j at random between 1 and n - i + 1 and sets xi equal to the jth largest of the unchosen numbers.
A simple algorithm to generate a permutation of n items uniformly at random without retries, known as the Fisher–Yates shuffle, is to start with any permutation (for example, the identity permutation), and then go through the positions 0 through n − 2 (we use a convention where the first element has index 0, and the last element has index n − 1), and for each position i swap the element currently there with a randomly chosen element from positions i through n − 1 (the end), inclusive. It's easy to verify that any permutation of n elements will be produced by this algorithm with probability exactly 1/n!, thus yielding a uniform distribution over all such permutations.
void initialize_and_permute(unsigned permutation[], unsigned n)
Note that if the uniform function is implemented simply as random % (m) then a bias in the results is introduced if the number of return values of random is not a multiple of m, but this becomes insignificant if the number of return values of random is orders of magnitude greater than m.
See main article: Rencontres numbers. The probability distribution of the number of fixed points in a uniformly distributed random permutation approaches a Poisson distribution with expected value 1 as n grows. In particular, it is an elegant application of the inclusion–exclusion principle to show that the probability that there are no fixed points approaches 1/e. When n is big enough, the probability distribution of fixed points is almost the Poisson distribution with expected value 1.[1] The first n moments of this distribution are exactly those of the Poisson distribution.
As with all random processes, the quality of the resulting distribution of an implementation of a randomized algorithm such as the Knuth shuffle (i.e., how close it is to the desired uniform distribution) depends on the quality of the underlying source of randomness, such as a pseudorandom number generator. There are many possible randomness tests for random permutations, such as some of the Diehard tests. A typical example of such a test is to take some permutation statistic for which the distribution is known and test whether the distribution of this statistic on a set of randomly generated permutations closely approximates the true distribution.