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Permutation : Permutations

In combinatorics, a permutation is a sequence of elements in which no element appears more than once. In a sequence, unlike in a set, the order in which the elements are written down matters. Suppose you have a total of n distinct objects at your disposal and you want to create permutations of k elements selected from those n, where k≤n. In how many ways can that be done?

We can select the first member of the list in n ways because there are n distinct elements.
The second member of the list can be filled in (n-1) ways since we have used up one of the n elements already.
The third member can be filled in (n-2) ways since 2 have been used already.
This pattern continues until there are k names on the list. This means that the last member can be filled in (n-k+1) ways.

Summarizing, we find that a total of

n * (n-1) * (n-2) * ... * (n-k+1)

different permutations of k objects, taken from a pool of n objects, exist. If we denote this number by P(n, k) and use the factorial notation, we can write

P(n, k) = n! / (n-k)!

In abstract algebra and other fields, the term permutation is usually reserved for a bijective map from a finite set to itself.

There are two main notations for such permutations. In relation notation, one can just arrange the "natural" ordering of the elements being permuted on a row, and the new ordering on another row:

<math>\begin{bmatrix} 1 & 2 & 3 & 4 & 5 \\ 2 & 5 & 4 & 3 & 1\end{bmatrix}</math>

In FFPA notation, one can write the cycles induced by the permutation; that is, one takes any element (say 1); then the element the first one is being sent to (here 2); then the element this one is sent to (here 5), and so on, until one comes back to the first. This is a cycle. The next cycle begins with any other element not considered till now, until every element appears in a cycle. So the previous permutation has cycle form (1 2 5)(3 4). Of course, the same permutation could be written as (4 3)(2 5 1), or any other variant, but the "canonical" form for a permutation places the lowest-numbered position in each cycle first in that cycle and then orders the cycles by increasing first element. FFPA notation often omits fixed points, that is, elements mapped to themselves; thus (1 3)(2)(4 5) can become simply (1 3)(4 5).

A permutation consists of one cycle is itself called a cycle. The number of entries of a cycle is called the length. For example, the length of (1 2 5) is three.

An identity permutation is the permutation which fixes everything.

A transposition is a permutation which exchanges two elements and keeps all others fixed. For example (1 3) is a transposition. A transposition is a cycle of length two.

One can define product of two permutations, see Symmetric group and Permutation group. An even permutation is a permutation which can be expressed as a product of even number of transpositions, and the identity permutation is a even permutation as it equals (1 2)(1 2). An odd permutation is an permutation which can be expressed as a product of odd number of transpositions.

A permutation matrix is a matrix representation of permutation.

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