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Permutations

Introduction

A permutation is a particular ordering of a set of objects. Formally, a permutation of a finite set S is an ordered sequence all the elements with each one appearing once.

How many different ways can you order {1,2,3}?

Answer -- 6:

  • {1,2,3}
  • {1,3,2}
  • {2,1,3}
  • {2,3,1}
  • {3,1,2}
  • {3,2,1}

For a set of length n, there are n! permutations. So 3! = 6.

Why is this?

  • For each number n,
  • you can keep n in place,
  • and permute the remaining n - 1
  • then keep (n-1) in place and permute (n-2)
  • keep going...
  • n = 2 has 2 permutations
  • n = 1 has 1 permutations

(n)(n-1)(n-2)...(2)(1) is the definition of factorial of n.

We can also talk about the k-permutation of a set: P(n,r) , or nPr = n!/(n-r)!

Generating Permutations

Since O(permutations) is n! we have something that won't run in our lifetime for any decent size n...

Many algorithms seek the best way to order a set of objects:

  • traveling salesman problem -- the least-cord order to visit n cities
  • bandwidth minimization -- order vertices of a graph on a line so as to minimize the length of the longest edge
  • determining graph isomorphism -- oder the vertices of one graph so that it is identical to another

The lexiographic order of the permutations of {1,2,3} are:

  • {1,2,3}
  • {1,3,2}
  • {2,1,3}
  • {2,3,1}
  • {3,1,2}
  • {3,2,1}

REFERENCES

Permutations. Sal Khan. Kahn Academy.

The Algorithm Design Manual. Second Addition. Steven Skiena. 2008.