Permutation Lex

Progressively generates all the permutations of the given size, in lexicographic order.

Controller: CodeCogs

Contents

Interface
Class PermutationLex
Page Comments

Interface

C++

Class PermutationLex

Consider the permutations of size n

$\sigma = \left( \begin{array}{cccc} 1 & 2 & \ldots & n \cr \sigma(1) & \sigma(2) & \ldots & \sigma(n) \end{array} \right)$

(2)

and

$\tau = \left( \begin{array}{cccc} 1 & 2 & \ldots & n \cr \tau(1) & \tau(2) & \ldots & \tau(n) \end{array} \right)$

(3)

Now consider the next two numbers in the numerical base $\inline n + 1$ , corresponding to each permutation

$N_{\sigma} = \sum_{i = 0}^{n - 1} (n + 1)^i \sigma(n - i) \qquad N_{\tau} = \sum_{i = 0}^{n - 1} (n + 1)^i \tau(n - i)$

(4)

Then $\inline \tau$ is said to be the lexicographic succesor of $\inline \sigma$ if and only if $\inline N_{\tau} > N_{\sigma}$ .

This class progressively generates all the permutations of the given size, in lexicographic order, starting with the identical permutation.

Example:

#include <codecogs/maths/combinatorics/permutations/permutationlex.h>
#include <iostream>
int main()
{
  Maths::Combinatorics::Permutations::PermutationLex P(7);
  std::cout << "The first 5 lexicographic permutations of 7 elements:";
  std::cout << std::endl;
  for (int i = 0; i < 5; i++)
  {
    std::vector<int> alpha = P.getNext();
    for (int j = 0; j < alpha.size(); j++)
      std::cout << alpha[j] << " ";
    std::cout << "\t rank = " << P.getRank();
    std::cout << std::endl;
  }
  return 0;
}

Output:

The first 5 lexicographic permutations of 7 elements:
1 2 3 4 5 6 7    rank = 1
1 2 3 4 5 7 6    rank = 2
1 2 3 4 6 5 7    rank = 3
1 2 3 4 6 7 5    rank = 4
1 2 3 4 7 5 6    rank = 5

References:

SUBSET, a C++ library of combinatorial routines, http://www.csit.fsu.edu/~burkardt/cpp_src/subset/subset.html

Authors

Lucian Bentea (August 2005)

Source Code

Source code is available when you buy a Commercial licence.

Not a member, then Register with CodeCogs. Already a Member, then Login.