PermutationLex

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

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Interface

#include <codecogs/maths/combinatorics/permutations/permutationlex.h>

using namespace Maths::Combinatorics::Permutations;

class PermutationLex

PermutationLex(int n) : m_n(n) constructor
std::vector<int>getNext()
int getRank()

Overview

Consider the permutations of size n

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

(1)

and

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

(2)

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

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

(3)

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

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

Members of PermutationLex

CONSTRUCTOR

Parameters

the size of the permutations to generate

CLASS METHOD

getNext

generates the next permutation in lexicographic order; if all have been considered, it starts again from the first

CLASS METHOD

getRank

returns the rank of the current permutation