Inversion_Sequence — Permutations

Computes the inversion sequence of a permutation.

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Interface

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

using namespace Maths::Combinatorics::Permutations;

std::vector<int> inversion_sequence(int n, int *p)

For a given permutation

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

(1)

the inversion sequence $( \alpha )$ is defined as

$$\alpha(i) = \mathrm{card} \{ j < i | \sigma(i) < \sigma(j) \}, \quad \forall i = \overline{1, n}$$

(2)

with $\alpha(1) = 0$.

This function calculates the inversion sequence of a permutation based on the above formula and returns its value as a C++ vector object.

Parameters

the size of the permutation

the actual permutation given as an array

Returns

the inversion sequence of the given permutation

Example:

#include <codecogs/maths/combinatorics/permutations/inversion_sequence.h>
#include <iostream>
int main()
{
  int sigma[5] = {3, 5, 1, 4, 2};
  std::vector<int> result = Maths::Combinatorics::Permutations::inversion_sequence(5, sigma);
  std::cout << "The inversion sequence of the Sigma permutation: ";
  std::cout << std::endl;
  for (int i = 0; i < 5; i++)
    std::cout << result[i] << "  ";
  std::cout << std::endl;
  return 0;
}

Output:

The inversion sequence of the Sigma permutation:
0  0  2  1  3

References

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