fixed_permutation — Sequences

Calculates the number of permutations of \e n objects with \e m fixed.

Interface

#include <codecogs/maths/combinatorics/sequences/fixed_permutation.h>

using namespace Maths::Combinatorics::Sequences;

int fixed_permutation(int n, int m)

A permutation of n objects with m fixed is a permutation in which exactly m of the objects retain their original positions. In more formal terms, consider the following permutation

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

(1)

then $\sigma$ has exactly m fixed objects if

$$\mathrm{card} \{ i \,|\,\sigma(i) = i, \quad i = \overline{1, n} \} = m$$

(2)

If $m = 0$, the permutation is a derangement, while if $m = n$, the permutation is the identity.

The number of permutations of n objects with m fixed is given by

$$F(n, m) = (n! / m!) \cdot (1 - 1/1! + 1/2! + \cdots + (-1)^{n-m}/(n-m)!)$$

(3)

$$F(n, m) = \left( \begin{array}{c} n \cr m \end{array} \right) \cdot D_{n - m}$$

(4)

where $D_{n - m}$ is the number of derangements of $n - m$ objects.

This function calculates the value of $F(n, m)$ based on the above formula.

References

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

Example 1

#include <codecogs/maths/combinatorics/sequences/fixed_permutation.h>
#include <iostream>
int main()
{
  std::cout << "The number of permutations of 10 elements with 5 fixed is: ";
  std::cout << Maths::Combinatorics::Sequences::fixed_permutation(10, 5) << std::endl;
  return 0;
}

Output

The number of permutations of 10 elements with 5 fixed is: 11088

Parameters

the size of the permutation

the number of fixed objects

Returns

the number of permutations of n items with m fixed

Interactive Calculator

Result –