Derangements_Number — Sequences

Calculates the number of derangements of \e n objects.

Interface

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

using namespace Maths::Combinatorics::Sequences;

int derangements_number(int n)

A derangement of n objects is a permutation with no fixed points. If we symbolize the permutation by $\sigma$, then for a derangment, $\sigma(i)$ is never equal to $i$.

The number of derangements of n objects is given by the following formula

$$d(n) = n! \cdot (1 - 1/1! + 1/2! - 1/3! + \ldots (-1)^n/n!)$$

(1)

Based on the inclusion/exclusion law we are allowed to write

$$d(n) = \lceil n! / e \rceil$$

(2)

where $\lceil x \rceil$ is the ceiling function.

Parameters

the number of objects

Returns

the number of derangements of n objects

Example:

#include <codecogs/maths/combinatorics/sequences/derangements_number.h>
#include <iostream>
int main()
{
  for (int i = 0; i < 10; i++)
    std::cout << i << " " << Maths::Combinatorics::Sequences::derangements_number(i) << std::endl;
  return 0;
}

Output:

References

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

Interactive Calculator

Result –