Derangements Number

Calculates the number of derangements of \e n objects.

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Interface
Derangements Number
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Interface

C++

Excel

Derangements Number

intderangements_number( int n )

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

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

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

(2)

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

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

(3)

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

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

Parameters

n	the number of objects

Returns

the number of derangements of n objects

Authors

Lucian Bentea (August 2005)

Source Code

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