falling factorial

Calculates the falling factorial with arguments \e x and \e n.

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Interface
Falling Factorial
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Interface

C++

Excel

Falling Factorial

doublefalling_factorial(	double	`x`
	int	`n`	)

The falling factorial has the following formula

$[x]_n = \prod_{i = 0}^{n - 1} (x - i) = x! / n!$

(2)

Note that the number of <em> injections </em> or 1-to-1 mappings from a set of n elements to a set of m elements is $\inline [m]_n$ . The number of permutations of n objects out of m is $\inline [m]_n$ . Moreover, the Stirling numbers of the first kind can be used to convert a falling factorial into a polynomial, as follows:

$[x]_n = \sum_{i = 0}^n S^i_n x^i$

(3)

Example:

#include <codecogs/maths/combinatorics/arithmetic/falling_factorial.h>
#include <iostream>
int main()
{
  std::cout << Maths::Combinatorics::Arithmetic::falling_factorial(4, 2) << 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

x	the first falling factorial argument
n	the second falling factorial argument

Returns

the falling factorial of the pair of values x and n

Authors

Lucian Bentea (August 2005)

Source Code

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