falling_factorial — Arithmetic

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

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Interface

#include <codecogs/maths/combinatorics/arithmetic/falling_factorial.h>

using namespace Maths::Combinatorics::Arithmetic;

double falling_factorial(double x, int n)

The falling factorial has the following formula

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

(1)

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 $[m]_n$. The number of permutations of n objects out of m is $[m]_n$. Moreover, the Stirling numbers of the first kind can be used to convert a falling factorial into a polynomial, as follows:

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

(2)

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

the first falling factorial argument

the second falling factorial argument

Returns

the falling factorial of the pair of values x and n

Interactive Calculator

Result –