Subfactorial
Subfactorial[n]
gives the number of permutations of n objects that leave no object fixed.
Details

- Mathematical function, suitable for both symbolic and numerical manipulation.
- For noninteger n, the numerical value of Subfactorial[n] is given by Gamma[n+1,-1]/E.
- Subfactorial can be evaluated to arbitrary numerical precision.
- A permutation in which no object appears in its natural place is called a derangement.
- Subfactorial automatically threads over lists.
- Subfactorial[0] gives 1.
- Subfactorial can be used with CenteredInterval objects. »
Examples
open allclose allBasic Examples (5)
Scope (24)
Numerical Evaluation (5)
The precision of the output tracks the precision of the input:
Evaluate efficiently at high precision:
Subfactorial can be used with CenteredInterval objects:
Specific Values (5)
Values of Subfactorial at fixed points:
Find a value of for which the real part of Subfactorial[x] is equal to 5:
Visualization (2)
Plot the absolute value of Subfactorial:
Plot the real part of Subfactorial[x+ y]:
Plot the imaginary part of Subfactorial[x+ y]:
Function Properties (7)
Real domain of Subfactorial:
Subfactorial automatically threads over lists:
Subfactorial is not an analytic function on :
In fact, it is singular and discontinuous everywhere on the reals:
However, it is analytic in the complex plane:
The absolute value of Subfactorial is not injective:
The absolute value of Subfactorial is not surjective:
Subfactorial is neither non-negative nor non-positive:
Subfactorial is neither convex nor concave:
Differentiation (2)
Applications (1)
Properties & Relations (5)
Subfactorial[n] is given by :
Recurrence relations satisfied by Subfactorial:
Subfactorial can be represented as a DifferenceRoot:
FindSequenceFunction can recognize the Subfactorial sequence:
The exponential generating function for Subfactorial:
Text
Wolfram Research (2007), Subfactorial, Wolfram Language function, https://reference.wolfram.com/language/ref/Subfactorial.html.
CMS
Wolfram Language. 2007. "Subfactorial." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/Subfactorial.html.
APA
Wolfram Language. (2007). Subfactorial. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Subfactorial.html