Subfactorial

The subfactorial or left factorial, written \(!n\), is the number of ways that n objects can be arranged where no object appears in its natural position (known as "derangements.")Subfactorial -- from Wolfram MathWorld There are many formulas for \(!n\):
\begin{eqnarray*}
!n &=& n! \displaystyle\sum^{n}_{i = 0} \frac{(-1)^i}{i!}\\
&=& \displaystyle\sum^{n}_{i = 0} i! (-1)^{n - i} \binom{n}{i}\\
&=& \displaystyle\frac{\Gamma(n + 1, -1)}{e}\\
&=& \left[\frac{n!}{e}\right] \text{ (only for $n > 0$)}
\end{eqnarray*}
In the last formula, [n] means the nearest integer to n. (It is a direct consequence of the first formula — the summation converges to \(1/e\).)
The first few values of !n for n = 0, 1, 2, 3, 4, 5, etc. are 1, 0, 1, 2, 9, 44, 265, 1,854, 14,833, ...
In base 10, only one number is equal to the sum of the subfactorials of its digits: 148,349 = !1 + !4 + !8 + !3 + !4 + !9 = 109 × 1,361.
== Pseudocode ==
 // Standard factorial function
 function factorial(z):
 result := 1
 for i from 1 to z:
 result := result * i
 return result
 
 // Subfactorial
 function subfactorial(z):
 return floor(factorial(z) / e + 0.5)
== Sources ==
== See also ==
Category:Functions
Category:Factorials