Definition:Subfactorial

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Definition

Let $n \in \Z_{\ge0}$ be a (strictly) positive integer.

The subfactorial of $n$ is defined and denoted as:

\(\displaystyle !n\) \(:=\) \(\displaystyle n! \sum_{k \mathop = 0}^n \frac {\left({-1}\right)^k} {k!}\)
\(\displaystyle \) \(=\) \(\displaystyle n! \left({1 - \dfrac 1 {1!} + \dfrac 1 {2!} - \dfrac 1 {3!} + \cdots + \dfrac {\left({-1}\right)^n} {n!} }\right)\)


It arises as the number of derangements of $n$ distinct objects.


Sequence of Subfactorials

The sequence of subfactorials begins:

$\begin{array}{r|r} n & !n \\ \hline 0 & 1 \\ 1 & 0 \\ 2 & 1 \\ 3 & 2 \\ 4 & 9 \\ 5 & 44 \\ 6 & 265 \\ 7 & 1 \, 854 \\ 8 & 14 \, 833 \\ \end{array}$


Also denoted as

The subfactorial can also be denoted as $D_n$, where the $D$ derives from derangements.


Also see


Sources