Definition:Subfactorial

Definition

Let $n \in \Z_{\ge 0}$ be a (strictly) positive integer.

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

\(\ds !n\)

\(:=\)

\(\ds n! \sum_{k \mathop = 0}^n \frac {\paren {-1}^k} {k!}\)

\(\ds \)

\(=\)

\(\ds n! \paren {1 - \dfrac 1 {1!} + \dfrac 1 {2!} - \dfrac 1 {3!} + \cdots + \dfrac {\paren {-1}^n} {n!} }\)

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 of $n$ can also be denoted as:

$D_n$

$d_n$

$\map d n$

where the $D$ or $d$ derives from derangements.

Also see

• Closed Form for Number of Derangements on Finite Set
• Definition:Derangement
• Hat-Check Problem

• Results about subfactorials can be found here.

Sources

• 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $9$
• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $9$
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): subfactorial
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): subfactorial

• Weisstein, Eric W. "Subfactorial." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Subfactorial.html