The 
th subfactorial (also called the derangement
number; Goulden and Jackson 1983, p. 48; Graham et al. 2003, p. 1050)
is the number of permutations of 
objects in which no object appears in its natural place (i.e.,
"derangements").
The term "subfactorial "was introduced by Whitworth (1867 or 1878; Cajori 1993, p. 77). Euler (1809) calculated the first ten terms.
The first few values of 
for 
, 2, ... are 0, 1, 2, 9, 44, 265, 1854,
14833, ... (OEIS A000166). For example, the
only derangements of 
are 
and 
, so 
. Similarly, the derangements
of 
are 
, 
, 
, 
, 
, 
, 
, 
, and 
, so 
.
Sums and formulas for 
include
 |  |  |
(1)
|
 |  |  |
(2)
|
 |  |  |
(3)
|
 |  |  |
(4)
|
where 
is a factorial, 
is a binomial coefficient, and 
is the incomplete
gamma function.
Subfactorials are implemented in the Wolfram Language as Subfactorial[n].
A plot the real and imaginary parts of the subfactorial generalized to any real argument is illustrated above, with the usual integer-valued subfactorial corresponding to
nonnegative integer 
.
The subfactorials are also called the rencontres numbers and satisfy the recurrence relations
 |  |  |
(5)
|
 |  | ![(n-1)[!(n-2)+!(n-1)].](/images/equations/Subfactorial/Inline42.svg) |
(6)
|
The subfactorial can be considered a special case of a restricted rooks problem.
The subfactorial has generating function
 |  |  |
(7)
|
 |  |  |
(8)
|
 |  |  |
(9)
|
where 
is the exponential integral, and exponential
generating function
 |  |  |
(10)
|
 |  |  |
(11)
|
 |  |  |
(12)
|
Subfactorials are commonly denoted 
, 
(Graham et al. 2003, p. 194), 
(Dörrie 1965, p. 19), 
(Pemmaraju and Skiena 2003, p. 106), 
(Goulden and Jackson 1983, p. 48; van Lint and Wilson
1992, p. 90), or 
(Riordan 1980, p. 59; Stanley 1997, p. 489), the latter being especially
used when viewing them as derangements.
Another equation is given by
![!n=[(n!)/e],](/images/equations/Subfactorial/NumberedEquation1.svg) |
(13)
|
where 
is the usual factorial and ![[x]](/images/equations/Subfactorial/Inline69.svg)
is the nearest integer
function. M. Hassani (pers. comm., Oct. 28, 2004) gave the forms
 |
(14)
|
for 
and
 |
(15)
|
for 
,
where 
is the floor function.
An integral for 
is given by
 |
(16)
|
A continued fraction for 
is given by
 |
(17)
|
The numbers of decimal digits in 
for 
, 1, ... are 7, 158, 2568, 35660, 456574, 5565709, 65657059,
... (OEIS A114485).
The only prime subfactorial is 
.
The only number equal to the sum of subfactorials of its digits is
 |
(18)
|
(Madachy 1979).
The subfactorial may be analytically continued to the complex plane, as illustrated above.
