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 
distinct objects in which no object appears in its natural
place (i.e., "derangements"), where 
is a nonnegative
integer and 
.
The subfactorials are also called rencontres numbers. The same name is used more generally for the partial derangement counts
of permutations
with exactly 
fixed points. In this notation, 
.
The term "subfactorial" was introduced by Whitworth (1867 or 1878; Cajori 1993, p. 77). Euler (1811) 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 of the real part and imaginary part of the subfactorial generalized to any real
argument is illustrated above, with the usual integer-valued subfactorial corresponding
to nonnegative integer 
.
The subfactorials satisfy the recurrence relations, valid for 
and 
,
respectively,
 |  |  |
(5)
|
 |  | ![(n-1)[!(n-2)+!(n-1)].](/images/equations/Subfactorial/Inline49.svg) |
(6)
|
The subfactorial can be considered a special case of a restricted rooks problem.
The subfactorial has the formal generating function
 |  |  |
(7)
|
 |  |  |
(8)
|
This formal power series has radius of convergence zero. The function
 |
(9)
|
where 
is the exponential integral, has the asymptotic
series 
as 
. The exponential
generating function, convergent for 
, is
 |  |  |
(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 counting derangements.
For 
,
another equation is given by
![!n=[(n!)/e],](/images/equations/Subfactorial/NumberedEquation2.svg) |
(13)
|
where 
is the usual factorial and ![[x]](/images/equations/Subfactorial/Inline77.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)
|
For 
,
the error in the approximation 
has the integral representation
(Hassani 2020, p. 2)
 |
(17)
|
For each fixed positive integer 
, the asymptotic expansion
as 
is (Hassani 2020, p. 3)
 |
(18)
|
where 
is a Bell number and 
denotes Landau symbols.
The coefficients 
for 
,
2, ... are 1, 2, 5, 15, 52, ... (OEIS A000110).
A continued fraction for 
is given by
 |
(19)
|
The numbers of decimal digits in 
for 
, 2, ... 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
 |
(20)
|
(Madachy 1979).
The formula 
,
using the principal branch of the incomplete
gamma function, interpolates the subfactorials in the complex
plane, as illustrated above.
