*
*

The

*
th subfactorial (also called the derangement number; Goulden and Jackson 1983, p.48; Grayêu thích et al. 2003, p.1050) is the number of permutations of
*
objects in which no object appears in its natural place (i.e., "derangements").

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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
*
&
*
, so
*
. Similarly, the derangements of
*
are
*
,
*
,
*
,
*
,
*
,
*
,
*
,
*
, and
*
, so
*
.

Sums và formulas for

*
include

*
*
*

(1)
*
*
*

(2)
*
*
*

(3)
*
*
*

(4)

where

*
is a factorial,
*
is a binomial coefficient, và
*
is the incomplete gamma function.

Subfactorials are implemented in the bigbiglands.comLanguage as Subfactorial.

*

A plot the real and imaginary parts of the subfactorial generalized khổng lồ any real argument is illustrated above, with the usual integer-valued subfactorial corresponding khổng lồ nonnegative integer

*
.

The subfactorials are also called the rencontres numbers và satisfy the recurrencerelations

*
*
*

(5)
*
*
*
." />

(6)

The subfactorial can be considered a special case of a restricted rooksproblem.

The subfactorial has generating function

*
*
*

(7)
*
*
*

(8)
(9)

where

*
is the exponential integral, và exponential generating function

*
*
*

(10)
(11)
(12)

(OEIS A053557 và A053556).

Subfactorials are commonly denoted

*
,
*
(Gramê man et al. 2003, p.194),
*
(Dörrie 1965, p.19),
*
(Pemmaraju & Skiemãng cầu 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


(13)

where

*
is the usual factorial và
*
" /> is the nearest integer function. M.Hassani (pers. comilimet., Oct.28, 2004) gave sầu the forms


(14)

for

*
=1" /> 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 lớn the sum of subfactorials of its digits is


(18)

(Madachy 1979).

*
*

The subfactorial may be analytically continued lớn the complex plane, as illustrated above sầu.


SEE ALSO: Derangement, Factorial, Married Couples Problem, Rooks Problem, Superfactorial
REFERENCES:

Cajori, F. A History of Mathematical Notations, Vol.2. New York: Cosimo Classics, 2007.

Dörrie, H. §6 in 100 Great Problems of Elementary Mathematics: Their History & Solutions. New York: Dover, pp.19-21, 1965.

Euler, L. "Solution quaestionis curiosae ex doctrina combinationum." Mémoires Académie Sciences St. Pétersbourg 3, 57-64, 1809. Reprinted in Opera Omnia, Series Prima, Vol.7. Leipzig, Germany: Teubner, pp.435-440, 1915.

Goulden, I.P. and Jackson, D.M. CombinatorialEnumeration. New York: Wiley, 1983.

Grayêu thích, R.L.; Grötschel, M.; và Lovász, L. (Eds.). Handbookof Combinatorics, Vol.2. Cambridge, MA: MIT Press, 2003.

Xem thêm: Tiểu Sử Ca Sĩ Bùi Lê Mận : Tôi Muốn "Là Của Lạ" Của Chồng Làm Anh Chơi Vơi

Madachy, J.S. Madachy"sMathematical Recreations. New York: Dover, p.167, 1979.

Pemmaraju, S. và Skiemãng cầu, S. Computational Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Cambridge, England: Cambridge University Press, 2003.

Riordan, J. AnIntroduction khổng lồ Combinatorial Analysis. New York: Wiley, 1980.

Sloane, N.J.A. Sequences A000166/M1937, A053556, A053557, và A114485 in "The On-Line Encyclopedia of Integer Sequences."

Sloane, N.J.A. & Plouffe, S. Figure M1937 in TheEncyclopedia of Integer Sequences. San Diego: Academic Press, 1995.

Stanley, R.P. Enumerative Combinatorics, Vol.1. Cambridge, England: Cambridge University Press, p.67, 1997.

van Lint, J.H. và Wilson, R.M. ACourse in Combinatorics. New York: Cambridge University Press, 1992.

Wells, D. The Penguin Dictionary of Curious & Interesting Numbers. Middlesex, England: Penguin Books, p.27, 1986.

Whitworth, W.A. Choice and Chance, Two Chapters of Arithmetic, with an Appendix Containing the Algebraical Treatment of Permutations & Combinations Newly Set Forth. Cambridge, England: Deighton, Bell, 1867.

Whitworth, W.A. Messenger Math. 1878.


Referenced on bigbiglands.com|Alpha: Subfactorial
CITE THIS AS:

Weisstein, Eric W. "Subfactorial." Frombigbiglands.com--A bigbiglands.com Web Resource. https://bigbiglands.com/Subfactorial.html


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