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The number of (complete)
derangements, or
number of permutations with no rencontre,
[1] of
distinct objects (i.e. the number of
permutations of
distinct objects with no fixed point) is given by the
subfactorial of
. The
derangement numbers are given by the
subfactorial numbers.
Formula
-
!n := n! 1 − + − + ⋯ + (−1) n = n! . |
Subfactorial numbers
A000166 Subfactorial (
rencontres numbers),
[2] or
derangements: number of permutations of
elements with no fixed point.
-
{1, 0, 1, 2, 9, 44, 265, 1854, 14833, 133496, 1334961, 14684570, 176214841, 2290792932, 32071101049, 481066515734, 7697064251745, 130850092279664, 2355301661033953, ...}
The last digit (base
10) of
seems to follow the pattern (of length
10)
-
{1, 0, 1, 2, 9, 4, 5, 4, 3, 6}
Example
You have 6 balls in 6 different colors, and for every ball you have a box of the same color. How many derangements do you have, if no ball is in a box of the same color?
-
!6 = 6! ⋅ 1 − + − + − + = 265. |
Comparison of derangement, permutation and arrangement numbers
Comparison of derangement, permutation and arrangement numbers
|
Number of derangements
|
Number of permutations
|
Number of arrangements
|
|
A000166
|
A000142
|
A000522
|
0 |
1 |
1 |
1
|
1 |
0 |
1 |
2
|
2 |
1 |
2 |
5
|
3 |
2 |
6 |
16
|
4 |
9 |
24 |
65
|
5 |
44 |
120 |
326
|
6 |
265 |
720 |
1957
|
7 |
1854 |
5040 |
13700
|
8 |
14833 |
40320 |
109601
|
9 |
133496 |
362880 |
986410
|
10 |
1334961 |
3628800 |
9864101
|
11 |
14684570 |
39916800 |
108505112
|
12 |
176214841 |
479001600 |
1302061345
|
13 |
2290792932 |
6227020800 |
16926797486
|
14 |
32071101049 |
87178291200 |
236975164805
|
15 |
481066515734 |
1307674368000 |
3554627472076
|
16 |
7697064251745 |
20922789888000 |
56874039553217
|
17 |
130850092279664 |
355687428096000 |
966858672404690
|
18 |
2355301661033953 |
6402373705728000 |
17403456103284421
|
19 |
44750731559645106 |
121645100408832000 |
330665665962404000
|
20 |
895014631192902121 |
2432902008176640000 |
6613313319248080001
|
|
|
|
|
|
|
A001044
|
A?????? (Add to OEIS?.) [3]
|
A?????? (Add to OEIS?.) [4]
|
0 |
1 |
1 |
0
|
1 |
1 |
0 |
1
|
2 |
4 |
5 |
− 1
|
3 |
36 |
32 |
4
|
4 |
576 |
585 |
− 9
|
5 |
14400 |
14344 |
56
|
6 |
518400 |
518605 |
− 205
|
7 |
25401600 |
25399800 |
1800
|
8 |
1625702400 |
|
|
9 |
131681894400 |
|
|
10 |
13168189440000 |
|
|
11 |
1593350922240000 |
|
|
12 |
229442532802560000 |
|
|
13 |
38775788043632640000 |
|
|
14 |
7600054456551997440000 |
|
|
15 |
1710012252724199424000000 |
|
|
16 |
437763136697395052544000000 |
|
|
17 |
126513546505547170185216000000 |
|
|
18 |
40990389067797283140009984000000 |
|
|
19 |
14797530453474819213543604224000000 |
|
|
20 |
5919012181389927685417441689600000000 |
|
|
|
Recurrences
-
!n = !(n − 1) ⋅ n + (−1) n, n ≥ 1. |
-
!n = (n − 1) ⋅ [!(n − 1) + !(n − 2)], n ≥ 2. |
Note that the factorial has a similar recurrence
-
n! = (n − 1) ⋅ [(n − 1)! + (n − 2)!], n ≥ 2. |
So, for
-
Other formulae
-
where
(Cf.
A001113) is
Euler’s number and
is the
incomplete gamma function.
A very good approximation is given by
-
If rounded, you get a perfect formula for
-
If you add 1 to the factorial, before dividing, you can truncate instead of rounding to get a perfect formula for
-
Comparison with approximations
|
|
|
|
|
|
0
|
1 |
0.3679 |
0 |
0.7358 |
0
|
1
|
0 |
0.3679 |
0 |
0.7358 |
0
|
2
|
1 |
0.7358 |
1 |
1.1036 |
1
|
3
|
2 |
2.2073 |
2 |
2.5752 |
2
|
4
|
9 |
8.8291 |
9 |
9.1970 |
9
|
5
|
44 |
44.1455 |
44 |
44.5134 |
44
|
6
|
265 |
264.8732 |
265 |
265.2411 |
265
|
7
|
1854 |
1854.1124 |
1854 |
1854.4803 |
1854
|
8
|
14833 |
14832.8991 |
14833 |
14833.2669 |
14833
|
9
|
133496 |
133496.0916 |
133496 |
133496.4595 |
133496
|
where
is the
round function and
is the
floor function.
There is a sequence (Cf.
A000255)
with the members
and the recursive rule:
-
an = n ⋅ an − 1 + (n − 1) ⋅ an − 2 . |
With this sequence you can calculate the subfactorial
-
|
1 |
0 |
1 |
2 |
9 |
44 |
265 |
1854 |
14833 |
133496 |
1334961
|
|
0
|
1
|
2
|
3
|
4
|
5
|
6
|
7
|
8
|
9
|
10
|
|
1 |
1 |
3 |
11 |
53 |
309 |
2119 |
16687 |
148329 |
1468457 |
16019531
|
Generating function
Ordinary generating function
-
G{!n}(x) ≡ !n x n = ⋅ Ei (1 + 1 / x) | e (1 + 1 / x) | , |
where
is the
exponential integral.
Exponential generating function
-
Asymptotic behaviour
The limit of the quotient of
factorial and
subfactorial converges to
(Cf.
A001113 and
Euler’s number)
-
Proof:
-
e x = , e = e 1 = , = e − 1 = , |
-
n → ∞limn → ∞ = n → ∞limn → ∞ = . |
The limit of the quotient of the
number of arrangements of any subset of
distinct objects over the
number of derangements of
distinct objects converges to
(Cf.
A072334)
-
The geometric mean of the number of arrangements and the number of derangements is asymptotic to the number of permutations
-
n → ∞limn → ∞ 2√ an dn = n! |
Permutations having at least one fixed point
The number of permutations
having at least one fixed point, thus not being (complete) derangements is given by
-
fn = n! − !n = n! − n! = n! , |
where the second summation gives the
empty sum (defined as the
additive identity, i.e.
0) for
.
Sequences
A000166 Subfactorial numbers (or
rencontres numbers, or
derangements: number of permutations of
elements with no fixed points)
-
{1, 0, 1, 2, 9, 44, 265, 1854, 14833, 133496, 1334961, 14684570, 176214841, 2290792932, 32071101049, 481066515734, 7697064251745, 130850092279664, 2355301661033953, ...}
A000255 a (n) = !(n + 1) + !n = = n a (n − 1) + (n − 1) a (n − 2), a (0) = 1, a (1) = 1. |
-
{1, 1, 3, 11, 53, 309, 2119, 16687, 148329, 1468457, 16019531, 190899411, 2467007773, 34361893981, 513137616783, 8178130767479, 138547156531409, 2486151753313617, ...}
A002467 Number
of permutations of
having a fixed point.
-
{0, 1, 1, 4, 15, 76, 455, 3186, 25487, 229384, 2293839, 25232230, 302786759, 3936227868, 55107190151, 826607852266, 13225725636255, 224837335816336, 4047072044694047, ...}
See also
Notes
- ↑ “Rencontre” being a french word meaning encounter.
- ↑ Should be called “rencontre-free” [permutation] numbers, which is french for “encounter-free” [permutation] numbers. Number of permutations with 0 rencontre (encounter), i.e. number of permutations with 0 fixed point.
- ↑ Add sequence to OEIS?
- ↑ Add sequence to OEIS?
References
- Mehdi Hassani, Derangements and Applications, Journal of Integer Sequences, 6(1), Article 03.1.2, 2003.
- Weisstein, Eric W., Derangement, from MathWorld—A Wolfram Web Resource., as of 2010-12-19.
- Weisstein, Eric W., Subfactorial, from MathWorld—A Wolfram Web Resource., as of 2010-12-19.