Sub-faculty
Sub-faculty | Faculty | |
---|---|---|
0 | 1 | 1 |
1 | 0 | 1 |
2 | 1 | 2 |
3 | 2 | 6th |
4th | 9 | 24 |
5 | 44 | 120 |
6th | 265 | 720 |
7th | 1,854 | 5,040 |
8th | 14,833 | 40,320 |
9 | 133,496 | 362.880 |
10 | 1,334,961 | 3,628,800 |
The sub-faculty is primarily a function that occurs in combinatorics . It indicates the number of fixed point-free permutations of a set with elements and is noted by. The sub-faculty is closely related to the faculty , which indicates the total number of permutations of a -element set. It is approximately equal to the quotient of the factorial and Euler's number .
definition
The Subfakultät a natural number is calculated using the faculty by
Are defined. The sub-faculty corresponds to the number of fixed point-free permutations (derangements) of a -element set, while the faculty indicates the number of all possible permutations .
example
Suppose you have six balls of different colors and a box in the matching color for each ball. You have to determine the number of possibilities to distribute the balls on the boxes so that each box contains exactly one different colored ball. There is exactly that
Options.
Further representations
Rounding representations
1 | 0.37 | 0 | 0.74 | 0 |
2 | 0.74 | 1 | 1.10 | 1 |
3 | 2.21 | 2 | 2.58 | 2 |
4th | 8.83 | 9 | 9.20 | 9 |
5 | 44.15 | 44 | 44.51 | 44 |
6th | 264.87 | 265 | 265.24 | 265 |
7th | 1,854.11 | 1,854 | 1,854.48 | 1,854 |
8th | 14,832.90 | 14,833 | 14,833.27 | 14,833 |
9 | 133,496.09 | 133,496 | 133,496.46 | 133,496 |
It applies
with Euler's number and the incomplete gamma function . A very good approximation is
- .
You even get the exact formula when rounded
- ,
where denotes the closest integer. If the number one is added in the last formula before the division, you save yourself the distinction between rounding up or down. Instead, you simply cut off the decimal point (see Gaussian brackets ) and you get :
- .
Recursive representations
1 | 1 | 1 | −1 | 0 |
2 | 0 | 0 | +1 | 1 |
3 | 1 | 3 | −1 | 2 |
4th | 2 | 8th | +1 | 9 |
5 | 9 | 45 | −1 | 44 |
6th | 44 | 264 | +1 | 265 |
7th | 265 | 1,855 | −1 | 1,854 |
8th | 1,854 | 14,832 | +1 | 14,833 |
9 | 14,833 | 133,497 | −1 | 133,496 |
The sub-faculty can also be determined using the two formulas
and
compute recursively. The term corresponds to the number of fixed-point-free permutations of a -element set for which an element is fixed (sequence A000255 in OEIS ).
Integral representation
The following integral representation generalizes the sub-faculty around its domain of definition from natural to complex numbers:
- .
Here is with .
Entertainment math
The only sub-facultative narcissistic number , i.e. the only number that is equal to the sum of its sub-faculty (decimal) digits, is
- .
In other number systems this is u. a. the case with 9:
In particular, 5 is the smallest base for which a number with this property exists.
Individual evidence
- ^ Mehdi Hassani: Derangements and Applications . In: Journal of Integer Sequences . Vol. 6, Article 03.1.2, 2003.
- ↑ Joseph S. Madachy: Madachy's Mathematical Recreations . Dover, New York NY 1979, ISBN 0-486-23762-1 , pp. 167 .
Web links
- Eric W. Weisstein : Sub-Faculty . In: MathWorld (English).
- Follow A000166 in OEIS