Sub-faculty

${\displaystyle n}$	Sub-faculty ${\displaystyle {!}n}$	Faculty ${\displaystyle n!}$
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 . ${\displaystyle n}$ ${\displaystyle {!}n}$ ${\displaystyle n!}$ ${\displaystyle n}$ ${\displaystyle e}$

definition

The Subfakultät a natural number is calculated using the faculty by ${\displaystyle n\geq 0}$

${\displaystyle {!}n=n!\sum _{k=0}^{n}{\frac {(-1)^{k}}{k!}}=n!\cdot \left(1-{\frac {1}{1!}}+{\frac {1}{2!}}-{\frac {1}{3!}}+\cdots +(-1)^{n}{\frac {1}{n!}}\right)}$

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 . ${\displaystyle {!}n}$ ${\displaystyle n}$ ${\displaystyle n!}$

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

${\displaystyle {!}6=6!\cdot \left(1-{\frac {1}{1!}}+{\frac {1}{2!}}-{\frac {1}{3!}}+{\frac {1}{4!}}-{\frac {1}{5!}}+{\frac {1}{6!}}\right)=265}$

Options.

Further representations

Rounding representations

Comparison of approximations of the sub-faculty

${\displaystyle n}$	${\displaystyle {\frac {n!}{e}}}$	${\displaystyle \left[{\frac {n!}{e}}\right]}$	${\displaystyle {\frac {n!\!+\!1}{e}}}$	${\displaystyle \left\lfloor {\frac {n!\!+\!1}{e}}\right\rfloor }$
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

${\displaystyle {!}n={\frac {\Gamma (n+1,-1)}{e}}}$

with Euler's number and the incomplete gamma function . A very good approximation is ${\displaystyle e}$ ${\displaystyle \Gamma }$

${\displaystyle {!}n\approx {\frac {n!}{e}}}$.

You even get the exact formula when rounded ${\displaystyle n\geq 1}$

${\displaystyle {!}n=\left[{\frac {n!}{e}}\right]}$,

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 : ${\displaystyle \left[x\right]}$ ${\displaystyle x}$ ${\displaystyle n\geq 1}$

${\displaystyle {!}n=\left\lfloor {\frac {n!+1}{e}}\right\rfloor }$.

Recursive representations

Recursive representation of the sub-faculty

${\displaystyle n}$	${\displaystyle {!}(n-1)}$	${\displaystyle n\cdot {!}(n-1)}$	${\displaystyle (-1)^{n}}$	${\displaystyle {!}n}$
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

${\displaystyle {!}n=n\cdot {!}(n-1)+(-1)^{n}}$

and

${\displaystyle {!}n=(n-1)\cdot ({!}(n-1)+{!}(n-2))}$

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 ). ${\displaystyle {!}(n-1)+{!}(n-2)}$ ${\displaystyle n}$

Integral representation

The following integral representation generalizes the sub-faculty around its domain of definition from natural to complex numbers:

${\displaystyle {!}z=\int _{0}^{\infty }\mathrm {e} ^{-t}(t-1)^{z}dt}$.

Here is with . ${\displaystyle z\in \mathbb {C} }$ ${\displaystyle \operatorname {Re} (z)>-1}$

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

${\displaystyle 148349={!}1+{!}4+{!}8+{!}3+{!}4+{!}9}$.

In other number systems this is u. a. the case with 9:

${\displaystyle 9=1\cdot 5+4={!}1+{!}4=0+9}$

In particular, 5 is the smallest base for which a number with this property exists.

Individual evidence

1. ^ Mehdi Hassani: Derangements and Applications . In: Journal of Integer Sequences . Vol. 6, Article 03.1.2, 2003.
2. ↑ Joseph S. Madachy: Madachy's Mathematical Recreations . Dover, New York NY 1979, ISBN 0-486-23762-1 , pp. 167 .

Web links

Wiktionary: Sub-faculty  - explanations of meanings, word origins, synonyms, translations

• Eric W. Weisstein : Sub-Faculty . In: MathWorld (English).
• Follow A000166 in OEIS