Tool to compute subfactorial. Subfactorial !n is the number of derangements of n objects, i.e. the number of permutations of n objects in order that no object stands in its original position.
Subfactorial - dCode
Tag(s) : Arithmetics
dCode is free and its tools are a valuable help in games, maths, geocaching, puzzles and problems to solve every day!
A suggestion ? a feedback ? a bug ? an idea ? Write to dCode!
Subfactorial
SubFactorial Calculator !N
Answers to Questions (FAQ)
How to calculate a subfactorial?
SubFactorial $ n $ is calculated using this formula: $$ !n = n! \sum_{k=0}^n \frac {(-1)^k}{k!} $$
Example: $$ \begin{align} !4 &= 4! ( \frac{(-1)^0}{0!} + \frac{(-1)^1}{1!} + \frac{(-1)^2}{2!} + \frac{(-1)^3}{3!} + \frac{(-1)^4}{4!} ) \\ &= 4! \times ( 1/1 - 1/1 + 1/2 - 1/6 + 1/24 ) \\ &= 24 \times 9/24 \\ &= 9 \end{align} $$
This formula is also used: $$ !n = \left [ \frac {n!}{e} \right ] $$ where brackets [] stands for rounding to the closest integer.
Example: $ 4! / e \approx 24/2.718 \approx 8.829 \Rightarrow !4 = 9 $
And a recurrence relationship : $$ !n = n \times !(n-1) + (-1)^n $$
What are the first values of the subfactorial function?
The first values for the first natural numbers are:
| !1 = 0 |
| !2 = 1 |
| !3 = 2 |
| !4 = 9 |
| !5 = 44 |
| !6 = 265 |
| !7 = 1854 |
| !8 = 14833 |
| !9 = 133496 |
| !10 = 1334961 |
How to write a subfactorial?
The subfactorial as the factorial, uses the exclamation mark as symbol but it is written to the left of the number: $ !n $
What is the precedence of the operator subfactorial (order of operations)?
By convention, postfixed operators have priority (the calculation goes first) over prefixed, so factorial (postfixed) has priority over subfactorial (prefixed)
Example: $ !3! = !(3!) $
How to calculate derangements
Derangements (or Rencontres) are permutations without the one with fixed points (no item is in its original place). The number of derangements for $ n $ elements is subfactorial of $ n $: $ !n $.
Example: The $ !4 = 9 $ derangements of {1,2,3,4} are {2,1,4,3}, {2,3,4,1}, {2,4,1,3}, {3,1,4,2}, {3,4,1,2}, {3,4,2,1}, {4,1,2,3}, {4,3,1,2}, and {4,3,2,1}.
Source code
dCode retains ownership of the "Subfactorial" source code. Except explicit open source licence (indicated Creative Commons / free), the "Subfactorial" algorithm, the applet or snippet (converter, solver, encryption / decryption, encoding / decoding, ciphering / deciphering, breaker, translator), or the "Subfactorial" functions (calculate, convert, solve, decrypt / encrypt, decipher / cipher, decode / encode, translate) written in any informatic language (Python, Java, PHP, C#, Javascript, Matlab, etc.) and all data download, script, or API access for "Subfactorial" are not public, same for offline use on PC, mobile, tablet, iPhone or Android app!
Reminder : dCode is free to use.
Cite dCode
The copy-paste of the page "Subfactorial" or any of its results, is allowed (even for commercial purposes) as long as you credit dCode!
Exporting results as a .csv or .txt file is free by clicking on the export icon
Cite as source (bibliography):
Subfactorial on dCode.fr [online website], retrieved on 2024-12-30,
