Superfactorials - A summary of Advanced Methods of Factorials
Herein, I present a summary of some advanced methods of factorials. These are some interesting methods of expressing factorials, which are not commonly known. Still, they can be useful in, rather more specific, cases.
0. Standard factorial
Suppose $n!$ for some $n \in \mathbb{N}$, then:
\[n!=n \times (n-1) \times (n-2) \dots 2 \times 1\]Which can be expressed more concisely as the following:
\[n!=\prod_{k=1}^{n} n\]Double factorial
The parity of $n$ affects the series. Suppose $n!!$ for some $n \in \mathbb{N}$, then:
\[n!!= \begin{cases} n \times (n-2) \dots 4 \times 2 \text{ | even} \\ n \times (n-2) \dots 3 \times 1 \text{ | odd} \\ \end{cases}\]Now, written more elegantly as the following expression:
\[n!!= \begin{cases} \prod_{k=1}^{\cfrac{n}{2}} (2 \times k) \text{ | even} \\ \prod_{k=1}^{\cfrac{n+1}{2}} (2 \times k - 1) \text{ | odd} \\ \end{cases}\]Subfactorial
A subfactorial of $n$, denoted as $!n$, expresses the number of de-arrangements of a set with $n$ terms, where $n \in \mathbb{N}$. That is, the number of ways in which $n$ elements can be arranged in a sequence, such that no element is in its original position. The following expression thereby holds:
\[!n = n! \times \big( 1-\frac{1}{1!}+\frac{1}{2!} \times \dots \times (-1)^{n} \times \frac{1}{n!}\big)\]Primorial
Suppose a primorial of $n$, denoted as n#, which express the series of prime numbers $p$, such that $p \leq n$. That is:
\[n\# = \prod_{p \leq n} p\]Super factorial (the Sloane definition)
Suppose a super factorial of $n$, denoted as $sf(n)$, which expresses the series of all $k!$, such that $k \leq n$, where $n, k \in \mathbb{N}$. Hence:
\[sf(n)=n! \times (n-1)! \times (n-2)! \times \dots \times 2! \times 1!\]Which is similarly expressed in the following form:
\[sf(n)=\prod_{k=1}^{n} k!\]Exponential factorial (also know as the hyperpower)
As the name suggests, the exponential factorial of $n$, denoted as n$, expresses $n$ raised to the power of $(n-1)$ that is raised to the power of $(n-2)$ and so forth. This relation is also defined with a recurrent sequence, such that $a_{n}= n^{a_{n-1}} \land a_{0}= 1$.
For instance, the exponential factorial of $4$ is $4^{3^{2^{1}}}$.
Hyper factorial
A hyper factorial of $n$, denoted as $H(n)$, expresses the following series:
\[H(n)=n^n \times (n-1)^{n-1} \times (n-2)^{n-2} \times \dots \times 2^2 \times 1^1\]Or, expressed in a more elegant form:
\[H(n)=\prod_{k=0}^{n} (n-k)^{n-k}\]These have been some interesting advanced methods for describing alternative forms of factorials. Indeed, there is
a lot more to explore - this is just the tip of the iceberg (i.e., a short summary). Still, there’s much to delve into, and some further applications may be covered in the future. For now, I hope you’ve enjoyed this post, and I hope you’ve learned something new. In case of some inconsistencies, please, do not hesitate to leave an issue on the GitHub repository. Thank you for reading!