# Factorial Numbers and Their Practical Applications
## 5 Conclusions  
This work's study of factorial numbers made it possible to display existing methods for obtaining factorial numbers and simplify them. Evaluating the noise immunity of factorial numbers allows them to be used for the efficient storage and transmission of information. Compressing factorial numbers by numbering them makes it possible to reduce their transmission time and storage capacity. Factorial arithmetic is noise-tolerant and therefore allows for performing arithmetic operations on factorial numbers more reliably.  
An important practical application of factorial numbers is the formation of permutations in ascending and descending order with the ability to perform computational operations on them. This allows permutations to be used in many practical applications, such as combinatorial optimization problems or constructing ciphers. In addition, permutations themselves have good noise immunity, and therefore, their practical use makes it possible to effectively detect and, in some cases, correct errors.  
A valuable property of factorial numbers is the possibility of enumerating them and, with it, enumerating permutations, both in the case of increasing the factorial numbers by one and adding any other integer, both positive and negative. This gives flexibility and versatility to the problem of enumerating permutations.  
Permutations that use factorial numbers for their generation provide good opportunities to hide useful information since they allow binary messages of almost unlimited length  
Figure 2: Graph of the probability of detected errors \(p\) as a function of the permutation length \(n\).  
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \(n\) & \(n!\) & \(n^{n}\) & \(n!/n^{n}\) & \(p\) & \(n\) & \(n!\) & \(n^{n}\) & \(n!/n^{n}\) & \(p\) \\ \hline \(1\) & \(1\) & \(1\) & \(0\) & \(0\) & \(5\) & \(120\) & \(3125\) & \(3005\) & \(0.96\) \\ \(2\) & \(2\) & \(4\) & \(2\) & \(0.5\) & \(6\) & \(720\) & \(46,656\) & \(45,936\) & \(0.99\) \\ \(3\) & \(6\) & \(27\) & \(21\) & \(0.78\) & \(7\) & \(5040\) & \(823,543\) & \(818,503\) & \(0.994\) \\ \(4\) & \(24\) & \(256\) & \(232\) & \(0.91\) & \(8\) & \(40,320\) & \(16,777,216\) & \(16,736,896\) & \(0.998\) \\ \hline \hline \end{tabular}
\end{table}
Table 5: Probability of detected errors \(p\).  
to be converted into them. Thus, the probabilities of letters and numbers in the alphabets of input messages are masked. In addition, their secrecy without additional time costs is accompanied by increased noise immunity.  
In the future, various restrictions may be introduced on permutations, which will require the introduction of additional restrictions on factorial numbers and, with them, the development of new methods for generating permutations with restrictions.  
Conceptualization, O.B. ([PERSON]), O.H. and S.M.; methodology, O.B. ([PERSON]) and O.H.; validation, O.B. ([PERSON]), S.M. and V.B; data curation, V.B.; writing--original draft preparation, O.B. ([PERSON]), O.H. and S.M.; writing--review and editing, O.B. ([PERSON]), S.M., O.H., O.B. ([PERSON]), A.N., T.S., J.P. and V.B.; visualization, O.H. and S.M.; supervision, [PERSON] ([PERSON]) and J.P.; project administration, J.P.; funding acquisition, J.P. All authors have read and agreed to the published version of the manuscript.  
This research was supported by the RRF project Latvian Quantum Technologies Initiative Nr. 2.3.1.1.1.0/1/22/I/CFLA/001.  
Not applicable.  
The original contributions presented in the study are included in the article, further inquiries can be directed to the corresponding author.  
The authors declare no conflict of interest.