Editing Factorial number system (section)

{{short description|Numeral system in combinatorics}}
{{numeral systems}}
{{Refimprove|date=March 2021}}
In [[combinatorics]], the '''factorial number system''' (also known as '''factoradic'''), is a [[mixed radix]] [[numeral system]] adapted to numbering [[permutation]]s.  It is also called '''factorial base''', although [[factorial]]s do not function as [[radix|base]], but as [[place value]] of digits. By converting a number less than ''n''! to factorial representation, one obtains a [[sequence]] of ''n'' digits that can be converted to a permutation of ''n'' elements in a straightforward way, either using them as [[Lehmer code]] or as [[inversion (discrete mathematics)|inversion]] table<ref>{{citation
| last=Knuth | first = D. E. | author-link = Donald Ervin Knuth
| title = [[The Art of Computer Programming]]
| contribution = Volume 3: Sorting and Searching | publisher = Addison-Wesley | year = 1973
| isbn = 0-201-89685-0
| pages = 12}}</ref> representation; in the
former case the resulting map from [[integer]]s to permutations of ''n'' elements lists them in [[lexicographical order]]. General mixed radix systems were studied by [[Georg Cantor]].<ref>{{citation
 | last = Cantor | first = G. | author-link= Georg Cantor
 | title = Zeitschrift für Mathematik und Physik | volume = 14 | year =1869}}.</ref>

The term "factorial number system" is used by [[Donald Knuth|Knuth]],<ref>{{citation
 | last = Knuth | first = D. E. | author-link = Donald Ervin Knuth
 | contribution = Volume 2: Seminumerical Algorithms
 | edition = 3rd
 | isbn = 0-201-89684-2
 | pages = 192
 | publisher = Addison-Wesley
 | title = The Art of Computer Programming
 | year = 1997}}.</ref>
while the French equivalent "numération factorielle" was first used in 1888.<ref>{{citation
 | last = Laisant | first = Charles-Ange | author-link = Charles-Ange Laisant
 | journal = Bulletin de la Société Mathématique de France
 | language = French
 | pages = 176–183
 | title = Sur la numération factorielle, application aux permutations
 | url = http://www.numdam.org/item?id=BSMF_1888__16__176_0
 | volume = 16
 | year = 1888}}.</ref> The term "factoradic", which is a [[portmanteau]] of factorial and mixed radix, appears to be of more recent date.<ref>The term "factoradic" is apparently introduced in {{citation
 | last = McCaffrey | first = James | author-link = James D. McCaffrey
 | publisher = Microsoft Developer Network
 | title = Using Permutations in .NET for Improved Systems Security
 | url = http://msdn2.microsoft.com/en-us/library/aa302371.aspx
 | year = 2003}}.</ref>