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In combinatorics, the factorial number system, also called factoradic, is a mixed radix numeral system adapted to numbering permutations. It is also called factorial base, although factorials do not function as 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 table representation; in theformer case the resulting map from integers to permutations of n elements lists them in lexicographical order. General mixed radix systems were studied by Georg Cantor.The term "factorial number system" is used by Knuth,while the French equivalent "numération factorielle" was first used in 1888.

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rdfs:label	Fakultätsbasiertes Zahlensystem (de) Factorádico (es) Factorial number system (en) Faktoradik (in) 계승진법 (ko) 階乗進法 (ja) Faculteitssysteem (nl) Silniowy system pozycyjny (pl) 阶乘进制 (zh)
rdfs:comment	In der Kombinatorik wird das fakultätsbasierte Zahlensystem verwendet, um einen eindeutigen Index für Permutationen zu erzeugen. (de) En combinatoria, factorádico (del inglés 'factoradic') es un sistema numérico especialmente construido. El sistema factorádico proporciona un índice lexicográfico para permutaciones por lo que tiene potencial aplicación en la seguridad informática. La idea del sistema factorádico está muy ligada al . Un artículo de documenta el índice factorádico para permutaciones con ejemplos de código escrito en C#. El término 'factorádico' es una combinación en inglés de los términos factorial y raíz mixta ('factorial' and 'mixed radix'). (es) Faktoradik adalah sebuah sistem bilangan yang setiap posisi angka memiliki basis sesuai dengan faktorial dari posisinya. Sistem bilangan ini memungkinkan untuk membangkitkan permutasi dalam urutan leksikografik. Faktoradik memiliki bentuk deretan bilanganan...a4a3a2a1a0, dengan setiap bilangan ai bersifat: dan (in) 組合せ数学において、階乗進法とは順列を数え上げるのに適する、複数の底が混在した位取り記数法である。 (ja) Het faculteitssysteem of faculteitsstelsel is een bijzonder talstelsel. Het is een positiestelsel, maar niet op de gebruikelijke wijze. In het faculteitssysteem vertegenwoordigt een positie niet een macht van een grondtal, maar de faculteit van de positie. Met oplopende positie kunnen dus steeds meer "cijfers" gebruikt worden. De bijdrage van het cijfer c op positie is dus . Om eenduidigheid te garanderen, mag op positie maximaal het "cijfer" gebruikt worden. Het getal dcba in dit stelsel is dus het getal: , waarin maximaal 4, maximaal 3, maximaal 2 en 0 of 1 kan zijn. (nl) 수학에서 계승진법(階乘進法, 영어: factorial number system, factoradic system)은 자연수를 계승들의 합으로 표기하는 표기법이다. 이를 통해 순열들의 집합 위의 전순서를 쉽게 매길 수 있다. 학교 내신 문제에서 주로 나오는 사전식 배열 문제를 풀 때도 용이하다. (ko) Silniowy system pozycyjny – pozycyjny system liczbowy w którym mnożniki poszczególnych pozycji nie są definiowane przez potęgę pewnej liczby (podstawy), lecz silnię kolejnych liczb naturalnych (z zerem), a liczba cyfr używanych na -tej pozycji wynosi Przykład: Stąd zapis silniowy, np. liczby 4600, wygląda następująco: Ze względu na to, iż na pozycji zerowej jest zawsze zero, istnieje odmiana bez tej pozycji, co nie wpływa na wartości zapisywanych liczb. Zapis jest jednoznaczny, tzn. każdą liczbę naturalną można zapisać w tylko jeden sposób i każdy zapis oddaje dokładnie jedną wartość. (pl) 在组合数学中，阶乘进制又稱階乘數字系統是一種適用於編號排列的的进制數字系統。階乘本身不做為底數，而是做為进制的位數值。若將一個小於n!的數轉換成阶乘进制可以得到一個n位的序列，該序列可以轉換成n的直接排列方式，也可以用於或作為逆序对表；在前一種情況下，從整數到n排列的映射結果將n的排列按字典順序列出。康托尔研究了一般的混合底數係統。術語「階乘數字系統」（factorial number system）由高德纳使用。例如3:4:1:0:1:0!代表354413021100，其值為： = 3×5! + 4×4! + 1×3! + 0×2! + 1×1! + 0×0! = ((((3×5 + 4)×4 + 1)×3 + 0)×2 + 1)×1 + 0= 46310. (zh) In combinatorics, the factorial number system, also called factoradic, is a mixed radix numeral system adapted to numbering permutations. It is also called factorial base, although factorials do not function as 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 table representation; in theformer case the resulting map from integers to permutations of n elements lists them in lexicographical order. General mixed radix systems were studied by Georg Cantor.The term "factorial number system" is used by Knuth,while the French equivalent "numération factorielle" was first used in 1888. (en)
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dcterms:subject	Factorial and binomial topics Non-standard positional numeral systems Combinatorics
Wikipage page ID	1113108 (xsd:integer)
Wikipage revision ID	1114431492 (xsd:integer)
Link from a Wikipage to another Wikipage	Decimal Permutation Inversion (discrete mathematics) 0.999... Factorial and binomial topics Concatenation Mathematical induction Quotient Function (mathematics) Georg Cantor Gray code Mixed Radix Combinatorial number system Combinatorics Portmanteau Non-standard positional numeral systems Colexicographical order Combinatorics Numeral system Lehmer code Radix Mixed radix Donald Ervin Knuth Arabic numeral Integer Sequence Factorial Lexicographical order Permutation group Steinhaus–Johnson–Trotter algorithm Telescoping series Lexicographical Group direct product Place value
Link from a Wikipage to an external page	http://www.dmtcs.org/volumes/abstracts/pdfpapers/dm040203.pdf http://www.jjj.de/fxt/%23fxtbook https://oeis.org/A046021 https://web.archive.org/web/20050115122923/http:/www-ang.kfunigraz.ac.at/~fripert/fga/k1lehm.html https://web.archive.org/web/20110524210428/http:/www.dmtcs.org/volumes/abstracts/pdfpapers/dm040203.pdf http://archive.numdam.org/ARCHIVE/BSMF/BSMF_1888__16_/BSMF_1888__16__176_0/BSMF_1888__16__176_0.pdf
sameAs	Factorial number system Factorial number system Factorial number system Factorial number system Factorial number system Factorial number system Factorial number system Factorial number system Factorial number system Factorial number system Factorial number system Factorial number system Factorial number system
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has abstract	In combinatorics, the factorial number system, also called factoradic, is a mixed radix numeral system adapted to numbering permutations. It is also called factorial base, although factorials do not function as 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 table representation; in theformer case the resulting map from integers to permutations of n elements lists them in lexicographical order. General mixed radix systems were studied by Georg Cantor.The term "factorial number system" is used by Knuth,while the French equivalent "numération factorielle" was first used in 1888. The term "factoradic", which is a portmanteau of factorial and mixed radix, appears to be of more recent date. (en) In der Kombinatorik wird das fakultätsbasierte Zahlensystem verwendet, um einen eindeutigen Index für Permutationen zu erzeugen. (de)

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