Factorial number system

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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^[1] representation; in the former 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.^[2]

The term "factorial number system" is used by Knuth,^[3] while the French equivalent "numération factorielle" was first used in 1888.^[4] The term "factoradic", which is a portmanteau of factorial and mixed radix, appears to be of more recent date.^[5]