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Factorial number system
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== Examples == The following sortable table shows the 24 permutations of four elements with different [[Inversion (discrete mathematics)|inversion]] related vectors. The left and right inversion counts <math>l</math> and <math>r</math> (the latter often called [[Lehmer code]]) are particularly eligible to be interpreted as factorial numbers. <math>l</math> gives the permutation's position in reverse [[colexicographical order|colexicographic]] order (the default order of this table), and the latter the position in [[lexicographical order|lexicographic]] order (both counted from 0). Sorting by a column that has the omissible 0 on the right makes the factorial numbers in that column correspond to the index numbers in the immovable column on the left. The small columns are reflections of the columns next to them, and can be used to bring those in colexicographic order. The rightmost column shows the digit sums of the factorial numbers ({{OEIS2C|A034968}} in the tables default order). [[File:Symmetric group 4; permutohedron 3D; Lehmer codes.svg|thumb|400px|The factorial numbers of a given length form a [[permutohedron]] when ordered by the bitwise <math>\le</math> relation<br/><br/>These are the right inversion counts (aka Lehmer codes) of the permutations of four elements.]] {{4-el perm inversions}} For another example, the greatest number that could be represented with six digits would be 543210<sub>!</sub> which equals 719 in [[decimal]]: :5Γ5! + 4Γ4! + 3x3! + 2Γ2! + 1Γ1! + 0Γ0!. Clearly the next factorial number representation after 5:4:3:2:1:0<sub>!</sub> is 1:0:0:0:0:0:0<sub>!</sub> which designates 6! = 720<sub>10</sub>, the place value for the radix-7 digit. So the former number, and its summed out expression above, is equal to: :6! β 1. The factorial number system provides a unique representation for each natural number, with the given restriction on the "digits" used. No number can be represented in more than one way because the sum of consecutive factorials multiplied by their index is always the next factorial minus one: :<math> \sum_{i=0}^n {i\cdot i!} = {(n+1)!} - 1. </math> This can be easily [[mathematical proof|proved]] with [[mathematical induction]], or simply by noticing that <math>\forall i, i\cdot i!=(i+1-1)\cdot i!=(i+1)!-i!</math>: subsequent terms cancel each other, leaving the first and last term (see [[Telescoping series]]). However, when using [[Arabic numerals]] to write the digits (and not including the subscripts as in the above examples), their simple concatenation becomes ambiguous for numbers having a "digit" greater than 9. The smallest such example is the number 10 Γ 10! = 36,288,000<sub>10</sub>, which may be written A0000000000<sub>!</sub>=10:0:0:0:0:0:0:0:0:0:0<sub>!</sub>, but not 100000000000<sub>!</sub> = 1:0:0:0:0:0:0:0:0:0:0:0<sub>!</sub> which denotes 11! = 39,916,800<sub>10</sub>. Thus using letters AβZ to denote digits 10, 11, 12, ..., 35 as in other base-''N'' make the largest representable number 36 Γ 36! β 1. For arbitrarily greater numbers one has to choose a base for representing individual digits, say decimal, and provide a separating mark between them (for instance by subscripting each digit by its base, also given in decimal, like 2<sub>4</sub>0<sub>3</sub>1<sub>2</sub>0<sub>1</sub>, this number also can be written as 2:0:1:0<sub>!</sub>). In fact the factorial number system itself is not truly a [[numeral system]] in the sense of providing a representation for all natural numbers using only a finite alphabet of symbols.
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