Reading the remainders backward gives 3:4:1:0:1:0!. In principle, this system may be extended to represent fractional numbers, though rather than the natural extension of place values (−1)!, (−2)!, etc., which are undefined, the symmetric choice of radix values n = 0, 1, 2, 3, 4, etc. after the point may be used instead. Again, the 0 and 1 places may be omitted as these are always zero. The corresponding place values are therefore 1/1, 1/1, 1/2, 1/6, 1/24, ..., 1/n!, etc. Examples The following sortable table shows the 24 permutations of four elements with different inversion related vectors. The left and right inversion counts and (the latter often called Lehmer code) are particularly eligible to be interpreted as factorial numbers. gives the permutation's position in reverse colexicographic order (the default order of this table), and the latter the position in 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 ( in the tables default order). For another example, the greatest number that could be represented with six digits would be 543210! which equals 719 in decimal: 5×5! + 4×4! + 3x3! + 2×2! + 1×1! + 0×0!. Clearly the next factorial number representation after 5:4:3:2:1:0! is 1:0:0:0:0:0:0! which designates 6! = 72010, the place value for the radix-7 digit. So the former number, and its summed out expression above, is equal to: 6! − 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: This can be easily proved with mathematical induction, or simply by noticing that : 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