Factorial Number System - Examples

Examples

Here are the first twenty-four numbers, counting from zero.

The table on the left shows permutations, and inversion vectors (which are reflected factorial numbers) below them. Another column shows the inversion sets. The digit sums of the inversion vectors (or factorial numbers) and the cardinalities of the inversion sets are equal (and have the same parity as the permutation). They form the sequence  A034968.

decimal factorial
0 0
1 10
2 100
3 110
4 200
5 210
6 1000
7 1010
8 1100
9 1110
10 1200
11 1210
12 2000
13 2010
14 2100
15 2110
16 2200
17 2210
18 3000
19 3010
20 3100
21 3110
22 3200
23 3210

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 543210! is 1000000! 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.

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 × 10! = 3628800010, which may be written A0000000000!, but not 100000000000! which denotes 11!=3991680010. Thus using letters A–Z to denote digits 10, ..., 35 as in other base-N make the largest representable number 36! − 1=37199332678990121746799944815083519999999910. 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). 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.

Read more about this topic:  Factorial Number System

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