The factorial based radix or factoradic is a factorial based mixed radix numeral scheme:

 radix: 5! 4! 3! 2! 1! decimal: 120 24 6 2 1 

In this numbering system, the rightmost digit may be 0 or 1, the next 0, 1, or 2, and so on. The numbers from 0 to 24 are

 decimal factoradic 0 0 1 1 2 10 3 11 4 20 5 21 6 100 7 101 8 110 9 111 10 120 11 121 12 200 13 201 14 210 15 211 16 220 17 221 18 300 19 301 20 310 21 311 22 320 23 321 24 1000 

For another example, the biggest number that could be represented with five digits would be 54321 which equals 719 in decimal:

5×5!+4×4!+3×3!+2×2!+1×1!.

It might not be clear at first sight but factorial based numbering system is also unambiguous. No number can be represented by more than one way because the sum of respective factorials multiplied by the index is always the next factorial minus one:

This can be easily proved with mathematical induction.

There is a natural mapping between the integers 1 .. n! (or equivalently the factoradic numbers with (n-1) digits) and permutations of n elements in lexicographic order, when the integers are expressed in factoradic form. For example, with n=3, such a mapping is

 decimal factoradic permutation 0 00 (0,1,2) 1 01 (0,2,1) 2 10 (1,0,2) 3 11 (2,0,1) 4 20 (1,2,0) 5 21 (2,1,0) 

where the leftmost factoradic digit (0..2) specifies the placement of the 0 in the permutation, the rightmost digit (0..1) specifies the placement of the 1 in the remaining two possible locations, and the 2 is put in the last open location in the permutation list.

A similar concept, combinadics, can be used to find combinations.

External links

• http://msdn.microsoft.com/library/en-us/dnnetsec/html/permutations.asp