× 10! = 36,288,00010, which may be written A0000000000!=10:0:0:0:0:0:0:0:0:0:0!, but not 100000000000! = 1:0:0:0:0:0:0:0:0:0:0:0! which denotes 11! = 39,916,80010. Thus using letters A–Z to denote digits 10, 11, 12, ..., 35 as in other base-N make the largest representable number 36 × 36! − 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 24031201, this number also can be written as 2:0:1:0!). 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, as it requires an additional separation mark. Permutations There is a natural mapping between the integers (or equivalently the numbers with n digits in factorial representation) and permutations of n elements in lexicographical order, when the integers are expressed in factoradic form. This mapping has been termed the Lehmer code (or inversion table). For example, with , such a mapping is In each case, calculating the permutation proceeds by using the leftmost factoradic digit (here, 0, 1, or 2) as the first permutation digit, then removing it from the list of choices (0, 1, and 2). Think of this new list of choices as zero indexed, and use each successive factoradic digit to choose from its remaining elements. If the second factoradic digit is "0" then the first element of the list is selected for the second permutation digit and is then removed from the list. Similarly, if the second factoradic digit is "1", the second is selected and then removed. The final factoradic digit is always "0", and since the list now contains only one element, it is selected as the last permutation digit. The process may become clearer with a longer example. Let's say we want the 2982nd permutation of the numbers 0 through 6. The number 2982 is 4:0:4:1:0:0:0! in factoradic, and that number picks out digits (4,0,6,2,1,3,5) in turn, via indexing a dwindling ordered set of digits and picking out each digit from the set at each turn: 4:0:4:1:0:0:0! ─► (4,0,6,2,1,3,5) factoradic: 4 : 0 : 4 : 1 : 0 : 0 : 0! ├─┬─┬─┬─┐ │ ├─┬─┬─┬─┐ ├─┐ │ │ │ sets: (0,1,2,3,4,5,6) ─► (0,1,2,3,5,6) ─► (1,2,3,5,6) ─► (1,2,3,5) ─► (1,3,5) ─► (3,5)