factorial number system

Another proposal is the so-called factorial number system:

The factorial number system provides a unique representation for each natural number, with the given restriction on the "digits" used.

Factorial number system (also called factoradics)

In a mixed radix system such as the factorial number system, the weights form a sequence where each weight is an integral multiple of the previous one.

The first step then is simply expression of N in the factorial number system, which is just a particular mixed radix representation, where for numbers up to n!

The factorial number system uses a varying radix, giving factorials as place values; they are related to Chinese remainder theorem and Residue number system enumerations.

Consecutive permutations in the sequence generated by the Steinhaus-Johnson-Trotter algorithm have numbers of inversions that differ by one, forming a Gray code for the factorial number system.

An obvious way to generate permutations of n is to generate values for the Lehmer code (possibly using the factorial number system representation of integers up to n!)

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.

Even for ordinary permutations it is significantly more efficient than generating values for the Lehmer code in lexicographic order (possibly using the factorial number system) and converting those to permutations.

The sum of the numbers in the factorial number system representation gives the number of inversions of the permutation, and the parity of that sum gives the signature of the permutation.

Unlike the factorial number system, the combinatorial number system of degree k is not a mixed radix system: the part of the number N represented by a "digit" c is not obtained from it by simply multiplying by a place value.