On number theory and enumerative combinatorics (Reviewed)

I've {re-}defined "natural permutation" as the following abstract set:

Given a natural number greater than the unit, "m", it determines univocally (when it is read in decimal or base-10) a set of smaller numbers for which it is itself the cardinal; Such set is isomorphically identical to a set of conventional symbols (usually called digits or numerals) that might be used for write integer numbers with a positional convention. For each case of both sets (either the conventional symbols used for writing numbers or the values assigned to each symbol) there might be assigned a third set containing all the possible permutations without repetitions (m! of them) for those entities.

The less expensive choice is to read those permutations precisely in the same base or radix that defines them, but also it is possible to read them for a greater radix.

This "possibility" additionally might be considered the connection between the positional number systems and the concept of permutation without repetitions due to the polynomial remainder theorem in its Diophantine interpretation.

Regards,

R. J. Cano (Sun January 6th 2013).