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- 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).
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