Geneva, november 2010
The permutational representation of numbers is a unique representation for each integer. Permutations of distinct elements - that one can also call natural permutations - require, when they are numbers, as many figures as the base in which these numbers are expressed. Number 27 has a proper, or adequate, permutational representation in base 4 because it is expressed as 123, or rather 0123, and uses all the figures in that base. There is no other base in which this phenomenon reproduces itself for number 27. Thus we know that while handling permutations of the digits in base 4 we shall find number 27 sooner or later. We decide to name Pnat[b] this set of all the permutations of distinct figures in a base b. But of course this set is lacunary, for example, the smallest permutation greater than 0123 is 0132 whose value is 30 (in base 10). There is here a gap of 3 numbers which can be filled by no permutation of distinct elements, this is why it appears necessary to include permutations with non-distinct elements if one wishes to obtain a permutational representation for each number.
The first measure to be taken towards permutations with non-distinct elements is to regard them as extensions of permutations of distinct elements. We will then work on fixed bases which will determine the range of concerned numbers. Indeed, the functions which enable us to handle permutations of non-distinct elements are the same ones as those which enabled us to handle distinct elements, except that from now on some of their actions will remain without effect. It is for instance ineffective to permute the last 2 figures of number 0122.
It is observed then that among all the permutations with non-distinct elements from a given base, there exists always a smallest number and a greateast number. In base 4 these numbers are 0000 (value 0) and 3333 (value 255 in base 10). So all our permutational representations in base 4 will appear between these two terms. However, as permutational representations of numbers in base 3 start with value 0 and finish with 222 (value 26) we notice an overlapping of these sets. So we are led to decide, in application of an economy principle, that the set of adequate permutational representations in a given base starts at the number that follows the value of the greatest permutation of non distinct elements in the base immediately lower, and stops at the value of the greatest permutation of non distinct elements in its own base. In base 4 thus, the fork ranges from 0123 to 3333, i.e. values 27 to 255 in base 10.
We can name Permin the lower term and Permax the higher one for any given set of permutational representations. One ends up this way with a single permutational representation for each number in N. This representation can then be called the adequate or proper permutational representation, in order to distinguish it from the valid permutational representations that do not obey the principle of economy, like permutational representations of numbers from 0 to 26 in base 4 for instance. The whole set of adequate permutational representations in a given base b will be called Pad[b].
Since we can allot each integer to its own permutational set, we know in which base to express it. Remains then the difficulty in managing to build this set by means of appropriate functions. Indeed, we wish to obtain a permutational set in the functional sense and not a collection of permutations, it is necessary thus that we have functions which make it possible to produce this set without appealing to arithmetic. We don't want to take the Permin for example and iteratively add one to it until obtaining its Permax. We want to modify the set of figures of Permin a certain number of times so as to obtain Permax. From this point of view we wish to know first of all how the set Pad[b] is constructed. Its major characteristic as a set of permutations of elements which are not always distinct, is that this non-distinctness can be expressed in various ways. Thus in a base 4 number, we can have 4 distinct elements, or 2 similar ones and two distinct, or 3 similar and 1 distinct, or 2 times 2 similar elements, or even 4 similar elements. That makes 5 manners of choosing distinct elements among other elements. These various manners of producing a set are called the partitions of the set and it will be the first characteristic which we will retain for the classification of the permutational representations of numbers.
Now that we have these partitions, it seems that we can already progress a little. By taking for example the first of them where the 4 digits are distinct and by permuting these digits, one obtains a subset of Pad , which is precisely Pnat . If we take the second partition which we mentioned, the things become more complicated however because as it contains two similar elements and two distinct ones; only 3 digits will be used on the 4 of our base and it will be necessary to choose them before being able to make them permute. For these 3 figures the possible combinations are:
(0012, 0013, 0023, 1102, 1103, 1123, 2201, 2203, 2213, 3301, 3302, 3312)
which make 12 possibilities of choice. Each partition is expressed by a certain amount of such combinations.
We can see now that creating the subset of permutations for each filling of each partition produces the set Pad[b] we were looking for. Partitions, combinations and permutations can all be represented by permutations belonging to Pad [b], consequently these functions define a relation and an order. During this research we thus obtained an additional and unexpected result: the permutational representation of numbers is likely to be expressed adequately on a graph with 3 dimensions. Indeed, these 3 parameters can be regarded as the 3 axes of a graph as shown in the following images representing the successor function among the 26 permutationnal integers in base 3:
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