Re: Randomness of ten digits

On 2008-11-26 03:43:33 -0400, Stig Holmquist <stigfjorden@xxxxxxxxxxx> said:

On Mon, 24 Nov 2008 06:18:51 -0500, Stig Holmquist
<stigfjorden@xxxxxxxxxxx> wrote:

On 24 Nov 2008 13:36:28 GMT, Martin Eisenberg
<martin.eisenberg@xxxxxxx> wrote:

Stig Holmquist wrote:

Your post is very interesting as I too have looked at the sums
of absolut differences but have discovered it can be misleading.
Using your set of seven digits one can make four sequences all
yielding a sum of 23 as follows with the sum of squares next and
the sum of cubes last:

3-5-2-7-1-6-4 103 509

4-6-2-7-1-5-3 101 485

3-6-2-7-1-5-4 103 497

3-7-1-5-2-6-4 97 443

4-5-2-7-1-6-3 105 521

Based on the above sums I would suggest the set with the highest
sum of cubes is the most random.

So, why stop at cubes? Note that with ever larger exponents, you
approach choosing by max-norm; i.e., those permutations containing
the largest single step win.


Martin

Each of the five permutations has a 7-1 gap but yield different sums
of cubes but only one lacks a 7-2 gap. If you must test thousands of
permutations it seems least demanding to just add cubes to find the
most random or disordered set. It's a form of entropy test.

Has this method of measuring disorder in a sequence been printed?

Stig

Using a random sequence generator I obtained 25 sets of permutations,
one per trial.
The sum of cubes ranged from 54 to 486 with a few duplications. Three
of them were 92 so I decided to use the fourth power of the gaps and
now they differed greatly. They became 502 ,1412 and 262.

Also note that the lowest sum of cubes becomes 42 for the sequence:
1-3-5-7-6-4-2 if you treat the set as a circle.

Do you know of a method or software that can generate all 5040 sets of
permutations of 1-7?

In times past permutation generators were a staple item of the Collected
Algorithms of ACM. The question was the order of presentation and the
amount of overhead computation. It should be in an book on combinatorial
methods.

The treating of the sequence as a circle is an important issue. One would
expect that shifting things along by one position should not influence
the measure. The measure being discussed is a 2-d measure and can also
be looked at by plotting the adjacant pairs in the plane. A small number
of lines will go through all the 7 points.

Stig


.

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