Sanity Check II

[wouter meeussen]

different Abs of vector sums generated by n vectors choosen from
2n (equally spaced) points around the unit circle should be:

1, 2, 3, 6, 11, 20, 54, 130, 200, 725

Grouping the bracelets from A006840 into equivalence classes (??)
gives

{{1,1}}
{{2,1}}
{{3,1}}
{{5,1},{1,2}}
{{10,1},{1,3}}
{{14,1},{1,2},{1,3},{4,4}}
{{40,1},{1,2},{12,3},{1,7}}
{{78,1},{6,2},{38,3},{3,6},{5,7}}
{{91,1},{10,2},{32,3},{21,4},{6,5},{17,6},{1,10},
        {6,12},{8,13},{1,14},{1,17},{2,18},{3,19},{1,32}}
{{306,1},{15,2},{265,3},{14,4},{2,5},{24,6},{4,8},{3,9},
      {64,10},{17,13},{1,14},{1,19},{2,20},{5,24},{2,32}}

whith "{{5,1},{1,2}}" to be read as
5 (sets of) bracelets with unique Abs vector sums, and
1 set of 2 bracelets having the same vector sum.

In physics, the 'Abs vector sums' would be called 'dipole
moments'.
(cfr A045611)

The calculation is hard on Mathematica :
the vector sums must be sorted & grouped using the accurate
floating point values of the sums over ugly radicals.

Forgive me for coming back to these "Physical Analoges" with
'charges', 'energy states' and 'dipole moments' etc. applied to
'clean bracelets', but I consider it an intruiging fact that
'equivalences' seem to appear that were not yet looked at.
(Is this so???)
And I guess they are 'hard to count' combinatorially.

Wouter Meeussen
wouter.meeussen at pandora.be