Charges : new surprise

[Wouter Meeussen]

hi SeqFanners,

I hit on a unexpected ambiguity in my earlier posts "Electric Charges Solved".

There I propose a series 
1,3,7,23,71,252,890,3299,12283,..
for the number of configurations, excluding reflection and black/white
interchange, of n black and n white beads on a string.

I still believe this to be correct as it stands.


It is however wrong to assume that this sequence counts the number of energy
levels for an analogous system of positive and negative electric charges.
There is a difference that turns up at n=18.

When counting energy levels, we must define an 18*18 adjacency matrix

w1 w2 w3 w4 ...   w17 w18
w2 w1 w2 w3 ...   w16 w17
w3 w2 w1 w2 ...   w15 w16
...
w17 w16     ...   w1  w2
w18 w17     ...   w2  w1

multiply it by left and by right with a charges-array {z[1], ..,z[18]}

resulting in 

z1*w1*z1  z1*w2*z2 ...
z2*w2*z1  z2*w1*z2 ...
...
z18*w18*z1         ... z18*w1*z18

this matrix can be flattened (sum of all its elements) to provide a measure
of of the interaction energies between the charges. Equal charges contribute
positive, and unequal charges negative energy contributions.

All we have to do is let the array z assume all possible permutations of 9
(-1) and 9 (+1) charges.

For each permutation we get a linear combination of the w[i].
Needless to say that in the 'physical' problem the w[i] stands for 1/r in
casu (1/(i-1)) with w[1] set to 0 (charges don't see themselves). This
approach is valid for charges on a straight line and with unit intervals.


*** THE SNAG ***


with 9 (+) and 9 (-) charges, the degeneracy bins come out as :

256*1 + 12009*2 + 9*4

this means:
256 of the permutations give rise to unique linear combinations of the w[i]
since they are their own (reverse & b/w exchange).

12009 permutations have distinct (reverse & b/w exchange). So they occur in
pairs.

There are 9 permutations that form 'double pairs': sets of 
	(a permutation,
	 its (rev &b&w exchange),
	 a completely unrelated one,
	 and this one's (rev &b&w exchange).


So there are 'accidental degeneracies' at work here!
the 9 sets of 4 permutations are:


{-1,-1,-1,-1,1,1,1,1,1,1,-1,-1,1,-1,1,-1,-1,1},
{-1,-1,1,1,1,1,1,1,-1,-1,-1,-1,1,-1,-1,1,-1,1},
{-1,1,-1,1,1,-1,1,1,1,1,-1,-1,-1,-1,-1,-1,1,1},
{-1,1,1,-1,1,-1,1,1,-1,-1,-1,-1,-1,-1,1,1,1,1},

{-1,-1,1,-1,-1,1,1,1,-1,1,1,-1,1,1,-1,-1,-1,1},
{-1,-1,1,1,1,-1,1,1,-1,1,1,-1,-1,-1,1,-1,-1,1},
{-1,1,1,-1,1,1,1,-1,-1,1,-1,-1,1,-1,-1,-1,1,1},
{-1,1,1,1,-1,-1,1,-1,-1,1,-1,-1,-1,1,1,-1,1,1},

{-1,-1,1,1,1,-1,-1,-1,1,1,1,-1,1,1,-1,-1,-1,1},
{-1,-1,1,1,1,-1,1,1,-1,-1,-1,1,1,1,-1,-1,-1,1},
{-1,1,1,1,-1,-1,-1,1,1,1,-1,-1,1,-1,-1,-1,1,1},
{-1,1,1,1,-1,-1,1,-1,-1,-1,1,1,1,-1,-1,-1,1,1},

{-1,1,-1,-1,1,-1,1,-1,1,1,1,-1,1,1,-1,-1,-1,1},
{-1,1,-1,1,-1,1,1,-1,1,1,1,-1,-1,-1,1,-1,-1,1},
{-1,1,1,-1,1,1,1,-1,-1,-1,1,-1,-1,1,-1,1,-1,1},
{-1,1,1,1,-1,-1,1,-1,-1,-1,1,-1,1,-1,1,1,-1,1},

{-1,1,-1,1,1,-1,-1,1,-1,1,1,-1,1,-1,1,-1,-1,1},
{-1,1,-1,1,1,-1,1,-1,1,-1,-1,1,1,-1,1,-1,-1,1},
{-1,1,1,-1,1,-1,-1,1,1,-1,1,-1,1,-1,-1,1,-1,1},
{-1,1,1,-1,1,-1,1,-1,-1,1,-1,1,1,-1,-1,1,-1,1},

{-1,-1,1,-1,-1,1,1,1,-1,1,1,-1,-1,-1,1,1,1,-1},
{-1,-1,1,1,1,-1,-1,-1,1,-1,-1,1,1,1,-1,1,1,-1},
{-1,1,1,-1,1,1,1,-1,-1,1,-1,-1,-1,1,1,1,-1,-1},
{-1,1,1,1,-1,-1,-1,1,1,-1,1,1,1,-1,-1,1,-1,-1},

{-1,-1,-1,-1,1,1,1,1,1,1,-1,-1,-1,1,-1,1,1,-1},
{-1,-1,1,1,1,1,1,1,-1,-1,-1,-1,-1,1,1,-1,1,-1},
{-1,1,-1,1,1,-1,-1,-1,-1,-1,1,1,1,1,1,1,-1,-1},
{-1,1,1,-1,1,-1,-1,-1,1,1,1,1,1,1,-1,-1,-1,-1},

{-1,-1,1,-1,-1,1,1,1,-1,1,-1,1,1,-1,1,-1,1,-1},
{-1,-1,1,1,1,-1,1,1,-1,1,-1,1,-1,1,-1,-1,1,-1},
{-1,1,-1,-1,1,-1,1,-1,1,-1,1,1,-1,1,1,1,-1,-1},
{-1,1,-1,1,-1,1,1,-1,1,-1,1,1,1,-1,-1,1,-1,-1},

{-1,-1,1,1,-1,1,-1,-1,1,1,-1,1,1,1,-1,-1,1,-1},
{-1,-1,1,1,-1,1,1,1,-1,-1,1,-1,1,1,-1,-1,1,-1},
{-1,1,-1,-1,1,1,-1,1,-1,-1,1,1,1,-1,1,1,-1,-1},
{-1,1,-1,-1,1,1,1,-1,1,1,-1,-1,1,-1,1,1,-1,-1}}


*************************************************
to me this was completely unexpected.
Of course, this is independent of the "measure of distance" we use for w[i].
When using 'nice' distance measures like w[i]=(i-1) or even 1/(i-1)?? the
degeneracies can increase. But never will they decrease!
**************************************************

This in fact all turns around the definition of 'equivalent configurations'.
I wouldn't be a bit surprised if my error of reasoning turned out to be
rather common. After all, it worked nice upto 16 charges.


wouter.

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