[SeqFan] more Electric Charges

Wouter Meeussen w.meeussen.vdmcc at vandemoortele.be
Fri Aug 7 17:56:19 CEST 1998

```dear Seq's & Fans,

It is hard to believe that a Jurassic (=quite old) trick like
counting configurations of black & white beads on a necklace or
on a string can still hold surprises.

A minor variation, introduction of a white/black symmetry, causes the
standard counting schemes to "fail".

An example:

count all arrangements of 4 "1" & 4 "0" on a line, starting with "1" :
you get:
1 1 1 1 0 0 0 0
1 1 1 0 1 0 0 0
1 1 1 0 0 1 0 0
...
1 0 0 0 1 0 1 1
1 0 0 0 0 1 1 1

Of course, we get (8 choose 4)/2 = 35 arrangements.

*** the white/black symmetry ***

now suppose we are counting electric charges, a "0" representinting q=-1,
and a "1" representing q=1. In that case, many properties remain unchanged
by a interchange of all positive & negative charges (eq. the electrostatic
energy).

In our counting scheme, this causes some arrangements to become equivalent,
(giving rise to two arrangements with the same energy, or twofold degenerate
energy levels as they are sometimes called.

for example: these two are equivalent:

1 1 1 0 0 1 0 0
reverse & interchange 1 & 0 :
1 1 0 1 1 0 0 0

Using this strategy, I found for 2 n "charges" on a line :

n	levels	degeneracy 'bins'
1	1	1*1
2	3	3*1
3	7	4*1+3*2
4	23	11*1+12*2
5	71	16*1+55*2
6	252	42*1+210*2
7	890	64*1+826*2
8	3299	163*1+3136*2

it isn't in EIS, although A029891 has
1,3,7,23,70,242,832,2983, ...

*** Question 1 ***
could SeqFan members come up with a better (shorter?) description than
"
"	number of configurations, excluding reflection and black-white
"	interchange, of n black and n white beads on a string
"

*** Question 2 ***
would this black-white symmetry lead to more "new" sequences?
Has this never been taken into account?
Could a symbolic counting scheme (as oposed to "generate all & group'm ")
be constructed?

%I A000000
%S A000000 1,2,3,7,23,71,252,890,3299
%N A000000 number of configurations, excluding reflection and black-
white interchange, of n black and n white beads on a string
%R A000000
%A A000000 w.meeussen.vdmcc at vandemoortele.be
%O A000000 0,2
%K A000000 nonn

wouter.

NV Vandemoortele Coordination Center
Oils & Fats Applied Research
Prins Albertlaan 79
Postbus 40
B-8870 Izegem (Belgium)
Tel: +/32/51/33 21 11
Fax: +/32/51/33 21 75
vdmcc at vandemoortele.be

```