[SeqFan] Electric Charges solved

[Wouter Meeussen]

not sooo hard :


Table[1/4*(2^n + Binomial[2*n, n] + 
     2*Binomial[-1 + n, 1/2*(-2 + n)]*Mod[1 + n, 2]), {n, 24}]
Out[]=
{1,3,7,23,71,252,890,3299,12283,46508,176870,677294,2602198,10034104,38787572,
  150289699,583434323,2268861516,8836447022,34461940538,134564992898,
  526025965864,2058359779052,8061905791118}

degeneracy bins:


Table[2^(n) +Mod[n,2]Binomial[ n,n/2-1/2],{n,0,10}]
Out[]=
{1,3,4,11,16,42,64,163,256,638,1024}

now since the total number of levels is Binomial[2n,n]/2, we substract the
previous table from that, and divide by two to find the "double degenerates":

(Table[Binomial[2n,n]/2 -(2^(n-1) +Mod[n+1,2]Binomial[ n-1,n/2-1]),{n,10}])/2
Out[]=
{0,0,3,12,55,210,826,3136,12027,45870}


That's it. The trick is that the self-symmetric configurations (under
reverse+b/w exchenge) make up the non-degenerate bin.

For a necklace, it will be quite a bit harder.

wouter.




*****************************************************************************
At 17:56 7-08-98 +0200, Wouter Meeussen wrote:
>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
>
>
>
Dr. Wouter L. J. MEEUSSEN
w.meeussen.vdmcc at vandemoortele.be
eu000949 at pophost.eunet.be