[SeqFan] final : summary on "charged necklaces"

Thu Aug 13 17:51:27 CEST 1998

since I sent mail while the issue was still foggy in my head, I would like
to close my interventions on the subject with a clean summary.

to Neil J.A. Sloane : I'll format them & send 'm in, and signal where my
descriptions sofar might have been misleading or plain wrong (:-(
It turns out to be necessary to distinguish between counting "equivalent
states under given symmetry operations" and "counting energy levels".
This difference is what I called "accidental degeneracies" in analogy with
Quantum Theory usage.

**** the summary ****

L I N E	  (string)					

I found for n positive and n negative "charges" on a line :					(not in EIS) 	

n	levels	states	(k1*d1+k2*d2+...)				
1	1	1	1*1				
2	3	3	3*1				
3	7	10	4*1+3*2				
4	23	35	11*1+12*2				
5	71	126	16*1+55*2				
6	252	462	42*1+210*2				
7	890	1716	64*1+826*2				
8	3299	6435	163*1+3136*2				
9****	12274	24310	256*1+12009*2+9*4				

for the permutations of n "1" s and n "-1" s on a line,	
with sign change and reversal :							
(already in EIS, correct as it stands)

n	levels	states	degen=1	degen=2			

1	1	1	1	0			
2	3	3	3	0			
3	7	10	4	3			
4	23	35	11	12			
5	71	126	16	55			
6	252	462	42	210			
7	890	1716	64	826			
8	3299	6435	163	3136			
9****	12283	24310	256	12027			
10	46508	92378	638	45870			

states = Binomial[2n,n]/2
degen1 = Table[2^(n) +Mod[n,2]Binomial[ n,n/2-1/2],{n,0,10}]			
degen2 = (Table[Binomial[2n,n]/2 -(2^(n-1)
+Mod[n+1,2]Binomial[n-1,n/2-1]),{n,10}])/2	

Difference is caused by the last bin for n=9:
I found nine pairs of "accidental degeneracies" 
These states are in no way symmetric, but give the same energy levels
on the adjacency matrix.
This causes the last bin of "18*2" to be a contraction of "9*4",
changing the nr of levels to be 12274 in stead of 12283. 				

acci1
{-1,-1,-1,-1,1,1,1,1,1,1,-1,-1,1,-1,1,-1,-1,1}
{-1,-1,1,1,1,1,1,1,-1,-1,-1,-1,1,-1,-1,1,-1,1}
acci2
{-1,-1,1,-1,-1,1,1,1,-1,1,1,-1,1,1,-1,-1,-1,1}
{-1,-1,1,1,1,-1,1,1,-1,1,1,-1,-1,-1,1,-1,-1,1}
acci3
{-1,-1,1,1,1,-1,-1,-1,1,1,1,-1,1,1,-1,-1,-1,1}
{-1,-1,1,1,1,-1,1,1,-1,-1,-1,1,1,1,-1,-1,-1,1}
acci4
{-1,1,-1,-1,1,-1,1,-1,1,1,1,-1,1,1,-1,-1,-1,1}
{-1,1,-1,1,-1,1,1,-1,1,1,1,-1,-1,-1,1,-1,-1,1}
acci5
{-1,1,-1,1,1,-1,-1,1,-1,1,1,-1,1,-1,1,-1,-1,1}
{-1,1,-1,1,1,-1,1,-1,1,-1,-1,1,1,-1,1,-1,-1,1}
acci6
{-1,-1,1,-1,-1,1,1,1,-1,1,1,-1,-1,-1,1,1,1,-1}
{-1,-1,1,1,1,-1,-1,-1,1,-1,-1,1,1,1,-1,1,1,-1}
acci7
{-1,-1,-1,-1,1,1,1,1,1,1,-1,-1,-1,1,-1,1,1,-1}
{-1,-1,1,1,1,1,1,1,-1,-1,-1,-1,-1,1,1,-1,1,-1}
acci8
{-1,-1,1,-1,-1,1,1,1,-1,1,-1,1,1,-1,1,-1,1,-1}
{-1,-1,1,1,1,-1,1,1,-1,1,-1,1,-1,1,-1,-1,1,-1}
acci9
{-1,-1,1,1,-1,1,-1,-1,1,1,-1,1,1,1,-1,-1,1,-1}
{-1,-1,1,1,-1,1,1,1,-1,-1,1,-1,1,1,-1,-1,1,-1}

R I N G	S (necklaces)

number of energy levels of n positive and n negative charges equally spaced
on a ring		(already in EIS, description MUST refer to charges & energy-levels)

			(count k of E-levels with degeneracy d)			
n	levels	states	(k1*d1+k2*d2+...)			
1	1	1	1*1			
2	2	3	1*1+1*2			
3	3	10	1*1+1*3+1*6			
4	7	35	1*1+1*2+2*4+3*8			
5	13	126	1*1+3*5+7*10+2*20			
6	35	462	1*1+1*2+1*3+4*6+20*12+8*24			
7	85	1716	1*1+7*7+35*14+42*28			
8****	254	6435	1*1+1*2+2*4+11*8+88*16+148*32+3*64			
9	701	24310	1*1+1*3+1*6+14*9+136*18+488*36+9*54+51*72			
10	2337	92378	1*1+1*2+3*5+22*10+362*20+1855*40+1*60+130*80+2*120			

for the permutations of n 1's and n -1's on a ring,						
rotation, sign change and reversal turn out as 						
n   # distinct	states	count*degeneracy			
1	1	1	1*1			
2	2	6	1*2+1*4			
3	3	20	1*2+1*6+1*12			
4	7	70	1*2+1*4+2*8+3*16			
5	13	252	1*2+3*10+7*20+2*40			
6	35	924	1*2+1*4+1*6+4*12+20*24+8*48			
7	85	3432	1*2+7*14+35*28+42*56			
8****	257	12870	1*2+1*4+2*8+11*16+88*32+154*64			

This is (most likely) :											
%I	A006840	M0837									
%S	A006840	1,2,3,7,13,35,85,257,765,2518						
%N	A006840	Binary sequences of period 2n with n 1's per period.	
%R	A006840	JAuMS A33 14 1982. CN 40 89 1983.			
%O	A006840	1,2									
%A	A006840	njas									
%K	A006840	nonn		

The difference is caused by the last bin for n=8:
I found three pairs of "accidental degeneracies" 
These states are in no way symmetric, but give the same energy levels
on the adjacency matrix.
This causes the last bin of "3*64" to be a contraction of "6*32",
changing the nr of levels to be 254 in stead of 257.
acci1=
{-1,-1,-1,-1,1,-1,1,1,-1,1,1,-1,-1,1,1,1}
{-1,-1,-1,-1,1,1,-1,1,-1,-1,1,1,1,-1,1,1}
acci2=
{-1,-1,-1,-1,-1,1,1,-1,1,-1,1,1,-1,1,1,1}}
{-1,-1,-1,-1,1,-1,1,1,-1,-1,1,1,1,1,-1,1}}
acci3=
{-1,-1,-1,1,-1,-1,1,1,-1,1,-1,1,-1,1,1,1}}
{-1,-1,-1,1,-1,1,-1,1,1,-1,-1,1,1,1,-1,1}}

WITHOUT REVERSALS:	(turning necklace over)									
symmetry conditions of necklace, n "1"'s and n "0"'s,
with rotation but no reversals,				
but with (1) <--> (0) exchange as symmetry operation	
new to EIS : 1,2,3,7,15,44,128,415,1367

n  # distinct	tot	count*degeneracy								
1	1	2	1*2								
2	2	6	1*2+1*4					
3	3	20	1*2+1*6+1*12					
4	7	70	1*2+1*4+2*8+3*16					
5	15	252	1*2+3*10+11*20					
6	44	924	1*2+1*4+1*6+6*12+35*24					
7	128	3432	1*2+9*14+118*28					
8	415	12870	1*2+1*4+2*8+19*16+392*32					
9	1367	48620	1*2+1*6+1*12+28*18+1336*36					

TWO NECKLACES

two facing necklaces, one above the other, each n positive and n negative
charges.		
Count the number of energy levels (inter, not intra).
new to EIS : 2,5,18,71,538

n    E levs	states	(level_count*degeneracy+...)	
1	2	4	{{2,2}}					
2	5	36	{{2,2},{2,4},{1,24}}
3	18	400	{{2,2},{2,6},{4,12},{4,24},{4,36},{2,48}}
4	71	4900
{{2,2},{2,4},{4,8},{14,16},{12,32},{4,48},{18,64},{2,96},{8,128},{2,160},{2,
320},{1,728}}
5	538	63504
{{2,2},{6,10},{34,20},{92,40},{8,60},{268,80},{2,100},{4,120},{56,160},{4,24
0},{16,280},{16,320},{4,380},{2,400},{2,440},{12,480},{4,640},{4,720},{2,1280}}

wouter.

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