A095268 Graphical Partitions - Extend?

[Gordon Royle]

On 17/08/2006, at 9:40 PM, franktaw at netscape.net wrote:

> Two points, neither answering Paul's questions.
>
> The comment in A095268 could be improved.  Instead of "the number  
> of these graphs having distinct degree sequences", it should read  
> something like "the number of distinct degree sequences these  
> graphs possess".  If two graphs have the same degree sequence, then  
> neither of them has a _distinct_ degree sequence; but we do want to  
> count that degree sequence once.
>
> Second, as I'm sure Paul knows, a sequence has a self-convolution  
> sequare-root iff all the odd-index terms are even.  So far, A095268  
> has a strict alternation of parity.  Does that continue?
>

Well, I haven't coded Frank Rs super fast algorithm, but I had an old  
naive one (ie backtrack and check Havel/Hakimi condition) lying  
around in my files which I fired up this morning.;

So I get the following (confirmation would be welcomed)

Total number distinct degree sequences for n vertex graphs with no  
isolated vertices

2 1
3 2
4 7
5 20
6 71
7 240
8 871
9 3148
10 11655
11 43332
12 162769
13 614198
14 2330537
15 8875768
16 33924859
17 130038230
18 499753855

So the alternating pattern seems to hold, which would be an  
intriguing result if it were true....

(I have the numbers partitioned according to numbers of edges, and  
also the actual degree sequences themselves for n<=16 if anyone  
interested..)

Cheers

Gordon