Series/Parallel Networks - A058387

[Gordon Royle]

I am confused (as usual) about exactly what is being counted in this  
sequence...

http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/ 
eisA.cgi?Anum=A058387

I can't seem to make the numbers work whether I interpret it as  
"2-terminal series-parallel networks" or "series-parallel graphs"  
unless I do something weird with isomorphisms...


Let's suppose that it is 2-terminal series/parallel networks that we  
are counting ...  so on one edge we have

	s---t

On two edges we are forced to join 2 of these in parallel (as multiple  
edges are not permitted for this sequence)

	s---x---t

On three edges, we can either join one edge in series, or one in  
parallel.

	s---x---x---t     s---x---t
                                    \------/


So far, so good... but it is when n=4 that I can't seem to make things  
work...

I can make

	s---x---x---x---t

Now, can I make one or two graphs by doing a series connection of K_2  
with the triangle?


                             s---x---x---t
                                    \------/


                                   s---x---x---t
                                    \------/


And what about these two graphs

	s---x---x---t
           \-----------/

and

	s---x---t---x
           \----------/


Now I've got FIVE graphs...

The only way that I can see to get four graphs (as required) is to  
distinguish two vertices as s/t (thus allowing isomorphic underlying  
*graphs*) but then to count an exchange of s with t as being isomorphic  
to the original....