# What's a series-parallel network? (Explanation for OEIS A000084)

The explanation for OEIS A000084 states:

Number of series-parallel networks with n unlabeled edges. Also called yoke-chains by Cayley and MacMahon.

In the example for 3 unlabeled edges, it gives the following as examples:

               /\
o-o-o-o o-o=o o--o o-o-o
\/   \_/


A fifth graph can be created from 3 unlabeled edges:

  o
|
o
/ \
o   o


Since this is not included as an example (and $a(3)=4$), that means that this isn't a series-parallel network.

My question is: What is a series-parallel network?

I found the following Wolfram|Alpha explanation: http://mathworld.wolfram.com/Series-ParallelNetwork.html

There, it states:

An 'essentially series network' is a network which can be broken down into two or more "subnetworks" in series.
An 'essentially parallel network' is a network which can be broken down into two or more "subnetworks" in parallel.

This doesn't answer my question though. What does a "series-parallel" network mean?

• The fifth graph is obtained by connecting three series-parallel networks (specifically the unitary one), and cannot be obtained by joining only two, and therefore is not series-parallel Aug 8, 2017 at 2:07
• From the OEIS: This is a series-parallel network: o-o; all other series-parallel networks are obtained by connecting two series-parallel networks in series or in parallel. Aug 8, 2017 at 2:08
• @somebody ._. whoops I'm blind. thanks Aug 8, 2017 at 2:14

The use of graphical networks is sometimes a bit fussy,but there is an algebraic interpretation. Consider the free algebraic system with two commutative associative operators $(x+y)$ and $(x*y)$ and one generator $A$. The number of elements with $n$ occurrences of the generator is $a(n)$. The use of the two operators is equivalent to series and parallel but is algebraically precise.
Examples: $n=1\!: A;\; n=2\!: A+A, A*A;\; n=3\!: A+A+A, A+(A*A), A*(A+A), A*A*A.$