In combinatorial mathematics, the series-parallel networks problem asks for the number of networks that can be formed using a given number of edges. The edges can be distinguishable or indistinguishable.

When the edges are indistinguishable, consider the problem of enumerating the number of topologically different networks on edges, where multiple edges are allowed. The idea is to break-down
the problem by classifying the networks as essentially series and essentially parallel
networks.

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

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

By the duality of networks, it can be proved that the number of essentially series networks is equal to the number of essentially parallel networks. Thus for all , the number of networks in edges is twice the number of essentially
series networks. For , the number of networks is 1.

Define
as the number of series-parallel networks on indistinguishable edges and as the number of essentially series networks. Then

(1)

can be found out by enumerating the partitions of . Consider a partition of , i.e.,

(2)

Then the number of essentially series networks can be computed as . Hence

(3)

where the summation is over all partitions of excluding the trivial partition . This gives a recurrence for computing from which can be computed as above.

In addition, the sequence satisfies

(4)

or more explicitly,

(5)

The first few values of for , 2, ... are 1, 2, 4, 10, 24, 66, 180, 522, 1532, 4624, ...
(OEIS A000084), and for are 1, 1, 2, 5, 12, 33, 90, 261, 766, 2312, ... (OEIS A000669).

Ellis-Monaghan, J. A. and Merino, C. "Graph Polynomials and Their Applications I: The Tutte Polynomial." 28 Jun 2008. http://arxiv.org/abs/0803.3079.Eppstein,
D. "Parallel Recognition of Series-Parallel Graphs." Information and
Computation98, 41-55, 1992.Finch, S. "Series-Parallel
Networks." July 7, 2003. http://algo.inria.fr/bsolve/.Sloane,
N. J. A. Sequences A000084 and A000669 in "The On-Line Encyclopedia of Integer
Sequences."Valdes, J. "Parsing Flowcharts and Series-Parallel
Graphs." Ph.D. thesis. Also technical report STAN-CS-78-682, Computer Science
Department. Stanford, CA: Stanford University, 1978.