“Series-parallel networks problem”的意思、由来-开放百科全书

词条

Series-parallel networks problem

释义

Indistinguishable edges
Solution
External links

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

Indistinguishable edges

When the edges are indistinguishable, we look for the number of topologically different networks on n edges.

Solution

The idea is to break-down the problem by classifying the networks as essentially series and essentially parallel networks.

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.

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 n > 1, the number of networks in n edges is twice the number of essentially series networks. For n = 1, the number of networks is 1.

Define

as the number of series-parallel networks on n indistinguishable edges.
as the number of essentially series networks.

Then

can be found out by enumerating the partitions of .

Consider a partition, of n:

Consider the essentially series networks whose components correspond to the partition above. That is the number of components with i edges is . The number of such networks can be computed as

Hence

where the summation is over all partitions, of n except for the trivial partition consisting of only n.

This gives a recurrence for computing . Now can be computed as above.

[TODO: Generating function for and are explained in the external links.]

External links

{{OEIS el|sequencenumber=A000084|name=Number of series-parallel networks with n unlabeled edges|formalname=Number of series-parallel networks with n unlabeled edges. Also called yoke-chains by Cayley and MacMahon}}
{{OEIS el|sequencenumber=A000669|name=Number of series-reduced planted trees with n leaves|formalname=Number of series-reduced planted trees with n leaves. Also the number of essentially series series-parallel networks with n edges; also the number of essentially parallel series-parallel networks with n edges}}
[https://arxiv.org/abs/cond-mat/9707023 Asymptotic behavior of two-terminal series-parallel networks]

1 : Graph enumeration

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