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From the standard procedure for constructing Feynman vacuum graphs in$\phi^4$ theory from the generating functional $Z$, we find a relation withsets of certain combinatorial matrices, which allows us to generate the set ofall Feynman graphs and the respective multiplicities in an equivalentcombinatoric way. These combinatorial matrices are explicitly related with thepermutation group, which facilitates the construction of the vacuum Feynmangraphs. Various insights in this combinatoric problem are proposed, which inprinciple provide an efficient way to compute the Feynman vacuum graphs and itsmultiplicities.

A combinatorial matrix approach for the generation of vacuum Feynman graphs multiplicities in $\phi^4$ theory