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