Cubic two-point Wick-pairing classification

At second order in the coupling constant of a real phi cubed theory, the two-point numerator has eight fields, hence $7!!=105$ Wick contractions. Of these, six give a free external line times a vacuum theta graph, nine give a free line times a vacuum dumbbell, 36 give a two-line bubble, 36 give a connected tadpole attachment, and 18 give two separate one-point tadpoles. Dividing by $2!(3!{)}^2=72$ gives weights $1/12,1/8,1/2,1/2,1/4$. The normalized vacuum generating functional cancels the first two contributions, while the last remains in the full two-point correlation function as the product of one-point functions. Thus cancellation of vacuum bubbles does not, by itself, take the connected part.