Vacuum pairing weights for two even scalar vertices ($N_{n,r}={\left(\genfrac{}{}{0ex}{}{n}{r}\right)}^{2}r![(n-r-1)!!{]}^2$)

For two identical even-degree $n$ scalar Feynman vertices, choose $r$ half-edges on each and pair them across the vertices. The remaining $n-r$ half-edges pair locally. This gives the displayed number of Wick contractions, with even $r$ and $(-1)!!=1$. The coefficient in the unnormalized partition function is $N_{n,r}/[2!(n!{)}^2]$. The $r=0$ contribution is disconnected and is removed from the logarithm, but remains in the partition function itself. This rule produces the quartic and sextic oscillator vacuum graph coefficients without drawing-dependent symmetry-factor guesses.