= Vacuum pairing weights for two even scalar vertices
{title2=$N_{n,r}=\binom nr^2r![(n-r-1)!!]^2$}
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.
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