I'm trying to understand an example that Stanley gives in his article "Subdivisions and local h vectors". It is example 2.3 part d).
If #V=3 and $h(\Gamma,x)=h_0+h_1x+h_2x^2+h_3x^3$ (so $h_0=1$ and $h_3=0)$, then $$\ell_V(\Gamma,x)=h_2x+h_2x^2.$$
I was trying using
$$\ell_V(\Gamma,x)=\sum_{W\subseteq V}(-1)^{|V-W|}h(\Gamma_W,x).$$ Then I have \begin{split} \ell_V(\Gamma,x)&=-1+3-(h(\Gamma_{1,2},x)+h(\Gamma_{1,3},x)+h(\Gamma_{2,3},x))+(h_0+h_1x+h_2x^2+h_3x^3)\\ &=-x(h(\Gamma_{1,2})+h(\Gamma_{1,3})+h(\Gamma_{2,3}))+h_1x+h_2x^2. \end{split}
And I know that $h(\Gamma_{1,2})+h(\Gamma_{1,3})+h(\Gamma_{2,3})$ are the interior vertices of $\Gamma$, but it is not clear to me how to reach the final conclusion. I really appreciate any help or ideas.