Exponential Hilbert series and hierarchical log-linear models

Consider a hierarchical log-linear model, given by a simplicial complex, $\Gamma$, and integer matrix $A_\Gamma$. We give a new characterization of the rank of $A_\Gamma$ given by a logarithmic transformation on the exponential Hilbert series of $\Gamma$. We show that, if each random variable in $X$ has the same number of possible outcomes, then this formula reduces to a simple description in terms of the face vector of $\Gamma$. If $\Gamma$ further satisfies the Dehn-Sommerville relations, then we give an exceptionally simple formula for computing the rank of $A_\Gamma$, and thus the dimension and the number of degrees of freedom of the model.