On the Denominator of the Poincar\'e series for monomial quotient rings

Let $S=\Bbbk[x_1,..., x_n]$ be a polynomial ring over a field $\Bbbk$ and $I$ a monomial ideal of $S$. It is well known that the Poincar\'e series of $\Bbbk$ over $S/I$ is rational. We describe the coefficients of the denominator of the series and study the multigraded homotopy Lie algebra of $S/I$.