  We consider asymptotics of power series coefficients of rational functions of
the form $1/Q$ where $Q$ is a symmetric multilinear polynomial. We review a
number of such cases from the literature, chiefly concerned either with
positivity of coefficients or diagonal asymptotics. We then analyze coefficient
asymptotics using ACSV (Analytic Combinatorics in Several Variables) methods.
While ACSV sometimes requires considerable overhead and geometric computation,
in the case of symmetric multilinear rational functions there are some
reductions that streamline the analysis. Our results include diagonal
asymptotics across entire classes of functions, for example the general
3-variable case and the Gillis-Reznick-Zeilberger (GRZ) case, where the
denominator in terms of elementary symmetric functions is $1 - e_1 + c e_d$ in
any number $d$ of variables. The ACSV analysis also explains a discontinuous
drop in exponential growth rate for the GRZ class at the parameter value $c =
(d-1)^{d-1}$, previously observed for $d=4$ only by separately computing
diagonal recurrences for critical and noncritical values of $c$.
