  Recently, a class of partition functions associated with higher rank rational
and trigonometric integrable models were introduced by Foda and Manabe. We use
the dynamical $R$-matrix of the elliptic quantum group $E_{\tau,\eta}(gl_3)$ to
introduce an elliptic analogue of the partition functions associated with
$E_{\tau,\eta}(gl_3)$. We investigate the partition functions of Foda-Manabe
type by developing a nested version of the elliptic Izergin-Korepin analysis,
and present the explicit forms as symmetrization of multivariable elliptic
functions. We show that special cases are essentially the elliptic weights
functions introduced in the works by Rim\'anyi-Tarasov-Varchenko, Konno,
Felder-Rim\'anyi-Varchenko.
