A class of partition functions associated with $E_{\tau,\eta}(gl_3)$ by Izergin-Korepin analysis

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.