We consider maps on genus-$g$ surfaces with $n$ (labeled) faces of prescribed even degrees. It is known since work Norbury that, if one disallows vertices degree one, the enumeration such related to counting lattice point in moduli space curves labeled points and given by a symmetric polynomial $N_{g,n}(\ell_1,\ldots,\ell_n)$ face degrees $2\ell_1, \ldots, 2\ell_n$. generalize this restricting ...