  Let $\mathfrak{u}_\zeta^\vee$ denote the small quantum group associated with
a simple Lie algebra $\mathfrak{g}^\vee$ and a root of unity $\zeta$. Based on
the geometric realization of the center of $\mathfrak{u}_\zeta^\vee$ in [8], we
use a combinatorial method to derive a formula for the dimension of a
subalgebra in the $G^\vee$-invariant part of the center
$Z(\mathfrak{u}_\zeta^\vee)^{G^\vee}$ of $\mathfrak{u}_\zeta^\vee$, that
conjecturally coincides with the whole $G^\vee$-invariant center. In case
$G=SL_n$ we study a refinement of the obtained dimension formula provided by
two geometrically defined gradings.
