  We obtain the fermionic formulas for the characters of (k, r)-admissible
configurations in the case of r=2 and r=3. This combinatorial object appears as
a label of a basis of certain subspace $W(\Lambda)$ of level-$k$ integrable
highest weight module of $\hat{sl}_{r}$. The dual space of $W(\Lambda)$ is
embedded into the space of symmetric polynomials. We introduce a filtration on
this space and determine the components of the associated graded space
explicitly by using vertex operators. This implies a fermionic formula for the
character of $W(\Lambda)$.
