Fermionic formulas for (k, 3)-admissible configurations

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 $W(\mathrm{\Lambda })$ of level-$k$ integrable highest weight module of ${\widehat{sl}}_r$. The dual space of $W(\mathrm{\Lambda })$ 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 $W(\mathrm{\Lambda })$.