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We construct new monomial quasi-particle bases of Feigin-Stoyanovsky's typesubspaces for affine Lie algebra $\mathfrak{sl}(3,\mathbb{C})^{\widetilde{}}$from which the known fermionic-type formulas for $(k,3)$-admissibleconfigurations follow naturally. In the proof we use vertex operator algebrarelations for standard modules and coefficients of intertwining operators.

Quasi-particle fermionic formulas for $(k,3)$-admissible configurations