Plethystic Murnaghan–Nakayama rule via vertex operators

Based on the vertex operator realization of the Schur functions, a determinant-type plethystic Murnaghan–Nakayama rule is obtained and utilized to derive a general formula of the expansion coefficients of sν s ν in the plethysm product (pₙ∘ hₖ)sμ ( p n ∘ h k ) s μ . Meanwhile, the equivalence between our algebraic rule and the combinatorial one is also established. As an application, we provide a simple way to compute the generalized Waring formula.