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 $s_{\nu }$ in the plethysm product $(p_n\circ h_k)s_{\mu }$. 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.