A Plethysm Formula for $p_\mu(\underline{x}) \circ h_\lambda(\underline{x})$

Abstract

This paper gives a new formula for the plethysm of power-sum symmetric functions and complete symmetric functions. The form of the main result is that for $\mu \vdash b$ and $\lambda \vdash a$ with length $t$, then $$p_\mu(\underline{x}) \circ h_\lambda(\underline{x}) = \sum_T \underline{\omega}^{{\rm maj}_{\mu^t} (T)} s_{{\rm sh}(T)}(\underline{x}) $$ where the sum is over semistandard tableaux of weight $\lambda_1^b \lambda_2^b \dots \lambda_t^b$ and $\underline{\omega}^{{\rm maj}_{\mu^t} (T)}$ is a root of unity which depends on $\mu$, $t$, and $T$.