A plethysm formula for \(p_ \mu(\underline x)\circ h_ \lambda(\underline x)\) (Q1378503)

Summary: 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 \(t\), \[ p_\mu(\underline{x}) \circ h_\lambda(\underline{x}) = \sum_T \underline{\omega}^{\text{maj}_{\mu^t} (T)} s_{\text{sh}(T)} (\underline{x}) \] where the sum is over semistandard tableaux of weight \(\lambda_1^b \lambda_2^b \dots \lambda_t^b\) and \(\underline{\omega}^{\text{maj}_{\mu^t} (T)}\) is a root of unity which depends on \(\mu\), \(t\), and \(T\).