Constructive Method for Finding the Coefficients of a Divided Symmetrization

We consider a type of divided symmetrization $\overrightarrow{D}_{\lambda,G}$ where $\lambda$ is a nonincreasing partition on $n$ and where $G$ is a graph. We discover that in the case where $\lambda$ is a hook shape partition with first part equal to 2, we may determine the expansion of $\overrightarrow{D}_{\lambda,G}$ over the basis of Schur functions. We show a combinatorial construction for finding the terms of the expansion and a second construction that allows computation of the coefficients.