Consider the Hall-inner product of a Schur Function with a Skew-Schur function. Say for instance $\langle s_{(5,3,2,1)/(3,2,1)} , s_{(3,2)} \rangle$. What is a fast way to compute this?
Since $\langle s_{\lambda /\mu}, s_{\nu} \rangle = \langle s_{\lambda}, s_{\nu}s_{\mu} \rangle$, the $s_{\nu}s_{\mu}$ term could be expanded by using the Littlewood-Richardson coefficients, and then use the fact that the inner product is billinear along with the fact that the Schur functions are an orthonormal basis for the symmetric functions to get the answer. However, this seems pretty slow to me. Is there a better way to do this?