  Consider the symmetric pair $(\textbf{GL}_n, \textbf{O}_n)$ in the setting of
the Kostant-Rallis Theorem. We provide a combinatorial formula for the graded
multiplicity of an irreducible $\textbf{O}_n$-representation in the harmonic
polynomials on $\mathfrak{p}$. The results here are expected to extend to the
other classical symmetric pairs.
