Let $(G,K)$ be a Hermitian symmetric pair and let $\g$ and $\k$ denote the corresponding complexified Lie algebras. Let $\g=\k \oplus \p^+\oplus \p^-$ be the usual decomposition of $\g$ as a $\k$-module. $K$ acts on the symmetric algebra $S(\p^-)$. We determine the $K$-structure of all $K$-stable ideals of the algebra. Our results resemble the Pieri rule for Young diagrams. The result implies a branching rule for a class of finite dimensional representations that appear in the work of Enright and Willenbring (preprint, 2001) and Enright and Hunziker (preprint, 2002) on Hilbert series for unitarizable highest weight modules.