Past exam of the Lie algebras and representation theory of symmetric groups course of the University of Oxford / 2010 / B / 6 / 2 / Solution

Irreducible character orthogonality gives $⟨\chi ,\chi ⟩=1$. Under the proposed vanishing condition, only the identity contributes, soBut $\chi (1)$ is an integer dimension and $720$ is not a square. Thus no such partition exists.