Let $\lambda=(\lambda_1,\ldots,\lambda_n)$ be a partition of $d$. Then hook length formula gives us
$$dim(\lambda)=\chi_{1^d}^{\lambda}=\frac{d!}{\prod h_{\lambda}(i,j)}$$ where $\chi_{a}^{b}$ denote the character of symmetric group.
Say $d$ is even,does there exists hook-length formula for all $\lambda$ $$\chi_{2^{d/2}}^{\lambda}? $$