Zeros of Wronskians of Hermite polynomials and Young diagrams

For a certain class of partitions, a simple qualitative relation is observed between the shape of the Young diagram and the pattern of zeros of the Wronskian of the corresponding Hermite polynomials. In the case of the two-term Wronskian $W(H_n,H_{n+k})$, we give an explicit formula for the asymptotic shape of the zero set as $n\to \infty$. Some empirical asymptotic formulas are given for the zero sets of three-term and four-term Wronskians.