  We study Wronskians of Hermite polynomials labelled by partitions and use the
combinatorial concepts of cores and quotients to derive explicit expressions
for their coefficients. These coefficients can be expressed in terms of the
characters of irreducible representations of the symmetric group, and also in
terms of hook lengths. Further, we derive the asymptotic behaviour of the
Wronskian Hermite polynomials when the length of the core tends to infinity,
while fixing the quotient. Via this combinatorial setting, we obtain in a
natural way the generalization of the correspondence between Hermite and
Laguerre polynomials to Wronskian Hermite polynomials and Wronskians involving
Laguerre polynomials. Lastly, we generalize most of our results to polynomials
that have zeros on the $p$-star.
