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.

中文翻译：

---

Wronskian Hermite 多项式的系数

我们研究了由分区标记的 Hermite 多项式的 Wronskians，并使用核和商的组合概念来推导出其系数的显式表达式。这些系数可以用对称群的不可约表示的特征来表示，也可以用钩长度来表示。此外，当核的长度趋于无穷大时，我们推导出 Wronskian Hermite 多项式的渐近行为，同时固定商。通过这种组合设置，我们以自然的方式获得了 Hermite 和 Laguerre 多项式到 Wronskian Hermite 多项式和包含 Laguerre 多项式的 Wronskian 之间的对应关系的推广。最后，我们将大部分结果推广到 $p$-star 上为零的多项式。