Counting irreducible representations of general linear groups and unitary groups

Let $G$ be a general linear group over $\text{BR}$, $\text{BC}$, or $\text{BH}$, or a real unitary group. In this paper, we precisely describe the number of isomorphism classes of irreducible Casselman-Wallach representations of $G$ with a given infinitesimal character and a given associated variety, expressed in terms of certain combinatorial data called painted Young diagrams and assigned Young diagrams.