Title: Computing the Lusztig--Vogan Bijection,
Abstract: Let $G$ be a connected complex reductive algebraic group with Lie algebra
$\mathfrak{g}$. The Lusztig--Vogan bijection relates two bases for the bounded
derived category of $G$-equivariant coherent sheaves on the nilpotent cone
$\mathcal{N}$ of $\mathfrak{g}$. One basis is indexed by $\Lambda^+$, the set
of dominant weights of $G$, and the other by $\Omega$, the set of pairs
$(\mathcal{O}, \mathcal{E})$ consisting of a nilpotent orbit $\mathcal{O}
\subset \mathcal{N}$ and an irreducible $G$-equivariant vector bundle
$\mathcal{E} \rightarrow \mathcal{O}$. The existence of the Lusztig--Vogan
bijection $\gamma \colon \Omega \rightarrow \Lambda^+$ was proven by
Bezrukavnikov, and an algorithm computing $\gamma$ in type $A$ was given by
Achar. Herein we present a combinatorial description of $\gamma$ in type $A$
that subsumes and dramatically simplifies Achar's algorithm.