A visual approach to symmetric chain decompositions of finite Young lattices

The finite Young lattice $L(m, n)$ is rank-symmetric, rank-unimodal, and has the strong Sperner property. R. Stanley further conjectured that $L(m, n)$ admits a symmetric chain order. We show that the order structure on $L(m, n)$ is equivalent to a natural ordering on the lattice points of a dilated $n$-simplex, which in turn corresponds to a weight diagram for the root system of type $A_n$. Lindstr{\" o}m's symmetric chain decompositions for $L(3, n)$ are described completely through pictures.