  Young's partition lattice $L(m,n)$ consists of unordered partitions having
$m$ parts where each part is at most $n$. Using methods from complex algebraic
geometry, R. Stanley proved that $L(m,n)$ is rank-symmetric, unimodal, and
strongly Sperner. Moreover, he conjectured that $L(m,n)$ has a stronger
property called symmetric chain decomposition. Despite many efforts, this
conjecture has only been proved for $\min(m,n)\leq 4$. In this paper, we
decompose $L(m,n)$ into level sets for certain tropical polynomials derived
from the secant varieties of the rational normal curve in projective space, and
we find that the resulting subposets have an elementary raising and lowering
algorithm. As a corollary, we obtain a symmetric chain decomposition for the
subposet of $L(m,n)$ consisting of "sufficiently generic" partitions.
