Tropical decomposition of Young’s partition lattice

Young’s partition lattice L(m,n) consists of integer partitions having m parts where each part is at most n. Using methods from complex algebraic geometry, R. Stanley proved that this poset is rank-symmetric, unimodal, and strongly Sperner. Moreover, he conjectured that it has a symmetric chain decomposition, which is a stronger property. Despite many efforts, this conjecture has only been proved for min(m,n)≤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.