I will present a long standing open problem of finding a symmetric chain decomposition of the poset of partitions in a w x h rectangle. I will show how this problem is related to that of finding a combinatorial interpretation for the plethysm coefficients of an SL_2(C) representation and show that certain generating functions of these plethysm coefficients are rational. Finally, I will relate this problem to counting integer points in a polytope. This is joing work with Álvaro Gutiérrez, Rosa Orellana, Franco Saliola and Anne Schilling