A simple proof of associativity and commutativity of LR-coefficients (or   the hive ring)
  We give a simple bijective proof of associativity and commutativity of the Littlewood-Richardson coefficients or the hive ring. Specifically, we establish existence a polarized polymatroidal discretely concave functions on the tetrahedron with given boundary values at two adjoint faces. 
V. Danilov and G. Koshevoy