2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/97640We 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.18 pages, 9 figuresCombinatoricsA simple proof of associativity and commutativity of LR-coefficients (or the hive ring)text