  We extend the works of Loday-Ronco and Burgunder-Ronco on the tridendriform
decomposition of the shuffle product on the faces of associahedra and
permutohedra, to other families of hypergraph polytopes (or nestohedra),
including simplices, hypercubes and some new families. We also extend the
shuffle product to take more than two arguments, and define accordingly a new
algebraic structure, that we call polydendriform, from which the original
tridendriform equations can be crisply synthesized.
