On posets and polytopes attached to arbors

Starting from the data of an arbor, which is a rooted tree with vertices decorated by disjoint sets, we introduce a lattice polytope and a partial order on its lattice points. We give recursive formulas for various classical invariants of these polytopes and posets, using the tree structure. For linear arbors, we propose a conjecture exchanging the Ehrhart polynomial of the polytope with the Zeta polynomial of the poset for the reverse arbor. The general motivation comes from a transmutation operator acting on $M$-triangles, which should link the posets considered here with some kind of generalized noncrossing partitions and generalized associahedra. We give some evidence for this relationship in several cases, including notably some polytopes, namely halohedra and Hochschild polytopes.