Generalised permutohedra include the polytopes created from shrinking the square faces: so the associahedron and permutohedron in three dimensions become the $5$ and $6$ point pyramids.
The square pyramid, or octahedron, was the matroid polytope considered in MacPherson's classic paper on combinatorial differential manifolds. This is a polytope recently mentioned by Arkani-Hamed at KITP. The operad tree labels can, unsurprisingly, greatly clarify the connection to tree (and loop) diagrams for physics, since they are essentially the same thing. By removing the squares on the associahedron, we are throwing away the rooted trees with two ternary nodes, labeled by hexagons with a single chord. The permutohedron extends the associahedron by labeling trees with ordered nodes, which is the same as permuting the order of the areas between adjacent branches on the tree. Thus all matroid polytopes may be given additional labels using operads.
6 years ago