  In this note, we apply combinatorial techniques from our Ph.D. thesis to
study how generalized permutohedra may be represented functionally on
Parke-Tayor factors and related rational functions. In any functional
representation of polyhedral cones, in general certain homological information
may be lost. The combinatorial relations of the Parke-Taylor factors lift
homologically to generalized permutohedra.
  The 6-particle case contains several related layers of interesting geometric
data: the Newton polytope for the polynomial numerator lifts the permutohedron
in three variables, which is a hexagon, and the fraction itself provides a
functional representation of certain neighborhoods of a vertex of a
5-dimensional weight permutohedron. The lift from fraction to generalized
permutohedron was derived by comparing functional representations. We observe
additionally that the numerator and its permutations satisfy a degree 3
polynomial relation which defines a classical projective variety known as the
Segre cubic.
  We include in an extended Appendix selected Mathematica code which can be
used to verify our computations independently.
