  There are basically two interesting breeds of $E_2$ operads, those that
detect loop spaces and those that solve Deligne's conjecture. The former
deformation retract to Milgram's space obtained by gluing together permutahedra
at their faces. We show how the second breed can be covered by permutahedra as
well. Even more is true, the quotient is actually already an operad up to
homotopy, which induces the operad structure on cellular chains adapted to
prove Deligne's conjecture, while no such structure is known on Milgram's
space. We show, explicitely, that these two quotients are homotopy equivalent.
This gives a new topological proof that operads of this type are indeed of the
right homotopy type. It also furnishes a very nice clean description in terms
of polyhedra, and with it PL topology, for the whole story. The permutahedra
and partial orders play a central role. This, in turn, provides direct links to
other fields of mathematics. We for instance find a new cellular decomposition
of permutahedra using partial orders and that the permutahedra give the cells
for the Dyer--Lashof operations.
