  We provide a systematic enumerative and combinatorial study of geometric
cellular diagonals on the permutahedra.
  In the first part of the paper, we study the combinatorics of certain
hyperplane arrangements obtained as the union of $\ell$ generically translated
copies of the classical braid arrangement. Based on Zaslavsky's theory, we
derive enumerative results on the faces of these arrangements involving
combinatorial objects named partition forests and rainbow forests. This yields
in particular nice formulas for the number of regions and bounded regions in
terms of exponentials of generating functions of Fuss-Catalan numbers. By
duality, the specialization of these results to the case $\ell = 2$ gives the
enumeration of any geometric diagonal of the permutahedron.
  In the second part of the paper, we study diagonals which respect the
operadic structure on the family of permutahedra. We show that there are
exactly two such diagonals, which are moreover isomorphic. We describe their
facets by a simple rule on paths in partition trees, and their vertices as
pattern-avoiding pairs of permutations. We show that one of these diagonals is
a topological enhancement of the Sanbeblidze-Umble diagonal, and unravel a
natural lattice structure on their sets of facets.
  In the third part of the paper, we use the preceding results to show that
there are precisely two isomorphic topological cellular operadic structures on
the families of operahedra and multiplihedra, and exactly two
infinity-isomorphic geometric universal tensor products of homotopy operads and
A-infinity morphisms.
