Cellular diagonals of permutahedra

Abstract

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.