The s-Weak Order and s-Permutahedra II: The Combinatorial Complex of Pure Intervals

This paper introduces the geometric foundations for the study of the s-permutahedron and the s-associahedron, two objects that encode the underlying geometric structure of the s-weak order and the s-Tamari lattice. We introduce the s-permutahedron as the complex of pure intervals of the s-weak order, present enumerative results about its number of faces, and prove that it is a combinatorial complex. This leads, in particular, to an explicit combinatorial description of the intersection of two faces. We also introduce the s-associahedron as the complex of pure s-Tamari intervals of the s-Tamari lattice, show some enumerative results, and prove that it is isomorphic to a well chosen ν-associahedron. Finally, we present three polytopality conjectures, evidence supporting them, and some hints about potential generalizations to other finite Coxeter groups.