The s-weak order and s-permutahedra II: the combinatorial complex of pure intervals

Cesar Ceballos , Viviane Pons (1, 2, 3, 4)

1 LISN - Laboratoire Interdisciplinaire des Sciences du Numérique
2 GALaC - Graphes, Algorithmes et Combinatoire - LISN
3 CRM - Centre de recherches mathématiques (IRL3457)
4 IRL CRM-CNRS - International Research Lab

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.

Keywords: Weak order; Permutahedron; Tamari lattice; Associahedron

Domains: Combinatorics [math.CO]

Cesar Ceballos, Viviane Pons. The s-weak order and s-permutahedra II: the combinatorial complex of pure intervals. 2023. ⟨hal-04390514⟩