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

Cesar Ceballos, Viviane Pons

TL;DR

This work develops the geometric framework for the $s$-permutahedron and the $s$-associahedron by realizing the $s$-weak order and the $s$-Tamari lattice as edge graphs of polytopal complexes. It defines the $s$-permutahedron as the complex of pure intervals of the $s$-weak order, derives its $f$-polynomial via $f_s(t)=\sum_T (1+t)^{asc(T)}$, and proves it is a combinatorial complex with the intersection of faces again a face, all while characterizing pure intervals through variations and essential variations. It then introduces the $s$-associahedron as the complex of pure $s$-Tamari intervals, shows its isomorphism with the $\nu$-associahedron, and provides an $f$-polynomial and $s$-Narayana numbers, confirming a polytopal realization in key cases. The paper closes with three polytopality conjectures, discusses potential generalizations to other finite Coxeter groups, and outlines connections to recent work on flow polytopes, offering substantial groundwork for future geometric/topological explorations in these generalized Tamari-type structures.

Abstract

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.

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

TL;DR

This work develops the geometric framework for the -permutahedron and the -associahedron by realizing the -weak order and the -Tamari lattice as edge graphs of polytopal complexes. It defines the -permutahedron as the complex of pure intervals of the -weak order, derives its -polynomial via , and proves it is a combinatorial complex with the intersection of faces again a face, all while characterizing pure intervals through variations and essential variations. It then introduces the -associahedron as the complex of pure -Tamari intervals, shows its isomorphism with the -associahedron, and provides an -polynomial and -Narayana numbers, confirming a polytopal realization in key cases. The paper closes with three polytopality conjectures, discusses potential generalizations to other finite Coxeter groups, and outlines connections to recent work on flow polytopes, offering substantial groundwork for future geometric/topological explorations in these generalized Tamari-type structures.

Abstract

This paper introduces the geometric foundations for the study of the -permutahedron and the -associahedron, two objects that encode the underlying geometric structure of the -weak order and the -Tamari lattice. We introduce the -permutahedron as the complex of pure intervals of the -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 -associahedron as the complex of pure -Tamari intervals of the -Tamari lattice, show some enumerative results, and prove that it is isomorphic to a well chosen -associahedron. Finally, we present three polytopality conjectures, evidence supporting them, and some hints about potential generalizations to other finite Coxeter groups.
Paper Structure (6 sections, 3 theorems, 2 equations, 9 figures)

This paper contains 6 sections, 3 theorems, 2 equations, 9 figures.

Table of Contents

  1. The $s$-permutahedron
  2. The $s$-weak order (background)
  3. Lattice definition through $s$-decreasing trees and tree-inversions
  4. Tree-ascents and cover relations
  5. The $s$-permutahedron
  6. Pure intervals of the $s$-weak order

Key Result

Proposition 1.1.2

For $s$ a given weak composition, a multi-set of inversions is the multi-set of tree-inversions of an $s$-decreasing tree $T$ if and only if it is planar, transitive, and included in $\Sigma_s$. We then call it an $s$-tree-inversion set.

Figures (9)

  • Figure 1: The $s$-weak order as the edge graph of the $s$-permutahedron for $s=(0,2,2)$.
  • Figure 2: The $s$-weak order as the edge graph of the $s$-permutahedron for $s=(0,0,2)$.
  • Figure 3: The $s$-Tamari lattice as the edge graph of the $s$-associahedron for $s=(0,2,2)$
  • Figure 4: The $s$-permutahedron and the $s$-associahedron obtained from it by removing certain facets, for $s=(0,2,2,2)$, $s =(0,2,3,2)$, and $s=(0,3,3,3)$.
  • Figure 5: an $s$-decreasing tree and its tree-inversions (Figure 4 of CP22)
  • ...and 4 more figures

Theorems & Definitions (9)

  • Definition 1.1.1: Definitions 1.4 and 1.5 of CP22
  • Proposition 1.1.2: Proposition 1.6 of CP22
  • Definition 1.1.3: Definition 1.12 of CP22
  • Definition 1.1.4: Definition 1.4 of CP22
  • Definition 1.1.5: CP22
  • Theorem 1.1.6: Theorem 1.21 of CP22
  • Definition 1.1.7: Definition 1.24 from CP22
  • Proposition 1.1.8: Proposition 1.27 of CP22
  • Definition 1.2.1