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The $s$-weak order and $s$-permutahedra I: combinatorics and lattice structure

Cesar Ceballos, Viviane Pons

TL;DR

This work generalizes the classical weak order and Tamari lattices by introducing the $s$-weak order and $s$-Tamari lattice, indexed by a weak composition $s$. It builds the combinatorial framework of $s$-decreasing trees and $s$-tree inversions, proving the $s$-weak order is a polygonal, congruence-uniform lattice and describing its cover relations via $s$-tree rotations. The $s$-Tamari lattice emerges as a sublattice (and a quotient when $s$ has no zeros) of the $s$-weak order and is shown to be isomorphic to a $\nu$-Tamari lattice, connecting these generalized Tamari-type objects to $\nu$-Catalan theory; the paper also notes the $s$-weak order has $s!$ elements, with $s! := 1\cdot (s(n)+1)\cdot (s(n-1)+s(n)+1)\cdots (s(2)+\dots+s(n)+1)$. Overall, the results lay a solid combinatorial and lattice-theoretic foundation for generalized permutahedra-like and associahedron-like structures, with geometric realizations to be explored in follow-up work.

Abstract

This is the first contribution of a sequence of papers introducing the notions of $s$-weak order and $s$-permutahedra, certain discrete objects that are indexed by a sequence of non-negative integers $s$. In this first paper, we concentrate purely on the combinatorics and lattice structure of the $s$-weak order, a partial order on certain decreasing trees which generalizes the classical weak order on permutations. In particular, we show that the $s$-weak order is a semidistributive and congruence uniform lattice, generalizing known results for the classical weak order on permutations. Restricting the $s$-weak order to certain trees gives rise to the $s$-Tamari lattice, a sublattice which generalizes the classical Tamari lattice. We show that the $s$-Tamari lattice can be obtained as a quotient lattice of the $s$-weak order when $s$ has no zeros, and show that the $s$-Tamari lattices (for arbitrary $s$) are isomorphic to the $ν$-Tamari lattices of Préville-Ratelle and Viennot. The underlying geometric structure of the $s$-weak order will be studied in a sequel of this paper, where we introduce the notion of $s$-permutahedra.

The $s$-weak order and $s$-permutahedra I: combinatorics and lattice structure

TL;DR

This work generalizes the classical weak order and Tamari lattices by introducing the -weak order and -Tamari lattice, indexed by a weak composition . It builds the combinatorial framework of -decreasing trees and -tree inversions, proving the -weak order is a polygonal, congruence-uniform lattice and describing its cover relations via -tree rotations. The -Tamari lattice emerges as a sublattice (and a quotient when has no zeros) of the -weak order and is shown to be isomorphic to a -Tamari lattice, connecting these generalized Tamari-type objects to -Catalan theory; the paper also notes the -weak order has elements, with . Overall, the results lay a solid combinatorial and lattice-theoretic foundation for generalized permutahedra-like and associahedron-like structures, with geometric realizations to be explored in follow-up work.

Abstract

This is the first contribution of a sequence of papers introducing the notions of -weak order and -permutahedra, certain discrete objects that are indexed by a sequence of non-negative integers . In this first paper, we concentrate purely on the combinatorics and lattice structure of the -weak order, a partial order on certain decreasing trees which generalizes the classical weak order on permutations. In particular, we show that the -weak order is a semidistributive and congruence uniform lattice, generalizing known results for the classical weak order on permutations. Restricting the -weak order to certain trees gives rise to the -Tamari lattice, a sublattice which generalizes the classical Tamari lattice. We show that the -Tamari lattice can be obtained as a quotient lattice of the -weak order when has no zeros, and show that the -Tamari lattices (for arbitrary ) are isomorphic to the -Tamari lattices of Préville-Ratelle and Viennot. The underlying geometric structure of the -weak order will be studied in a sequel of this paper, where we introduce the notion of -permutahedra.
Paper Structure (8 sections, 17 theorems, 13 equations, 11 figures)

This paper contains 8 sections, 17 theorems, 13 equations, 11 figures.

Table of Contents

  1. The $s$-weak order
  2. $s$-permutations
  3. $s$-decreasing trees
  4. The $s$-weak order or the lattice of $s$-decreasing trees
  5. Cover relations
  6. The $s$-weak order is polygonal, semidistributive and congruence uniform
  7. The $s$-Tamari lattice
  8. The sublattice of $s$-Tamari trees

Key Result

Proposition 1.6

For $s$ a given weak composition, the multiset of tree-inversions of any $s$-decreasing tree is an $s$-tree-inversion set. Moreover, this gives a bijection between $s$-decreasing trees and $s$-tree-inversions sets.

Figures (11)

  • Figure 1: The $s$-weak order and the $s$-Tamari lattice for $s=(0,2,2)$. Their Hasse diagrams are the edge graphs of the $s$-permutahedron and the $s$-associahedron.
  • Figure 2: Geometric realizations of the 3-dimensional $s$-permutahedron and $s$-associahedron, for $s=(0,2,2,2)$. Their edge graphs realize the Hasse diagrams of the $s$-weak order and the $s$-Tamari lattice, respectively.
  • Figure 3: Bijection from $s$-decreasing trees to $121$-avoiding $s$-permutations, with $s = (1,1,2,2)$
  • Figure 4: an $s$-decreasing tree and its tree-inversions
  • Figure 5: Illustration of the transitivity and planarity conditions on $s$-tree inversion sets.
  • ...and 6 more figures

Theorems & Definitions (55)

  • Definition 1.1
  • Remark 1.2
  • Definition 1.3: Tree-inversions
  • Definition 1.4
  • Definition 1.5
  • Proposition 1.6
  • Lemma 1.7
  • proof : Proof of Proposition \ref{['prop:tree-inversions-bij']}
  • Definition 1.9
  • Lemma 1.10
  • ...and 45 more