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Maximal degree subposets of $ν$-Tamari lattices

Aram Dermenjian

TL;DR

This work investigates two natural subposets of ν-Tamari lattices corresponding to maximal out-degree and maximal in-degree, revealing deep structure through area-vector data. For m-Dyck paths of height $n$, the maximal-out subposet is isomorphic to the $ u'$-Tamari lattice with $ u'=(NE^{m-1})^n$, while the maximal-in subposet is isomorphic to $(m-1)$-Dyck paths endowed with a greedy order, via explicit bijections that shift staircase-structure by subtracting a maximal staircase or applying a greedy transformation. The paper also develops a parallel theory for arbitrary ν, showing that the elegant bijections can fail in general but hold in key families, and it establishes precise isomorphisms through left/right area vectors and the associated staircase-structure maps. These results connect Tamari-like lattices with Greedy-order posets, providing new tools to analyze subposets in ν-Tamari lattices and highlighting when such correspondences extend beyond the classical staircase settings. Overall, the findings give sharp characterizations of maximal-degree subposets and illuminate the combinatorial geometry underpinning ν-Tamari lattices with potential applications to diagonal harmonics and related algebraic structures.

Abstract

In this paper, we study two different subposets of the $ν$-Tamari lattice: one in which all elements have maximal in-degree and one in which all elements have maximal out-degree. The maximal in-degree and maximal out-degree of a $ν$-Dyck path turns out to be the size of the maximal staircase shape path that fits weakly above $ν$. For $m$-Dyck paths of height $n$, we further show that the maximal out-degree poset is poset isomorphic to the $ν$-Tamari lattice of $(m-1)$-Dyck paths of height $n$, and the maximal in-degree poset is poset isomorphic to the $(m-1)$-Dyck paths of height $n$ together with a greedy order. We show these two isomorphisms and give some properties on $ν$-Tamari lattices along the way.

Maximal degree subposets of $ν$-Tamari lattices

TL;DR

This work investigates two natural subposets of ν-Tamari lattices corresponding to maximal out-degree and maximal in-degree, revealing deep structure through area-vector data. For m-Dyck paths of height , the maximal-out subposet is isomorphic to the -Tamari lattice with , while the maximal-in subposet is isomorphic to -Dyck paths endowed with a greedy order, via explicit bijections that shift staircase-structure by subtracting a maximal staircase or applying a greedy transformation. The paper also develops a parallel theory for arbitrary ν, showing that the elegant bijections can fail in general but hold in key families, and it establishes precise isomorphisms through left/right area vectors and the associated staircase-structure maps. These results connect Tamari-like lattices with Greedy-order posets, providing new tools to analyze subposets in ν-Tamari lattices and highlighting when such correspondences extend beyond the classical staircase settings. Overall, the findings give sharp characterizations of maximal-degree subposets and illuminate the combinatorial geometry underpinning ν-Tamari lattices with potential applications to diagonal harmonics and related algebraic structures.

Abstract

In this paper, we study two different subposets of the -Tamari lattice: one in which all elements have maximal in-degree and one in which all elements have maximal out-degree. The maximal in-degree and maximal out-degree of a -Dyck path turns out to be the size of the maximal staircase shape path that fits weakly above . For -Dyck paths of height , we further show that the maximal out-degree poset is poset isomorphic to the -Tamari lattice of -Dyck paths of height , and the maximal in-degree poset is poset isomorphic to the -Dyck paths of height together with a greedy order. We show these two isomorphisms and give some properties on -Tamari lattices along the way.
Paper Structure (18 sections, 27 theorems, 36 equations, 3 figures, 1 table)

This paper contains 18 sections, 27 theorems, 36 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Tamari orders
  3. v-Dyck paths
  4. v-Tamari order
  5. Reversing the v-Tamari order
  6. Subposets of the Tamari poset
  7. Area functions
  8. Staircase shape v-Dyck paths
  9. Two subposets of v-Tamari lattices
  10. Out-degree
  11. In-degree
  12. Isomorphisms
  13. The out-degree poset for m-Dyck paths of height n
  14. The in-degree poset for m-Dyck paths of height n
  15. G to T
  16. ...and 3 more sections

Key Result

lemma 1

Let $D$ be a $\nu$-Dyck path and $\delta_{i}\left( D \right)$ be defined as above. Let $r_i^D$, $t_i^D$, and $h_i^D$ be the $i$-th right hand point, touch point and hit point of $D$ respectively. Then $\delta_{i}\left( D \right)$ exists and $\delta_{i}\left( D \right) \cover_T D$ if and only if $t_i

Figures (3)

  • Figure 1: The path on the bottom is $D$ and the path on top is $\gamma_{i}\left( D \right)$ where $D \cover_G \gamma_{i}\left( D \right)$. The dashed lines indicate points with the same horizontal distance. The path $\nu$ is not shown in this figure.
  • Figure 2: In this figure there are two paths, $D$ (below) and $\gamma_{i}\left( D \right)$ (above) where $D \cover_G \gamma_{i}\left( D \right)$. The two paths coincide on every point before $x_i^D$ and after $h_i^D$.
  • Figure 3: An example of the algorithm for $\bar{\phi}$ on the left and for $\phi$ on the right.

Theorems & Definitions (50)

  • remark 1
  • lemma 1
  • proof
  • lemma 2
  • proof
  • proposition 1
  • proof
  • corollary 1
  • proof
  • lemma 3
  • ...and 40 more