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A lattice on Dyck paths close to the Tamari lattice

Jean-Luc Baril, Sergey Kirgizov, Mehdi Naima

TL;DR

The paper defines a new poset on Dyck paths, the pyramid lattice, via the restricted Tamari-like covering $DU^kD^k \rightarrow U^kD^kD$ and proves this transitive closure yields a lattice. It develops a rich enumeration framework: a trivariate generating function $A(x,y,z)$ for semilength and edge statistics $(s,t)$ that is symmetric in $y$ and $z$, a total-edge function $E(x)$, and closed forms for meet/join irreducibles; it also constructs an involution $\phi$ that swaps $(s,t)$. Intervals are counted through generating functions $I(x,y)$ and $J(x,y)$ using a kernel method, yielding explicit formulas and asymptotics, and the results are contrasted with the Tamari lattice. The work closes with several open questions and directions, including generalizations to pattern-avoiding Tamari posets and $m$-Dyck paths, and connections to outerplanar maps and Möbius functions, highlighting the structural richness and potential applications of this near-Tamari lattice. $\text{(Key results include: }J(x,1)=\frac{1-\sqrt{1-8x}}{4},\ I(x,1)=\frac{1-2x-\sqrt{1-8x}}{2(x+1)})$.

Abstract

We introduce a new poset structure on Dyck paths where the covering relation is a particular case of the relation inducing the Tamari lattice. We prove that the transitive closure of this relation endows Dyck paths with a lattice structure. We provide a trivariate generating function counting the number of Dyck paths with respect to the semilength, the numbers of outgoing and incoming edges in the Hasse diagram. We deduce the numbers of coverings, meet and join irreducible elements. As a byproduct, we present a new involution on Dyck paths that transports the bistatistic of the numbers of outgoing and incoming edges into its reverse. Finally, we give a generating function for the number of intervals, and we compare this number with the number of intervals in the Tamari lattice.

A lattice on Dyck paths close to the Tamari lattice

TL;DR

The paper defines a new poset on Dyck paths, the pyramid lattice, via the restricted Tamari-like covering and proves this transitive closure yields a lattice. It develops a rich enumeration framework: a trivariate generating function for semilength and edge statistics that is symmetric in and , a total-edge function , and closed forms for meet/join irreducibles; it also constructs an involution that swaps . Intervals are counted through generating functions and using a kernel method, yielding explicit formulas and asymptotics, and the results are contrasted with the Tamari lattice. The work closes with several open questions and directions, including generalizations to pattern-avoiding Tamari posets and -Dyck paths, and connections to outerplanar maps and Möbius functions, highlighting the structural richness and potential applications of this near-Tamari lattice. .

Abstract

We introduce a new poset structure on Dyck paths where the covering relation is a particular case of the relation inducing the Tamari lattice. We prove that the transitive closure of this relation endows Dyck paths with a lattice structure. We provide a trivariate generating function counting the number of Dyck paths with respect to the semilength, the numbers of outgoing and incoming edges in the Hasse diagram. We deduce the numbers of coverings, meet and join irreducible elements. As a byproduct, we present a new involution on Dyck paths that transports the bistatistic of the numbers of outgoing and incoming edges into its reverse. Finally, we give a generating function for the number of intervals, and we compare this number with the number of intervals in the Tamari lattice.
Paper Structure (6 sections, 16 theorems, 34 equations, 7 figures)

This paper contains 6 sections, 16 theorems, 34 equations, 7 figures.

Table of Contents

  1. Introduction and motivation
  2. Notation and definitions
  3. Lattice structure of $\mathcal{S}_n=(\mathcal{D}_n,\leq)$.
  4. Coverings, join and meet irreducible elements
  5. Enumeration of intervals
  6. Going further

Key Result

Lemma 1

For $n\geq 2$, any Dyck path $P\in\mathcal{D}_n$, $P\neq U^nD^n$, contains at least one occurrence of $DU^kD^k$ for some $k\geq 1$.

Figures (7)

  • Figure 1: ($i$) corresponds to a covering relation for the Tamari Lattice, while ($ii$) corresponds to the covering relation for the new lattice of this study.
  • Figure 2: The Hasse diagram of $\mathcal{S}_4=(\mathcal{D}_4,\leq)$. The Tamari lattice $\mathcal{T}_4=(\mathcal{D}_4,\preceq)$ can be viewed by considering the dotted edge (in red).
  • Figure 3: An illustration of the construction $Q=UR'DUS_1DS_2\leq UR'DU^kD^kS_2\rightarrow UR'U^kD^kDS_2$ of the case ($ii$) in the proof of Theorem \ref{['tla']}.
  • Figure 4: Bijection between $\mathcal{L}$ and binary words avoiding consecutive 1s except the tail 011.
  • Figure 5: An illustration of the involution $\phi$ for the cases ($ii$)-($v$).
  • ...and 2 more figures

Theorems & Definitions (22)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Lemma 1
  • Lemma 2
  • Proposition 1
  • Proposition 2
  • Corollary 1
  • ...and 12 more