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The ascent lattice on Dyck paths

Jean-Luc Baril, Mireille Bousquet-Mélou, Sergey Kirgizov, Mehdi Naima

TL;DR

This work introduces and analyzes the ascent lattice on Dyck paths, defining a greedy cover relation and proving it yields a lattice that interplays with the Nadeau–Tewari NT poset. It provides exact and asymptotic enumerations for intervals via a two-catalytic-variable framework and invariant methods tied to quadrant-walk models. In the base case m=1, the interval generating function is algebraic of degree 3, while for m>1 the generating functions fail to be D-finite, with growth governed by quadrant-walk asymptotics. The study reveals deep connections to sylvester classes of $m$-parking functions and to walks confined to a quadrant, highlighting rich algebraic and analytic structure with broader implications in lattice-path combinatorics and poset theory.

Abstract

In the Stanley lattice defined on Dyck paths of size $n$, cover relations are obtained by replacing a valley $DU$ by a peak $UD$. We investigate a greedy version of this lattice, first introduced by Chenevière, where cover relations replace a factor $DU^k D$ by $U^kD^2$. By relating this poset to another poset recently defined by Nadeau and Tewari, we prove that this still yields a lattice, which we call the ascent lattice, $L_n$. We then count intervals in $L_n$. Their generating function is found to be algebraic of degree $3$. The proof is based on a recursive decomposition of intervals involving two catalytic parameters. The solution of the corresponding functional equation is inspired by recent work on the enumeration of walks confined to a quadrant. We also consider the order induced in $L_{mn}$ on $m$-Dyck paths, that is, paths in which all ascent lengths are multiples of $m$, and on mirrored $m$-Dyck paths, in which all descent lengths are multiples of $m$. The first poset $L_{m,n}$ is still a lattice for any $m$, while the second poset $L'_{m,n}$ is only a join semilattice when $m>1$. In both cases, the enumeration of intervals is still described by an equation in two catalytic variables. Interesting connections arise with the sylvester congruence of Hivert, Novelli and Thibon, and again with walks confined to a quadrant. We combine the latter connection with probabilistic results to give asymptotic estimates of the number of intervals in both $L_{m,n}$ and $L'_{m,n}$. Their form implies that the generating functions of intervals are no longer algebraic, nor even D-finite, when $m>1$.

The ascent lattice on Dyck paths

TL;DR

This work introduces and analyzes the ascent lattice on Dyck paths, defining a greedy cover relation and proving it yields a lattice that interplays with the Nadeau–Tewari NT poset. It provides exact and asymptotic enumerations for intervals via a two-catalytic-variable framework and invariant methods tied to quadrant-walk models. In the base case m=1, the interval generating function is algebraic of degree 3, while for m>1 the generating functions fail to be D-finite, with growth governed by quadrant-walk asymptotics. The study reveals deep connections to sylvester classes of -parking functions and to walks confined to a quadrant, highlighting rich algebraic and analytic structure with broader implications in lattice-path combinatorics and poset theory.

Abstract

In the Stanley lattice defined on Dyck paths of size , cover relations are obtained by replacing a valley by a peak . We investigate a greedy version of this lattice, first introduced by Chenevière, where cover relations replace a factor by . By relating this poset to another poset recently defined by Nadeau and Tewari, we prove that this still yields a lattice, which we call the ascent lattice, . We then count intervals in . Their generating function is found to be algebraic of degree . The proof is based on a recursive decomposition of intervals involving two catalytic parameters. The solution of the corresponding functional equation is inspired by recent work on the enumeration of walks confined to a quadrant. We also consider the order induced in on -Dyck paths, that is, paths in which all ascent lengths are multiples of , and on mirrored -Dyck paths, in which all descent lengths are multiples of . The first poset is still a lattice for any , while the second poset is only a join semilattice when . In both cases, the enumeration of intervals is still described by an equation in two catalytic variables. Interesting connections arise with the sylvester congruence of Hivert, Novelli and Thibon, and again with walks confined to a quadrant. We combine the latter connection with probabilistic results to give asymptotic estimates of the number of intervals in both and . Their form implies that the generating functions of intervals are no longer algebraic, nor even D-finite, when .
Paper Structure (26 sections, 22 theorems, 123 equations, 10 figures)

This paper contains 26 sections, 22 theorems, 123 equations, 10 figures.

Table of Contents

  1. Introduction and main results
  2. Ascent posets
  3. Definitions
  4. Dyck paths.
  5. Orders on Dyck paths.
  6. Formal power series.
  7. A characterisation of the ascent order
  8. The posets $\mathbb{D}_{m,n}$ and $\mathbb{D}_{m,n}'$
  9. Lattice properties
  10. Intervals in the Nadeau-Tewari poset and classes of the sylvester congruence
  11. Recursive construction of intervals in $\mathbb{D}_{m,n}$ and $\mathbb{D}'_{m,n}$
  12. Intervals in $\mathbb{D}_{m,n}$
  13. Intervals in $\mathbb{D}'_{m,n}$
  14. Exact enumeration of intervals in $\mathbb{D}_n$
  15. The first ascent of $\bm P$, and the statistics $\bm{r(P,Q)}$
  16. ...and 11 more sections

Key Result

Theorem 1

Let $g(n)$ be the number of intervals in the ascent lattice $\mathbb{D}_n$. The associated generating function $G :=\sum_{n\geqslant 1} g(n) t^n$ is where $Z$ is the only formal power series in $t$ satisfying $Z=t(1+Z)(1 + 2Z)^2$. In particular, the series $G$ is algebraic of degree $3$ over $\mathbb{Q}(t)$. As $n$ tends to infinity, the number of intervals in $\mathbb{D}_n$ is equivalent to

Figures (10)

  • Figure 1: Subposet-inclusion structure of some orders on Dyck paths.
  • Figure 2: A cover relation between two Dyck paths of size $6$.
  • Figure 3: The Hasse diagram of $\mathbb{D}_4=\mathbb{D}_{1,4}=\mathbb{D}'_{1,4}$.
  • Figure 4: From left to right, a 2-Dyck path of size $3$ (element of $\mathcal{D}_{2,3}$), and a mirrored 2-Dyck path of size $3$ (element of $\mathcal{D}'_{2,3}$).
  • Figure 5: On the left, the Hasse diagram of $\mathbb{D}_{2,3}$, and on the right, the Hasse diagram of $\mathbb{D}'_{2,3}$.
  • ...and 5 more figures

Theorems & Definitions (49)

  • Theorem 1
  • Proposition 2
  • Proposition 3
  • Proposition 4
  • proof
  • Corollary 5
  • Remark 6
  • Proposition 7
  • proof
  • Definition 8
  • ...and 39 more