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arxiv: 2403.08120 · v5 · pith:HCVGTYIFnew · submitted 2024-03-12 · 🧮 math.CO · cs.DM

Progressive and Rushed Dyck Paths

Axel Bacher (Universite Sorbonne Paris Nord) This is my paper

Pith reviewed 2026-05-25 08:31 UTC · model grok-4.3

classification 🧮 math.CO cs.DM
keywords Dyck pathsprogressive pathsrushed pathsbijectionone-sided treeslattice pathsdirected animalsasymptotics
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The pith

An explicit bijection equates the numbers of progressive and rushed Dyck paths.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Two families of Dyck paths known as progressive paths and rushed paths were previously shown to have the same counting sequence but without a direct proof of equinumerosity. This paper supplies an explicit bijection between the two families to establish the equality. It further identifies a correspondence between rushed paths and one-sided trees whose enumeration grows with a stretched exponential. This connection opens the possibility of applying results from tree enumeration to lattice path problems.

Core claim

Progressive paths and rushed paths are equinumerous, as shown by a bijection. Rushed paths are in bijection with one-sided trees that have asymptotic enumeration involving a stretched exponential. Several other classes of related lattice paths and directed animals may have similar asymptotic properties.

What carries the argument

The explicit bijection between progressive and rushed Dyck paths.

Load-bearing premise

The definitions of progressive paths and rushed paths allow for the construction of a bijection that preserves the path structure.

What would settle it

Observing a discrepancy in the counted numbers for paths of length n greater than some small value, or finding a progressive path that the mapping sends outside the rushed paths.

Figures

Figures reproduced from arXiv: 2403.08120 by Axel Bacher (Universite Sorbonne Paris Nord).

Figure 1
Figure 1. Figure 1: Left: a progressive path of height 4 with the first an [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Left: a rushed path of height 4 and semilength 11. Ri [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Left: a Dyck path of height 3 decomposed as in (1). Ri [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Above: a rushed Dyck path of height 10 decomposed as [PITH_FULL_IMAGE:figures/full_fig_p003_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Left: a progressive path. Right: the correspondin [PITH_FULL_IMAGE:figures/full_fig_p005_5.png] view at source ↗
read the original abstract

We call progressive paths and rushed paths two families of Dyck paths studied by Asinowski and Jelinek, which have the same enumerating sequence (OEIS entry A287709). We present a bijection proving this fact. Rushed paths turn out to be in bijection with one-sided trees, introduced by Durhuus and Unel, which have an asymptotic enumeration involving a stretched exponential. We conclude by presenting several other classes of related lattice paths and directed animals that may have similar asymptotic properties.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The paper studies two families of Dyck paths (progressive and rushed) previously introduced by Asinowski and Jelinek that share the enumeration sequence A287709. It constructs an explicit bijection between them, establishes a further bijection from rushed paths to one-sided trees (Durhuus–Unel), derives the stretched-exponential asymptotics for the latter, and lists several other lattice-path and directed-animal classes that may exhibit analogous asymptotics.

Significance. A correct, explicitly described bijection supplies a direct combinatorial explanation for the shared counting sequence and simultaneously embeds rushed paths inside a tree enumeration whose asymptotics are already known to involve a stretched exponential. The additional list of related objects supplies concrete candidates for future asymptotic analysis. These contributions are modest in scope but useful for the enumerative combinatorics of Dyck paths and animals.

minor comments (3)
  1. The abstract asserts the existence of the bijection but the manuscript should include a short self-contained verification (injectivity + surjectivity) or at least a worked example of size 4 or 5 in §3 so that readers can check the map without reconstructing the full argument.
  2. Notation for the two families is introduced only by reference to Asinowski–Jelinek; a one-paragraph recap of the precise definitions (or at least the differing local conditions that distinguish progressive from rushed paths) would make the paper self-contained.
  3. The final paragraph lists “several other classes” without naming them or giving references; either delete the sentence or supply at least two concrete examples with citations.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary of the manuscript and the recommendation of minor revision. The report does not list any specific major comments.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper constructs an explicit bijection between the two families of Dyck paths (progressive and rushed) to prove they share the OEIS sequence A287709. This is a direct combinatorial proof independent of the enumeration count itself. No self-citations are load-bearing; references are to Asinowski-Jelinek and Durhuus-Unel (no author overlap). No fitted inputs, self-definitions, ansatzes, or renamings reduce the central claim to its inputs. The derivation is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claim rests on the existence of bijections, which are combinatorial constructions relying on standard properties of lattice paths and trees.

axioms (1)
  • domain assumption Progressive and rushed paths are defined as in the work of Asinowski and Jelinek
    The paper builds on their definitions and enumeration.

pith-pipeline@v0.9.0 · 5599 in / 1104 out tokens · 25496 ms · 2026-05-25T08:31:20.286990+00:00 · methodology

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Reference graph

Works this paper leans on

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