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Progressive and Rushed Dyck Paths

Axel Bacher

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.

Progressive and Rushed Dyck Paths

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.
Paper Structure (5 sections, 4 theorems, 10 equations, 5 figures)

This paper contains 5 sections, 4 theorems, 10 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Bijections
  3. Enumeration and generating functions
  4. Asymptotics
  5. Perspectives

Key Result

Theorem 2

The mapping $F$ is a bijection between Dyck paths of length $2n$ and height $h$ and progressive culminating paths of length $2n+h+1$ and height $h+1$.

Figures (5)

  • Figure 1: Left: a progressive path of height $4$ with the first and second visits at each height marked with a red and blue dot, respectively. Right: a rushed path of height $4$. Note that progressive paths are allowed to visit their maximal height multiple times, while rushed paths are not.
  • Figure 2: Left: a rushed path of height $4$ and semilength $11$. Right: the corresponding one-sided tree of height $3$ with $10$ edges (the leftmost edge generated by the classical bijection is omitted).
  • Figure 3: Left: a Dyck path of height $3$ decomposed as in \ref{['f']}. Right: its image by $F$, a progressive culminating path of height $4$ decomposed as in \ref{['F']}.
  • Figure 4: Above: a rushed Dyck path of height $10$ decomposed as in \ref{['g']}, where $m = 4$ and $j = 6$ (so $A_6$ has height exactly $4$, while $A_7$, $A_8$ and $A_9$ have height at most $4$). Below: its image by $G$, a progressive Dyck path of height $5$ decomposed as in \ref{['G']}.
  • Figure 5: Left: a progressive path. Right: the corresponding acute half-animal: for every site not on the bottom row, there are at least two sites to the left in the row just below.

Theorems & Definitions (7)

  • Definition 1
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • Theorem 4
  • Theorem 5