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Bijections between Variants of Dyck Paths and Integer Compositions

Manosij Ghosh Dastidar, Michael Wallner

Abstract

We give bijective results between several variants of lattice paths of length $2n$ (or $2n-2$) and integer compositions of n, all enumerated by the seemingly innocuous formula $4^{n-1}$. These associations lead us to make new connections between these objects, such as congruence results.

Bijections between Variants of Dyck Paths and Integer Compositions

Abstract

We give bijective results between several variants of lattice paths of length (or ) and integer compositions of n, all enumerated by the seemingly innocuous formula . These associations lead us to make new connections between these objects, such as congruence results.

Paper Structure

This paper contains 4 sections, 7 theorems, 10 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Bijections involving integer compositions
  3. Bijections involving Dyck paths
  4. Conclusion and Outlook

Key Result

Proposition \oldthetheorem

There exists a natural bijection between pairs of compositions of $n$ and Dyck walks of length $2n-2$.

Figures (3)

  • Figure 1: Bijections proved in this paper of classes of paths and integer compositions that are all enumerated by $4^{n-1}$.
  • Figure 2: A Dyck path with a marked peak (red dot) at height 6 and its image under the bijection from Theorem \ref{['theo:Dyckpeak']} given by a Dyck bridge starting with a $\mathtt d$ step and $5=6-1$ crossings (red dots). The black steps are used in the last-passage (resp., first-passage) decomposition in the proof.
  • Figure 3: A Dyck path with a height-labeled peak (red dot) with label $4$ at height $6$ and its image under the bijection from Proposition \ref{['prop:LabeltoMaxima']} given by a Dyck bridge with marked strict left-to-right maximum (red dot) at height $4$ and $2=6-4$ crossings (red squares).

Theorems & Definitions (14)

  • Proposition \oldthetheorem
  • proof
  • Proposition \oldthetheorem
  • proof
  • Example \oldthetheorem
  • Theorem \oldthetheorem
  • proof
  • Proposition \oldthetheorem
  • proof
  • Proposition \oldthetheorem
  • ...and 4 more