Bijections between Variants of Dyck Paths and Integer Compositions

TL;DR

This paper establishes bijections between various lattice path variants and integer compositions of n, revealing new combinatorial connections and congruence properties, all counted by 4^{n-1}.

Contribution

It introduces novel bijections linking lattice path variants and integer compositions, uncovering new combinatorial relationships and congruence results.

Findings

Bijections between lattice paths and compositions are established.

New congruence properties of these combinatorial objects are discovered.

Enumeration of these objects is unified under the formula 4^{n-1}.

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.