Eulerian digraphs and Dyck words, a bijection

TL;DR

This paper introduces a bijection between Dyck words and Eulerian digraphs using a novel combinatorial structure called Dyck matrix, supported by algorithms for both directions.

Contribution

It presents the first explicit bijection between Dyck words and Eulerian digraphs via Dyck matrices, with algorithms for conversion.

Findings

Established a bijection between Dyck words and Eulerian digraphs.

Introduced Dyck matrices as a new combinatorial structure.

Provided algorithms for converting between Dyck words and Eulerian digraphs.

Abstract

The main goal of this work is to establish a bijection between Dyck words and a family of Eulerian digraphs. We do so by providing two algorithms implementing such bijection in both directions. The connection between Dyck words and Eulerian digraphs exploits a novel combinatorial structure: a binary matrix, we call Dyck matrix, representing the cycles of an Eulerian digraph.