Dyck Numbers, III. Enumeration and bijection with symmetric Dyck paths

TL;DR

This paper enumerates Dyck numbers, explores their properties, and constructs a bijection with symmetric Dyck paths, revealing an infinite forest structure and providing software tools for analysis.

Contribution

It introduces a new bijection between Dyck numbers and symmetric Dyck paths, and models this relationship as an infinite forest of trees with software support.

Findings

Enumeration of Dyck numbers matching OEIS A036991

Construction of a bijection with symmetric Dyck paths

Representation of the bijection as an infinite forest of trees

Abstract

Dyck paths (also balanced brackets and Dyck words) are among the most heavily studied Catalan families. This paper is a continuation of [2, 3]. In the paper we enumerate the terms of the OEIS A036991, Dyck numbers, and construct a concomitant bijection with symmetric Dyck paths. In the case of binary coding of Dyck paths we work with compact natural numbers after removing leading zeros. Analysis of binary suffixes, allowed us to obtain a bijection between arbitrary A036991 terms and symmetric A036991 terms which encode symmetric Dyck paths. The bijection generates a forest of unary non-intersecting infinite trees. The root of each bijection tree is an asymmetric term; the other nodes are symmetrical. There are an infinite number of such trees. The reader is offered a software package for working with bijection trees.