Naturalized bracket row and Motzkin triangle

TL;DR

This paper explores the structure of Motzkin words and their relation to natural numbers, introducing a novel graphical approach to analyze the Motzkin triangle and its decomposition properties.

Contribution

It presents a new graphical method for constructing and analyzing the Motzkin triangle based on bracket pair decompositions and natural number features.

Findings

Decomposition of parentheses into matched pairs analogous to prime numbers

Construction of a point triangle of bracket pairs

Development of a numerical Motzkin triangle using stars and bars

Abstract

The paper considers the sequence of the Motzkin words, which is constructed according to formal features of natural numbers. We investigate the decomposition of well-formed parentheses into the matched pairs of parentheses (analogous to Prime Numbers). The point triangle of bracket pairs is constructed and studied, as well as the numerical Motzkin triangle (similar to Pascal's triangle). To calculate the Motzkin triangle, the graphical method of stars and bars is used.