Generating function for Naturalized Series: The case of Ordered Motzkin Words

TL;DR

This paper derives a generating function for a restricted class of Motzkin words, which are balanced parentheses with zeros, under specific constraints like no leading zeros, expanding combinatorial enumeration methods.

Contribution

It introduces a new generating function for modified Motzkin words with restrictions, advancing the analytical tools for combinatorial object enumeration.

Findings

Derived the generating function for restricted Motzkin words

Analyzed the combinatorial properties of zero-restricted Motzkin words

Extended the application of generating functions to new combinatorial constraints

Abstract

We continue to consider the ordered lexicographic sequence, which is constructed according to the formal characteristics of a series of natural numbers. For analysis, we selected balanced parentheses with zeros, Motzkin words. As you know, generating functions allow you to work with combinatorial objects by analytical methods. Motzkin words are enumerated by Motzkin numbers, for the generation of which there is a corresponding generating function. In our case, restrictions are imposed on Motzkin words, for example, there are no leading zeros in bracket sets. The purpose of this article is to obtain the generating function of such modified Motzkin words.