Arithmetization of well-formed parenthesis strings. Motzkin Numbers of the Second Kind

TL;DR

This paper develops an arithmetization method for well-formed parenthesis strings and Motzkin paths, linking them to natural numbers and exploring special number sequences related to these structures.

Contribution

It introduces a novel arithmetization approach for parenthesis strings and Motzkin paths, connecting formal language structures to number theory.

Findings

Constructed a Motzkin series approximating natural numbers

Encoded parenthesis strings using ternary codes and natural numbers

Identified special numbers like $3^n+2$ and mirror numbers in the process

Abstract

In this paper, we perform an arithmetization of well-formed parenthesis strings with zeros (Motzkin words) and of corresponding Motzkin paths. The transformations used are reminiscent of G\"odel numbering for mathematical objects of some formal language. We construct a Motzkin series that is as close as possible to natural numbers by many formal features. Parenthesis strings are encoded by ternary codes and corresponding natural numbers, Motzkin numbers of the 2nd kind, which made it possible to formalize and simplify the analysis of the successor function, to specify and clarify the procedure for selecting a successor. In the process of arithmetization of well-wormed parenthesis strings, various special numbers appeared. In this regard, in conclusion we will talk a little about numbers of the form $3^n+2$ and mirror numbers $2*3^n+1$.