# Motzkin path on RNA abstract shapes

## TL;DR

This paper explores the combinatorial structures of RNA abstract shapes using Motzkin paths and Narayana numbers, providing new identities and classifications that enhance understanding of RNA shape enumeration.

## Contribution

It introduces a novel connection between RNA shapes and 2-Motzkin paths, offering combinatorial interpretations and asymptotic classifications of RNA shapes.

## Key findings

- Derived generating functions using Narayana numbers and 2-Motzkin paths.
- Established a correspondence between 1-Motzkin paths and RNA shapes.
- Classified RNA shapes by number of components and analyzed their asymptotic distributions.

## Abstract

We consider a certain abstract of RNA secondary structures, which is closely related to RNA shapes. The generating function counting the number of the abstract structures is obtained by means of Narayana numbers and 2-Motzkin paths, through which we provide an identity related to Narayana numbers and Motzkin polynomials. Furthermore, we show that a combinatorial interpretation on 2-Motzkin paths leads to the correspondence between 1-Motkzin paths and RNA shapes, which facilitates probing further classifications or abstractions of the shapes. In this paper, we classify the shapes with respect to the number of components and calculate their asymptotic distributions.