Motzkin path on RNA abstract shapes

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.