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Combinatorial Properties of RNA Secondary Structures

The secondary structure of an RNA molecule is of great importance and possesses influence, e.g., on the interaction of tRNA molecules with proteins or on the stabilization of mRNA molecules. The classification of secondary structures by means of their order proved useful with respect to numerous applications. In 1978, Waterman, who gave the first precise formal framework for the topic, suggested to determine the number a(n,p) of secondary structures of size n and given order p. Since then, no satisfactory result has been found. Based on an observation due to Viennot et al., we will derive generating functions for the secondary structures of order p from generating functions for binary tree structures with Horton-Strahler number p. These generating functions enable us to compute a precise asymptotic equivalent for a(n,p). Furthermore, we will determine the related number of structures when the number of unpaired bases shows up as an additional parameter. Our approach proves to be general enough to compute the average order of a secondary structure together with all the r-th moments and to enumerate substructures such as hairpins or bulges in dependence on the order of the secondary structures considered.