Combinatorics on the joint distribution of hairpins and stacks in RNA secondary structures

Enumerative study of RNA secondary structures is one of the most important topics in computational biology. However, most of the existing results are concerned with a single type of structural motifs and are asymptotic. Hairpins and stacks are among the most important motifs in secondary structures. Certain subsets of secondary structures characterized by the number of contained hairpins and the way how these hairpin loops are organized, for instance, cloverleaves (Waterman 1979), have been enumerated in a variety of works, mostly asymptotically. In this paper, we generalize these enumerations and combinatorially obtain exact formulae counting general RNA secondary structures by the joint distribution of hairpins and stacks.