Exact enumeration of RNA secondary structures by helices and loops

Abstract: Enumerative studies of RNA secondary structures were initiated four decades ago by Waterman and his coworkers. Since then, RNA secondary structures have been explored according to many different structural characteristics, for instance, helices, components and loops by Hofacker, Schuster and Stadler, orders by Nebel, saturated structures by Clote, the $5^{\prime}$-$3^{\prime}$ end distance by Clote, Ponty and Steyaert, and the rainbow spectrum by Li and Reidys. However, the majority of the contributions are asymptotic results, and it is harder to derive explicit formulas. In this paper, we obtain exact formulas counting RNA secondary structures with a given number of helices as well as a given joint size distribution of helices and loops, while some related asymptotic results due to Hofacker, Schuster and Stadler have been known for about twenty years. Our approach is combinatorial, analyzing a recent bijection between RNA secondary structures and plane trees discovered by the first author and proposing a variation of Chen's bijective approach of counting trees by forests of simple trees.