The RNA pseudoknot is a conserved secondary structure encountered in a number of ribozymes, which assume a central role in the RNA world hypothesis. However, RNA folding algorithms could not predict pseudoknots until recently. Analytic combinatorics, a newly arisen mathematical field, has introduced a way of enumerating different RNA configurations and quantifying RNA pseudoknot structure robustness and evolvability, two features that drive their molecular evolution. I will present a mathematician's viewpoint of RNA secondary structures, and explain how analytic combinatorics applied on RNA sequence to structure maps can represent a valuable tool for understanding RNA secondary structure evolution. Analytic combinatorics can be implemented for the optimization of RNA secondary structure prediction algorithms, the derivation of molecular evolution mathematical models, as well as in a number of biotechnological applications, such as biosensors, riboswitches etc. Moreover, it showcases how the integration of biology and mathematics can provide a different viewpoint into the RNA world.