Combinatorics of γ-Structures

In this article we study canonical gamma-structures, a class of RNA pseudoknot structures that plays a key role in the context of polynomial time folding of RNA pseudoknot structures. A c-structure is composed of specific building blocks that have topological genus less than or equal to gamma, where composition means concatenation and nesting of such blocks. Our main result is the derivation of the generating function of gamma-structures via symbolic enumeration using so called irreducible shadows. We furthermore recursively compute the generating polynomials of irreducible shadows of genus <= gamma. The gamma-structures are constructed via gamma-matchings. For 1 <= gamma <= 10, we compute Puiseux expansions at the unique, dominant singularities, allowing us to derive simple asymptotic formulas for the number of gamma-structures.