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Combinatorics of $γ$-structures

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