Combinatorics of $\gamma$-structures

TL;DR

This paper develops a mathematical framework to enumerate and analyze $oldsymbol{ extit{ extbf{ extgamma}}}$-structures, a class of RNA pseudoknot structures, using generating functions and asymptotic analysis, advancing understanding of their combinatorial properties.

Contribution

It derives the generating function for $ extgamma$-structures and computes asymptotic formulas for their enumeration, providing new tools for RNA structure analysis.

Findings

Derived the generating function of $ extgamma$-structures.

Computed asymptotic formulas for the number of $ extgamma$-structures.

Analyzed structures for $ extgamma$ up to 10.

Abstract

In this paper 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 $\gamma$-structure is composed by 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 $\le \gamma$. $\gamma$-structures are constructed via $\gamma$-matchings. For $1\le \gamma \le 10$, we compute Puiseux-expansions at the unique, dominant singularities, allowing us to derive simple asymptotic formulas for the number of $\gamma$-structures.