Pseudoknot RNA structures with arc-length $\ge 4$

TL;DR

This paper analyzes the combinatorial properties of pseudoknot RNA structures with minimum arc-length 4, deriving generating functions, asymptotic formulas, and growth rates for structures with up to 9 crossings.

Contribution

It provides the first explicit functional equations and asymptotic formulas for the enumeration of k-noncrossing RNA structures with arc-length at least 4.

Findings

Derived a functional equation for the generating function of these structures.

Established asymptotic formulas for the number of structures for 4 ≤ k ≤ 9.

Computed exponential growth rates for these RNA structures.

Abstract

In this paper we study $k$-noncrossing RNA structures with minimum arc-length 4 and at most $k-1$ mutually crossing bonds. Let ${\sf T}_{k}^{[4]}(n)$ denote the number of $k$-noncrossing RNA structures with arc-length $\ge 4$ over $n$ vertices. We prove (a) a functional equation for the generating function $\sum_{n\ge 0}{\sf T}_{k}^{[4]}(n)z^n$ and (b) derive for $k\le 9$ the asymptotic formula ${\sf T}_{k}^{[4]}(n)\sim c_k n^{-((k-1)^2+(k-1)/2)} \gamma_k^{-n}$. Furthermore we explicitly compute the exponential growth rates $\gamma_k^{-1}$ and asymptotic formulas for $4\le k\le 9$.