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

In this paper we study $k$-noncrossing RNA structures with minimum arc-length 4 and at most $k-1$ mutually crossing bonds. Let ${\mathsf{T}}_k^{[4]}(n)$ 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 ${\sum }_{n\ge 0}{\mathsf{T}}_k^{[4]}(n)z^n$ and (b) derive for $k\le 9$ the asymptotic formula ${\sf T}_{k}^{[4]}(n)\sim c_k n^{-((k-1)^2+(k-1)/2)} \gamma_k^{-n}$\mathrm{.}Furthermoreweexplicitlycomputetheexponentialgrowthrates$\gamma_k^{-1}$andasymptoticformulasfor$4\le k\le 9$.