In this paper we study $k$-noncrossing RNA structures with arc-length $\ge 3$, i.e. RNA molecules in which for any $i$, the nucleotides labeled $i$ and $i+j$ ($j=1,2$) cannot form a bond and in which there are at most $k-1$ mutually crossing arcs. Let ${ \sf S}_{ k,3}(n)$ denote their number. Based on a novel functional equation for the generating function $\sum_{ n\ge 0}{ \sf S}_{ k,3}(n)z^n$, we derive for arbitrary $k\ge 3$ exponential growth factors and for $k=3$ the subexponential factor. Our main result is the derivation of the formula ${ \sf S}_{ 3,3}(n) \sim \frac{ 6.11170\cdot 4!}{ n(n-1)...(n-4)} 4.54920^n$.
Categories: math.CO q-bio.BM