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 $4\le k\le 9$.
Categories: math.CO