$k$-noncrossing RNA structures with arc-length $\ge 3$

In this paper we enumerate $k$-noncrossing RNA pseudoknot structures with given minimum arc- and stack-length. That is, we study the numbers of RNA pseudoknot structures with arc-length $\ge 3$, stack-length $\ge \sigma$ and in which there are at most $k-1$ mutually crossing bonds, denoted by ${\sf T}{k,\sigma}^{[3]}(n)$. In particular we prove that the numbers of 3, 4 and 5-noncrossing RNA structures with arc-length $\ge 3$ and stack-length $\ge 2$ satisfy ${\mathsf{T}}_{3,2}^{[3]}(n{)}^{}\sim K_3{n}^{-5}{2.5723}^n$, ${\sf T}^{[3]}{4,2}(n)\sim K_4 n^{-{21/2}} 3.0306^n$,and${\sf T}^{[3]}_{5,2}(n)\sim K_5 n^{-18} 3.4092^n$,respectively,where$K_3,K_4,K_5$ are constants. Our results are of importance for prediction algorithms for RNA pseudoknot structures.