$k$-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 ${sf T}_{3,2}^{[3]}(n)^{}sim K_3 n^{-5} 2.5723^n$, ${sf T}^{[3]}_{4,2}(n)sim K_4 n^{-{21/2}} 3.0306^n$, 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.