RNA pseudoknot structures with arc-length ≥3≥3 and stack-length ≥σ≥σ

In this paper, we enumerate kk-noncrossing RNA pseudoknot structures with a given minimum arc- and stack-length. That is, we study the numbers of RNA pseudoknot structures with arc-length ≥3≥3, stack-length ≥σ≥σ and in which there are at most k−1k−1 mutually crossing bonds, denoted by Tk,σ[3](n). We prove that the numbers ofkk-noncrossing RNA structures with arc-length ≥3≥3 and stack-length ≥2≥2 satisfy Tk,2[3](n)∼Ckn−(k−1)2−k−12(γk,2[3])−n. In the case k=3,4,5k=3,4,5, we derive T3,2[3](n)∼C3n−52.5721n, T4,2[3](n)∼C4n−2123.0306n, and T5,2[3](n)∼C5n−183.4092n, respectively, where C3,C4,C5C3,C4,C5 are constants. Our results are of importance for prediction algorithms for RNA pseudoknot structures.