Pseudoknot RNA Structures with Arc-Length ≥4

Abstract

In this article, we study k-noncrossing RNA structures with minimum arc-length 4 and at most k − 1 mutually crossing bonds. Let T _k ^[4](n) denote the number of k-noncrossing RNA structures with arc-length ≥4 over n vertices. We (a) prove a functional equation for the generating function ∑_n≥0 T _k ^[4](n)z ⁿ and (b) derive for 4 ≤ k ≤ 9 the asymptotic formula T _k ^[4](n) ∼ c _k n^{−((k−1)2+(k−1)/2)} γ _k ⁻ⁿ. Furthermore, we explicitly compute the exponential growth rates γ _k ⁻¹ and asymptotic formulas for 4 ≤ k ≤ 9.