Combinatorics Of RNA Structures With Pseudoknots

In this paper we derive the generating function of RNA structures with pseudoknots. We enumerate all $k$-noncrossing RNA pseudoknot structures categorized by their maximal sets of mutually intersecting arcs. In addition we enumerate pseudoknot structures over circular RNA. For 3-noncrossing RNA structures and RNA secondary structures we present a novel 4-term recursion formula and a 2-term recursion, respectively. Furthermore we enumerate for arbitrary $k$ all $k$-noncrossing, restricted RNA structures i.e. $k$-noncrossing RNA structures without 2-arcs i.e. arcs of the form $(i,i+2)$, for $1\le i\le n-2$.

💡 Research Summary

The paper presents a rigorous combinatorial framework for counting RNA structures that contain pseudoknots, a class of tertiary interactions that go beyond the traditional secondary‑structure model. The authors start by representing an RNA conformation as a set of arcs over a linear or circular sequence, where each arc connects two nucleotides that are base‑paired. A structure is called k‑noncrossing if no collection of k arcs mutually cross; this condition limits the topological complexity of the diagram.

A novel classification is introduced: the maximal mutually intersecting arc sets. Instead of treating each crossing individually, the authors group arcs that all intersect each other into a single block. This block‑wise view allows the generating function for the whole class of structures to be built from the generating functions of individual blocks, exploiting their combinatorial independence. Consequently, the enumeration problem reduces to a product of simpler series, each of which can be expressed in terms of well‑known combinatorial numbers such as Catalan and Motzkin numbers.

For linear RNA, the authors derive explicit generating functions (G_k(x)) for arbitrary (k). In the case (k=3) (3‑noncrossing structures) they obtain a four‑term linear recurrence: \