Combinatorics Of RNA Structures With Pseudoknots

Abstract

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 $1\le i\le n-2$.