Stacks in canonical RNA pseudoknot structures

Abstract

In this paper we study the distribution of stacks in $k$-noncrossing, $\tau$-canonical RNA pseudoknot structures ($<k,\tau> $-structures). An RNA structure is called $k$-noncrossing if it has no more than $k-1$ mutually crossing arcs and $\tau$-canonical if each arc is contained in a stack of length at least $\tau$. Based on the ordinary generating function of $<k,\tau>$-structures \cite{Reidys:08ma} we derive the bivariate generating function ${\bf T}_{k,\tau}(x,u)=\sum_{n \geq 0} \sum_{0\leq t \leq \frac{n}{2}} {\sf T}_{k, \tau}^{} (n,t) u^t x^n$, where ${\sf T}_{k,\tau}(n,t)$ is the number of $<k,\tau>$-structures having exactly $t$ stacks and study its singularities. We show that for a certain parametrization of the variable $u$, ${\bf T}_{k,\tau}(x,u)$ has a unique, dominant singularity. The particular shift of this singularity parametrized by $u$ implies a central limit theorem for the distribution of stack-numbers. Our results are of importance for understanding the ``language'' of minimum-free energy RNA pseudoknot structures, generated by computer folding algorithms.