Stacks in canonical RNA pseudoknot structures

In this paper we study the distribution of stacks in \(k\)-noncrossing, \(\tau\)-canonical RNA pseudoknot structures (\( \)-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 \(\)-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 \(\)-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.