Stacks in canonical RNA pseudoknot structures

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.

💡 Research Summary

The paper investigates the combinatorial distribution of stacks in RNA pseudoknot structures that satisfy two constraints: k‑noncrossing (no more than k‑1 mutually crossing arcs) and τ‑canonical (every arc belongs to a stack of length at least τ). These constraints model biologically realistic RNA secondary and tertiary interactions while keeping the combinatorial class amenable to analytic methods.

Building on the previously derived ordinary generating function (OGF) for <k,τ>‑structures, the authors introduce a bivariate generating function
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