Stacks in canonical RNA pseudoknot structures

In this paper we study the distribution of stacks/loops in k  -non-crossing, ττ-canonical RNA pseudoknot structures (〈k,τ〉〈k,τ〉-structures). Here, an RNA structure is called k  -non-crossing if it has no more than k-1k-1 mutually crossing arcs and ττ-canonical if each arc is contained in a stack of length at least ττ. Based on the ordinary generating function of 〈k,τ〉〈k,τ〉-structures [G. Ma, C.M. Reidys, Canonical RNA pseudoknot structures, J. Comput. Biol. 15 (10) (2008) 1257] we derive the bivariate generating function Tk,τ(x,u)=∑n⩾0∑0⩽t⩽n2Tk,τ(n,t)utxn, where Tk,τ(n,t)Tk,τ(n,t) is the number of 〈k,τ〉〈k,τ〉-structures having exactly t stacks and study its singularities. We show that for a specific parametrization of the variable u  , Tk,τ(x,u)Tk,τ(x,u) exhibits 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.