Stacks in canonical RNA pseudoknot structures

In this paper we study the distribution of stacks/loops in k-non-crossing, τ -canonical RNA pseudoknot structures ( 〈 k , τ 〉 -structures). Here, an RNA structure is called k-non-crossing if it has no more than k - 1 mutually crossing arcs and τ -canonical if each arc is contained in a stack of length at least τ . Based on the ordinary generating function of 〈 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 T k , τ ( x , u ) = ∑ n ⩾ 0 ∑ 0 ⩽ t ⩽ n 2 T k , τ ( n , t ) u t x n , where T k , τ ( n , t ) is the number of 〈 k , τ 〉 -structures having exactly t stacks and study its singularities. We show that for a specific parametrization of the variable u, T k , 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.