Explicit enumeration formulas for m-regular simple stacks

Combinatorial enumeration of various RNA secondary structures and protein contact maps is of significant interest for both combinatorialists and computational biologists. Numerous results have been obtained, most of which are in terms of generating functions, recurrences or asymptotic formulas, few are of explicit formulas. This paper is mainly concerned with finding explicit enumeration formulas related to m-regular simple stacks, a classic combinatorial model for RNA secondary structures. By using the theories of noncrossing matching and Dyck path, we obtain explicit enumeration formulas for m-regular simple stacks with statistics on arcs, hairpins, components and visible vertices. The results can reduce to some classic formulas like Schmitt and Waterman's closed form formula for the number of RNA secondary structures. Furthermore, we study the enumeration of enhanced m-regular simple stacks, stimulated by the study of protein contact maps, in which the upper bound of the degrees of the two terminal vertices is relaxed to two, explicit formulas are obtained.