Modular, $k$-noncrossing diagrams

In this paper we compute the generating function of modular, $k$-noncrossing diagrams. A $k$-noncrossing diagram is called modular if it does not contains any isolated arcs and any arc has length at least four. Modular diagrams represent the deformation retracts of RNA pseudoknot structures \cite{Stadler:99,Reidys:07pseu,Reidys:07lego} and their properties reflect basic features of these bio-molecules. The particular case of modular noncrossing diagrams has been extensively studied \cite{Waterman:78b, Waterman:79,Waterman:93, Schuster:98}. Let ${\sf Q}_k(n)$ denote the number of modular $k$-noncrossing diagrams over $n$ vertices. We derive exact enumeration results as well as the asymptotic formula ${\sf Q}_k(n)\sim c_k n^{-(k-1)^2-\frac{k-1}{2}}\gamma_{k}^{-n}$ for $k=3,..., 9$ and derive a new proof of the formula ${\sf Q}_2(n)\sim 1.4848\, n^{-3/2}\,1.8489^{-n}$ \cite{Schuster:98}.