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}.
In this paper we study $k$-noncrossing matchings. A $k$-noncrossing matching is a labeled graph with vertex set ${1,...,2n}$ arranged in increasing order in a horizontal line and vertex-degree 1. The $n$ arcs are drawn in the upper halfplane subject to the condition that there exist no $k$ arcs that mutually intersect. We derive: (a) for arbitrary $k$, an asymptotic approximation of the exponential generating function of $k$-noncrossing matchings $F_k(z)$. (b) the asymptotic formula for the number of $k$-noncrossing matchings $f_{k}(n) sim c_k n^{-((k-1)^2+(k-1)/2)} (2(k-1))^{2n}$ for some $c_k>0$.
