模块化，$k$ -非交叉图
Modular, $k$-noncrossing diagrams
摘要：
本文计算了模$k$ -不交图的生成函数。一个$k$ -无交图称为模图，如果它不包含任何孤立弧，并且任何弧的长度至少为4。模块图表示RNA假结结构\cite{Stadler:99,Reidys:07pseu,Reidys:07lego}的形变回缩，其性质反映了这些生物分子的基本特征。模块不交叉图的特殊情况已经被广泛研究\cite{Waterman:78b, Waterman:79,Waterman:93, Schuster:98}。令${\sf Q}_k(n)$表示$n$顶点上的模$k$ -不交图的数目。我们导出了$k=3,..., 9$的精确计数结果和渐近公式${\sf Q}_k(n)\sim c_k n^{-(k-1)^2-\frac{k-1}{2}}γ_{k}^{-n}$，并给出了公式${\sf Q}_2(n)\sim 1.4848\, n^{-3/2}\,1.8489^{-n}$ \cite{Schuster:98}的一个新的证明.
英文摘要：
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 ${\sf Q}_k(n)\sim c_k n^{-(k-1)^2-\frac{k-1}{2}}γ_{k}^{-n}$ 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}.
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