Modular, $k$-noncrossing diagrams

• Main Authors: Reidys, Christian M, Wang, Rita R, Zhao, Y. Y
• Format: Journal Article
• Language: English
• Published: 13-03-2010
• Subjects: Mathematics - Combinatorics
• DOI: 10.48550/arxiv.1003.2710

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}.