Efficient Counting and Asymptotics of $k$-noncrossing tangled-diagrams

In this paper we enumerate $k$-noncrossing tangled-diagrams. A tangled-diagram is a labeled graph whose vertices are $1,...,n$ have degree $le 2$, and are arranged in increasing order in a horizontal line. Its arcs are drawn in the upper halfplane with a particular notion of crossings and nestings. Our main result is the asymptotic formula for the number of $k$-noncrossing tangled-diagrams $T_{k}(n) sim c_k n^{-((k-1)^2+(k-1)/2)} (4(k-1)^2+2(k-1)+1)^n$ for some $c_k>0$.

A combinatorial framework for RNA tertiary interaction

In this paper we show how to express RNA tertiary interactions via the concepts of tangled diagrams. Tangled diagrams allow to formulate RNA base triples and pseudoknot-interactions and to control the maximum number of mutually crossing arcs. In particular we study two subsets of tangled diagrams: 3-noncrossing tangled-diagrams with $ell$ vertices of degree two and 2-regular, 3-noncrossing partitions (i.e. without arcs of the form $(i,i+1)$). Our main result is an asymptotic formula for the number of 2-regular, 3-noncrossing partitions, denoted by $p_{3,2}(n)$, 3-noncrossing partitions over $[n]$. The asymptotic formula is derived by the analytic theory of singular difference equations due to Birkhoff-Trjitzinsky. Explicitly, we prove the formula $p_{3,2}(n+1)sim K 8^{n}n^{-7}(1+c_{1}/n+c_{2}/n^2+c_3/n^3)$ where $K,c_i$, $i=1,2,3$ are constants.