Pseudoknot RNA structures with arc-length $ge 4$

In this paper we study $k$-noncrossing RNA structures with minimum arc-length 4 and at most $k-1$ mutually crossing bonds. Let ${sf T}_{k}^{[4]}(n)$ denote the number of $k$-noncrossing RNA structures with arc-length $ge 4$ over $n$ vertices. We prove (a) a functional equation for the generating function $sum_{nge 0}{sf T}_{k}^{[4]}(n)z^n$ and (b) derive for $kle 9$ the asymptotic formula ${sf T}_{k}^{[4]}(n)sim c_k n^{-((k-1)^2+(k-1)/2)} gamma_k^{-n}$. Furthermore we explicitly compute the exponential growth rates $gamma_k^{-1}$ and asymptotic formulas for $4le kle 9$.

Large components in random induced subgraphs of n-cubes

In this paper we study random induced subgraphs of the binary $n$-cube, $Q_2^n$. This random graph is obtained by selecting each $Q_2^n$-vertex with independent probability $lambda_n$. Using a novel construction of subcomponents we study the largest component for $lambda_n=frac{1+chi_n}{n}$, where $epsilonge chi_nge n^{-{1/3}+ delta}$, $delta>0$. We prove that there exists a.s. a unique largest component $C_n^{(1)}$. We furthermore show that $chi_n=epsilon$, $| C_n^{(1)}|sim alpha(epsilon) frac{1+chi_n}{n} 2^n$ and for $o(1)=chi_nge n^{-{1/3}+delta}$, $| C_n^{(1)}| sim 2 chi_n frac{1+chi_n}{n} 2^n$ holds. This improves the result of cite{Bollobas:91} where constant $chi_n=chi$ is considered. In particular, in case of $lambda_n=frac{1+epsilon} {n}$, our analysis implies that a.s. a unique giant component exists.

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.