Efficient Counting and Asymptotics of k-Noncrossing Tangled Diagrams - Chen, William Y. C.; Qin, Jing; Reidys, Christian M.; Zeilberger, Doron (The Electronic Journal of Combinatorics, 2009)
In this paper, we enumerate k-noncrossing tangled diagrams. A tangled diagram is a labeled graph with vertices 1,…,n, having degree at most two, which are arranged in increasing order in a horizontal line. The 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) c_k n^-((k-1)^2+(k-1)/2) (4(k-1)^2+2(k-1)+1)^n for some c_k>0.