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

Abstract

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 $c_k>0$.

9 pages and 2 figures