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{\bf William Y. C. Chen, Jing Qin, Christian M. Reidys and Doron Zeilberger}
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{\bf Efficient Counting and Asymptotics of $k$-Noncrossing Tangled Diagrams}
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In this paper, we enumerate $k$-noncrossing tangled diagrams. A
tangled diagram is a labeled graph with vertices $1,\dots,n$, 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 $T_{k}(n) \, \sim \,c_k \, n^{-((k-1)^2+(k-1)/2)}\,
(4(k-1)^2+2(k-1)+1)^n$ for some $c_k>0$.

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