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 ≤ 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 knoncrossing tangled-diagrams Tk(n) ∼ ck n −((k−1)+(k−1)/2) (4(k − 1)2 + 2(k − 1) + 1)n for some ck > 0. 1. Tangled diagrams as molecules or walks In this paper we show how to compute the numbers of k-noncrossing tangled-diagrams and prove the asymptotic formula (1.1) Tk(n) ∼ ck n −((k−1)+(k−1)/2) (4(k − 1) + 2(k − 1) + 1), ck > 0 . This article is accompanied by a Maple package TANGLE, downloadable from the webpage http : //www.math.rutgers.edu/ ̃zeilberg/mamarim/mamarimhtml/tangled.html . k-noncrossing tangled-diagrams are motivated by studies of RNA molecules. They serve as combinatorial frames for searching molecular configurations and were recently studied [5] by the first three authors. Tangled-diagrams are labeled graphs over the vertices 1, . . . , n, drawn in a horizontal line in increasing order. Their arcs are drawn in the upper halfplane having the following types Date: February, 2008.