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

TL;DR

This paper develops an enumeration and asymptotic analysis of $k$-noncrossing tangled-diagrams, revealing their growth rate and providing formulas for their counts as the number of vertices increases.

Contribution

It introduces a novel enumeration of $k$-noncrossing tangled-diagrams and derives their asymptotic formulas, extending combinatorial understanding of these graph structures.

Findings

Asymptotic formula for $T_k(n)$ with explicit growth rate

Identification of the dominant exponential term in the enumeration

Establishment of the polynomial correction factor in the asymptotics

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 $c_k>0$.