Counting Bubbles in Linear Chord Diagrams

TL;DR

This paper introduces a formal method to count and analyze bubbles in linear chord diagrams, deriving generating functions and asymptotic distributions for bubble sizes, revealing their behavior in long diagrams.

Contribution

It provides the first exact generating function for bubble counts and characterizes the asymptotic distribution of bubble sizes in linear chord diagrams.

Findings

Exact generating function for bubble counts

Mean bubble size derived

Asymptotic distribution of bubble sizes in long diagrams

Abstract

In a linear chord diagram a short chord is one which joins adjacent vertices. We define a bubble to be a region in a linear chord diagram devoid of short chords. We derive a formal generating function counting bubbles by their size and find an exact result for the mean bubble size. We find that once one discards diagrams which have no short chords at all, the distribution of bubble sizes is given by a smooth function in the limit of long diagrams. Using a summation over short chords, the exact form of this asymptotic distribution is found.