Counting Bubbles in Linear Chord 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.