Linear chord diagrams with long chords

TL;DR

This paper studies linear chord diagrams with long chords, revealing that the counts along diagonals follow a geometric sequence and identifying when this pattern begins.

Contribution

It proves that the geometric sequence pattern holds for all diagonals in linear chord diagrams with long chords and determines the onset of this pattern.

Findings

Diagonal counts follow a geometric sequence

Pattern holds for all diagonals

Identifies when the geometric pattern starts

Abstract

A linear chord diagram of size $n$ is a partition of the set $\{1,2,\cdots,2n\}$ into sets of size two, called chords. From a table showing the number of linear chord diagrams of degree $n$ such that every chord has length at least $k$, we observe that if we proceed far enough along the diagonals, they are given by a geometric sequence. We prove that this holds for all diagonals, and identify when the effect starts.