Statistics on Linear Chord Diagrams

TL;DR

This paper analyzes the distribution of short chords in linear chord diagrams, extending known distributions, and proves properties like unimodality and log-concavity for related statistics.

Contribution

It introduces a combinatorial analysis of short chords in linear chord diagrams, generalizing the Narayana distribution and establishing new properties such as unimodality and log-concavity.

Findings

Distribution of short chords is unimodal with an expected value of one.

Number of specific chord pairs follows the second-order Eulerian triangle.

Distribution of chord pair statistics is log-concave.

Abstract

Linear chord diagrams are partitions of $\left[2n\right]$ into $n$ blocks of size two called chords. We refer to a block of the form $\{i,i+1\}$ as a short chord. In this paper, we study the distribution of the number of short chords on the set of linear chord diagrams, as a generalization of the Narayana distribution obtained when restricted to the set of noncrossing linear chord diagrams. We provide a combinatorial proof that this distribution is unimodal and has an expected value of one. We also study the number of pairs $(i,i+1)$ where $i$ is the minimal element of a chord and $i+1$ is the maximal element of a chord. We show that the distribution of this statistic on linear chord diagrams corresponds to the second-order Eulerian triangle and is log-concave.