Partial chord diagrams and matrix models

TL;DR

This paper develops matrix models to enumerate partial chord diagrams, incorporating various spectra and deriving differential equations for their generating functions, advancing combinatorial enumeration techniques.

Contribution

It introduces new matrix models that encode generating functions of partial chord diagrams with multiple spectra, unifying boundary length and point spectra.

Findings

Derived PDEs for generating functions using matrix models

Unified boundary length and point spectrum in a single framework

Enhanced enumeration methods for partial chord diagrams

Abstract

In this article, the enumeration of partial chord diagrams is discussed via matrix model techniques. In addition to the basic data such as the number of backbones and chords, we also consider the Euler characteristic, the backbone spectrum, the boundary point spectrum, and the boundary length spectrum. Furthermore, we consider the boundary length and point spectrum that unifies the last two types of spectra. We introduce matrix models that encode generating functions of partial chord diagrams filtered by each of these spectra. Using these matrix models, we derive partial differential equations - obtained independently by cut-and-join arguments in an earlier work - for the corresponding generating functions.