The boundary length and point spectrum enumeration of partial chord diagrams using cut and join recursion

TL;DR

This paper introduces a new spectrum for partial chord diagrams, establishes a recursion relation for counting them based on this spectrum, and links it to a nonlinear PDE for their generating function.

Contribution

It generalizes existing spectra for partial chord diagrams and derives a novel recursion relation and PDE connecting their enumeration to spectral data.

Findings

Derived a recursion relation for partial chord diagrams by boundary spectrum

Established the equivalence of the recursion to a nonlinear PDE for the generating function

Provided a unique determination of diagram counts using initial conditions

Abstract

We introduce the boundary length and point spectrum, as a joint generalization of the boundary length spectrum and boundary point spectrum in arXiv:1307.0967. We establish by cut-and-join methods that the number of partial chord diagrams filtered by the boundary length and point spectrum satisfies a recursion relation, which combined with an initial condition determines these numbers uniquely. This recursion relation is equivalent to a second order, non-linear, algebraic partial differential equation for the generating function of the numbers of partial chord diagrams filtered by the boundary length and point spectrum.