Enumeration of chord diagrams on many intervals and their non-orientable analogs

TL;DR

This paper develops algebraic and differential equation frameworks for enumerating orientable and non-orientable chord diagrams with applications to RNA modeling and matrix integrals, advancing combinatorial enumeration methods.

Contribution

It introduces new generating functions and differential equations for counting chord diagrams with various boundary cycle properties, including non-orientable cases, linking to integrable hierarchies.

Findings

Derived algebraic PDEs for generating functions of chord diagrams.

Established connections to the KP hierarchy and free probability.

Applied models to RNA interactions and matrix integrals.

Abstract

Two types of connected chord diagrams with chord endpoints lying in a collection of ordered and oriented real segments are considered here: the real segments may contain additional bivalent vertices in one model but not in the other. In the former case, we record in a generating function the number of fatgraph boundary cycles containing a fixed number of bivalent vertices while in the latter, we instead record the number of boundary cycles of each fixed length. Second order, non-linear, algebraic partial differential equations are derived which are satisfied by these generating functions in each case giving efficient enumerative schemes. Moreover, these generating functions provide multi-parameter families of solutions to the KP hierarchy. For each model, there is furthermore a non-orientable analog, and each such model likewise has its own associated differential equation. The enumerative problems we solve are interpreted in terms of certain polygon gluings. As specific applications, we discuss models of several interacting RNA molecules. We also study a matrix integral which computes numbers of chord diagrams in both orientable and non-orientable cases in the model with bivalent vertices, and the large-N limit is computed using techniques of free probability.