Counting non-crossing permutations on surfaces of any genus

TL;DR

This paper introduces a method to count non-crossing permutations on surfaces of any genus, revealing polynomial-like behaviors and connections to intersection numbers in moduli space.

Contribution

It extends the enumeration of non-crossing structures to arbitrary genus surfaces and links the counts to intersection theory on moduli spaces.

Findings

Count of polygon diagrams is almost polynomial in boundary points.

Leading coefficients relate to intersection numbers on moduli space.

Polygon diagram counts exhibit rich algebraic structures.

Abstract

Given a surface with boundary and some points on its boundary, a polygon diagram is a way to connect those points as vertices of non-overlapping polygons on the surface. Such polygon diagrams represent non-crossing permutations on a surface with any genus and number of boundary components. If only bigons are allowed, then it becomes an arc diagram. The count of arc diagrams is known to have a rich structure. We show that the count of polygon diagrams exhibits the same interesting behaviours, in particular it is almost polynomial in the number of points on the boundary components, and the leading coefficients of those polynomials are the intersection numbers on the compactified moduli space of curves.