Asymptotic Enumeration of Non-crossing Partitions on Surfaces

TL;DR

This paper extends the concept of non-crossing partitions from disks to general surfaces with boundary, showing that their exponential growth rate remains the same as Catalan numbers, using combinatorial and analytic techniques.

Contribution

It introduces a generalized notion of non-crossing partitions on arbitrary surfaces and proves their growth rate matches that of classical Catalan numbers.

Findings

Growth rate of non-crossing partitions on surfaces equals Catalan numbers

Uses bijective and analytic methods for enumeration

Results apply to general surfaces with boundary

Abstract

We generalize the notion of non-crossing partition on a disk to general surfaces with boundary. For this, we consider a surface $\Sigma$ and introduce the number $C_{\Sigma}(n)$ of non-crossing partitions of a set of $n$ points laying on the boundary of $\Sigma$. Our proofs use bijective techniques arising from map enumeration, joint with the symbolic method and singularity analysis on generating functions. An outcome of our results is that the exponential growth of $C_{\Sigma}(n)$ is the same as the one of the $n$-th Catalan number, i.e., does not change when we move from the case where $\Sigma$ is a disk to general surfaces with boundary.