Asymptotic Enumeration of Non-crossing Partitions on Surfaces

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.