Enumerating simplicial decompositions of surfaces with boundaries

TL;DR

This paper generalizes the enumeration of boundary-vertex simplicial decompositions from polygons to arbitrary surfaces with boundaries, providing asymptotic counts and limit laws for various face configurations.

Contribution

It extends known combinatorial enumeration results from polygons to general surfaces with boundaries, including asymptotic formulas and probabilistic limit laws.

Findings

Asymptotic enumeration formulas for simplicial decompositions of surfaces.

Results on the number of dissections with faces of specified degrees.

Limit laws for parameters of these dissections.

Abstract

It is well-known that the triangulations of the disc with $n+2$ vertices on its boundary are counted by the $n$th Catalan number $C(n)=\frac{1}{n+1}{2n \choose n}$. This paper deals with the generalisation of this problem to any arbitrary compact surface $S$ with boundaries. We obtain the asymptotic number of simplicial decompositions of the surface $S$ with $n$ vertices on its boundary. More generally, we determine the asymptotic number of dissections of $S$ when the faces are $\delta$-gons with $\delta$ belonging to a set of admissible degrees $\Delta\subseteq \{3,4,5,...\}$. We also give the limit laws of certain parameters of such dissections.