Enumerating simplicial decompositions of surfaces with boundaries

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 $δ$-gons with $δ$ belonging to a set of admissible degrees $Δ\subseteq \{3,4,5,...\}$. We also give the limit laws of certain parameters of such dissections.

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