Enumerating simplicial decompositions of surfaces with boundaries

Olivier Bernardi (1) , Juanjo Rué (2)

1 LM-Orsay - Laboratoire de Mathématiques d'Orsay
2 Applied Mathematics IV Department

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 $\Delta\subseteq \{3,4,5,\ldots\}$. We also give the limit laws of certain parameters of such dissections.

Domaines: Combinatoire [math.CO]

Olivier Bernardi, Juanjo Rué. Enumerating simplicial decompositions of surfaces with boundaries. European Journal of Combinatorics, 2012, 33, pp. 302-325. ⟨hal-00352316⟩