# Enumerating simplicial decompositions of surfaces with boundaries - Mathematics > Combinatorics

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Abstract: It is well-known that the triangulations of the disc with $n+2$ vertices onits boundary are counted by the $n$th Catalan number $Cn=\frac{1}{n+1}{2n\choose n}$. This paper deals with the generalisation of this problem to anyarbitrary compact surface $S$ with boundaries. We obtain the asymptotic numberof simplicial decompositions of the surface $S$ with $n$ vertices on itsboundary. More generally, we determine the asymptotic number of dissections of$S$ when the faces are $\delta$-gons with $\delta$ belonging to a set ofadmissible degrees $\Delta\subseteq \{3,4,5,

.\}$. We also give the limit lawsof certain parameters of such dissections.

Author: ** Olivier Bernardi LM-Orsay, Juanjo RuĂ©**

Source: https://arxiv.org/