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 nth Catalan number C(n)=1n+12nn. This paper deals with the generalisation of this problem to any 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⊆{3,4,5,…}. We also give the limit laws for certain parameters of such dissections.