Abstract
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) = 18n+1 (2n/n). 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 delta-gons with delta belonging to a set of admissible degrees Delta subset of {3,4,5, ...}. We also give the limit laws for certain parameters of such dissections. (C) 2011 Elsevier Ltd. All rights reserved.