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 ) = 1 n + 1 2 n 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 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.