We compute the generating function for triangulations on a cylinder, with the restriction that all vertices belong to its boundary and that the intersection of a pair of different faces is either empty, a vertex or an edge. We generalize these results to maps with either constant ({k}-dissections) or unrestricted (unrestricted dissections) face degree. We apply singularity analysis to the resulting generating functions to obtain asymptotic estimates for their coefficients, as well as limit distributions for natural parameters.