  Drmota and Stufler proved recently that the expected number of pattern
occurrences of a given map is asymptotically linear when the number of edges
goes to infinity. In this paper we improve their result by means of a different
method. Our method allows us to develop a systematic way for computing the
explicit constant of the linear (main) term and shows that it is a positive
rational number. Moreover, by extending our method, we also solve the
corresponding problem of submap occurrences.
