Enumeration of rooted planar halin maps

A Halin map is a kind of planar maps oriented by a tree. In this paper the rooted halin maps with the vertex, partitions as parameters are enumerated such that a famous results on rooted trees due to Harary, Prins, and Tutte is deduced as a special case. Further, by using Lagrangian inversion to obtain a number of summation free formulae directly, the various kinds of rooted Halin maps with up to three parameters have been counted. The paper supported by the National Natural Science Foundation of China (19701002).