ENUMERATIONS OF ONE-VERTEX MAPS WITH FACE PARTITION ON THE PLANE

In addition to the known method given in [1],authors provide other three methods to the enumeration of one-vertex maps with face partition on the plane.Correspondingly,there are four functional equations in the enufuntion .It is shown that the four equations are equivalent.Moreover,an explicit expression of the solution is found by expanding the powers of the matrix of infinite order directly.This is a new complement of what appeared in [1].