On the functional equations in rectilinear embedding — Counting rooted near quadrangulations on the sphere

This paper provides quartic functional equations satisfied by the enumerating functions of rooted planar near-quadrangulations with the size, the valency of the root-face and the number of non-rooted vertices. Rooted two edge-connected planar near-quadrangulations are also counted. Furthermore, explicit formulae for such two types of maps with above parameters are derived respectively after employing Lagrangian inversion. For two particular cases, the numbers of rooted planar trees and outerplanar quadrangulations are deduced directly.