I wish to enumerate all quadrangulations of a $2p$ gon with $n$ internal vertices. Quadrangles are required to have simple faces. Simple face means all four vertices of each quadrangle are distinct.
Is anything known about this?
I wish to enumerate all quadrangulations of a $2p$ gon with $n$ internal vertices. Quadrangles are required to have simple faces. Simple face means all four vertices of each quadrangle are distinct. Is anything known about this? 


There are a number of subtly different definitions out there. The following two papers enumerate types of quadrangulations of a disk, but I didn't check if the type you are interested in is included: W. G. Brown, Enumeration of quadrangular dissections of the disk, Canad. J. Math., 17 (1965) 302317. R. C. Mullin and P. J. Schellenberg, The enumeration of cnets via quadrangulations, J. Combinatorial Theory, 4 (1968) 259276. 

