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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.

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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) 302-317.

R. C. Mullin and P. J. Schellenberg, The enumeration of c-nets via quadrangulations, J. Combinatorial Theory, 4 (1968) 259-276.

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