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A recursive method is given for finding generating functions which enumeraterooted hypermaps by number of vertices, edges and faces for any given number ofdarts. It makes use of matrix-integral expressions arising from the study ofbipartite quantum systems. Direct evaluation of these generating functions isthen demonstrated through the enumeration of all rooted hypermaps with up to 13darts.

Matrix integrals and generating functions for enumerating rooted hypermaps by vertices, edges and faces for a given number of darts