A coarse Tutte polynomial for hypermaps

We give an analogue of the Tutte polynomial for hypermaps. This polynomial can be defined as either a sum over subhypermaps, or recursively through deletion-contraction relations where the base case consists of isolated vertices. Our Tutte polynomial extends the classical Tutte polynomial of a graph as well as the Tutte polynomial of an embedded graph (i.e., the ribbon graph polynomial). We examine relations between our polynomial and other hypermap polynomials. We give hypermap duality and partial duality identities for our polynomial, as well as some evaluations.