There is a remarkable relationship between the genus series for rooted maps and rooted quadrangulations that has been obtained by a character theoretic argument. Hitherto no combinatorial explanation for this has been given. In this paper we generalize this relationship to a larger set of maps for which such a relationship can be found and we ascertain some of the properties that a putative bijection must possess. Several examples of images of sets under this bijection are given. We give them in detail since they may contain useful information about the bijection.
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