Enumeration of unrooted odd-valent regular planar maps

We derive closed formulae for the numbers of rooted maps with a fixed number of vertices of the same odd degree except for the root vertex and one other exceptional vertex of degree 1. The same applies to the generating functions for these numbers. Similar results, but without the vertex of degree 1, were obtained by the first author and Rahman. We also show, by manipulating a recursion of Bouttier, Di Francesco and Guitter, that there are closed formulae when the exceptional vertex has arbitrary degree. We combine these formulae with results of the second author to count unrooted regular maps of odd degree. In this way we obtain, for each even n, a closed ∗Supported by University of Macau and NSERC. †Supported by the Belarusian RFFR (grant No. F05-227). ‡Supported by the Canada Research Chairs program, NSERC and the University of Macau.