Simple recurrence formulas to count maps on orientable surfaces

Our main formula is a consequence of the KP equation for the generating function of bipartite maps, coupled with a Tutte equation, and it was apparently unnoticed before. It is similar in appearance to the one discovered by Goulden and Jackson for triangulations, and indeed our method to go from the KP equation to the recurrence formula can be seen as a combinatorial simplification of Goulden and Jackson's approach (together with one additional combinatorial trick). All these formulas have a very combinatorial flavour, but finding a bijective interpretation is currently unsolved.