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A simple recurrence formula for the number of rooted maps on surfaces by edges and genus

Abstract : We establish a simple recurrence formula for the number $Q_g^n$ of rooted orientable maps counted by edges and genus. The 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 gives by far the fastest known way of computing these numbers, or the fixed-genus generating functions, especially for large $g$. The formula is similar in look to the one discovered by Goulden and Jackson for triangulations (although the latter does not rely on an additional Tutte equation). Both of them have a very combinatorial flavour, but finding a bijective interpretation is currently unsolved - should such an interpretation exist, the history of bijective methods for maps would tend to show that the case treated here is easier to start with than the one of triangulations.
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Sean Carrell, Guillaume Chapuy. A simple recurrence formula for the number of rooted maps on surfaces by edges and genus. 26th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2014), 2014, Chicago, United States. pp.573-584, ⟨10.46298/dmtcs.2424⟩. ⟨hal-01207551⟩



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