Carrell, Sean and Chapuy, Guillaume - A simple recurrence formula for the number of rooted maps on surfaces by edges and genus

dmtcs:2424 - Discrete Mathematics & Theoretical Computer Science, January 1, 2014, DMTCS Proceedings vol. AT, 26th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2014)
A simple recurrence formula for the number of rooted maps on surfaces by edges and genus

Authors: Carrell, Sean and Chapuy, Guillaume

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.


Source : oai:HAL:hal-01207551v1
Volume: DMTCS Proceedings vol. AT, 26th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2014)
Section: Proceedings
Published on: January 1, 2014
Submitted on: November 21, 2016
Keywords: maps on surfaces,Enumeration,quadrangulations,KP hierarchy,[INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM],[MATH.MATH-CO] Mathematics [math]/Combinatorics [math.CO]


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