Planar Maps as Labeled Mobiles

J. Bouttier, P. Di Francesco, E. Guitter
2004 Electronic Journal of Combinatorics  
We extend Schaeffer's bijection between rooted quadrangulations and well-labeled trees to the general case of Eulerian planar maps with prescribed face valences to obtain a bijection with a new class of labeled trees, which we call mobiles. Our bijection covers all the classes of maps previously enumerated by either the two-matrix model used by physicists or by the bijection with blossom trees used by combinatorists. Our bijection reduces the enumeration of maps to that, much simpler, of
more » ... and moreover keeps track of the geodesic distance within the initial maps via the mobiles' labels. Generating functions for mobiles are shown to obey systems of algebraic recursion relations.
doi:10.37236/1822 fatcat:5qllntsep5dv5geurhqrc5ijv4