https://hal-cea.archives-ouvertes.fr/cea-03109970Fusy, ÉricÉricFusyLIX - Laboratoire d'informatique de l'École polytechnique [Palaiseau] - X - École polytechnique - CNRS - Centre National de la Recherche ScientifiqueGuitter, EmmanuelEmmanuelGuitterIPHT - Institut de Physique Théorique - UMR CNRS 3681 - CEA - Commissariat à l'énergie atomique et aux énergies alternatives - Université Paris-Saclay - CNRS - Centre National de la Recherche ScientifiqueMaps of unfixed genus and blossoming treesHAL CCSD2021[MATH.MATH-CO] Mathematics [math]/Combinatorics [math.CO][MATH.MATH-MP] Mathematics [math]/Mathematical Physics [math-ph]Savelli, Bruno - Graphes, Algorithmes et TOpologie - - GATO2016 - ANR-16-CE40-0009 - AAPG2016 - VALID - - Combinatoire enumerative en interaction avec l'algebre, la theorie des nombres et la physique - - COMBINE2019 - ANR-19-CE48-0011 - AAPG2019 - VALID - 2021-01-14 10:52:422023-03-24 14:53:202021-01-14 10:55:57enJournal articleshttps://hal-cea.archives-ouvertes.fr/cea-03109970/documentapplication/pdf1We introduce bijections between families of rooted maps with unfixed genus and families of so-called blossoming trees endowed with an arbitrary forward matching of their leaves. We first focus on Eulerian maps with controlled vertex degrees. The mapping from blossoming trees to maps is a generalization to unfixed genus of Schaeffer's closing construction for planar Eulerian maps. The inverse mapping relies on the existence of canonical orientations which allow to equip the maps with canonical spanning trees, as proved by Bernardi. Our bijection gives in particular (here in the Eulerian case) a combinatorial explanation to the striking similarity between the (infinite) recursive system of equations which determines the partition function of maps with unfixed genus (as obtained via matrix models and orthogonal polynomials) and that determining the partition function of planar maps. All the functions in the recursive system get a combinatorial interpretation as generating functions for maps endowed with particular multiple markings of their edges. This allows us in particular to give a combinatorial proof of some differential identities satisfied by these functions. We also consider face-colored Eulerian maps with unfixed genus and derive some striking identities between their generating functions and those of properly weighted marked maps. The same methodology is then applied to deal with m-regular bipartite maps with unfixed genus, leading to similar results. The case of cubic maps is also briefly discussed.