A bijection for rooted maps on orientable surfaces

Guillaume Chapuy (1) , Michel Marcus (2) , Gilles Schaeffer (1)

1 LIX - Laboratoire d'informatique de l'École polytechnique [Palaiseau]
2 LaBRI - Laboratoire Bordelais de Recherche en Informatique

We use the Marcus and Schaeffer's bijection, that relates rooted maps on orientable surfaces to labelled unicellular maps, to perform the asymptotic enumeration of rooted maps of given genus. In particular, we derive in a combinatorial way the exponent 5/2(g-1) counting maps of genus g (a result already obtained by Bender and Canfield by an extension of Tutte's method, or by matrix integrals techniques).

Domaines: Combinatoire [math.CO]

Guillaume Chapuy, Michel Marcus, Gilles Schaeffer. A bijection for rooted maps on orientable surfaces. SIAM Journal on Discrete Mathematics, 2009, 33 (3), pp.1587--1611. ⟨10.1137/080720097⟩. ⟨hal-00713482⟩