2 research outputs found
On polynomials counting essentially irreducible maps
We consider maps on genus- surfaces with (labeled) faces of prescribed
even degrees. It is known since work of Norbury that, if one disallows vertices
of degree one, the enumeration of such maps is related to the counting of
lattice point in the moduli space of genus- curves with labeled points
and is given by a symmetric polynomial in the
face degrees . We generalize this by restricting to
genus- maps that are essentially -irreducible for , which
loosely speaking means that they are not allowed to possess contractible cycles
of length less than and each such cycle of length is required to
bound a face of degree . The enumeration of such maps is shown to be again
given by a symmetric polynomial in
the face degrees with a polynomial dependence on . These polynomials satisfy
(generalized) string and dilaton equations, which for uniquely
determine them. The proofs rely heavily on a substitution approach by Bouttier
and Guitter and the enumeration of planar maps on genus- surfaces.Comment: 37 pages, 5 figure
A bijection for essentially 3-connected toroidal maps
International audienceWe present a bijection for toroidal maps that are essentially 3-connected (3-connected in the periodic planar representation). Our construction actually proceeds on certain closely related bipartite toroidal maps with all faces of degree 4 except for a hexagonal root-face. We show that these maps are in bijection with certain well-characterized bipartite unicellular maps. Our bijection, closely related to the recent one by Bonichon and LĂ©vĂŞque for essentially 4-connected toroidal triangulations, can be seen as the toroidal counterpart of the one developed in the planar case by Fusy, Poulalhon and Schaeffer, and it extends the one recently proposed by Fusy and LĂ©vĂŞque for essentially simple toroidal triangulations. Moreover, we show that rooted essentially 3-connected toroidal maps can be decomposed into two pieces, a toroidal part that is treated by our bijection, and a planar part that is treated by the above-mentioned planar case bijection. This yields a combinatorial derivation for the bivariate generating function of rooted essentially 3-connected toroidal maps, counted by vertices and faces