Bijective link between Chapoton’s new intervals and bipartite planar maps

New bijective links on planar maps

This article describes new bijective links on planar maps, which are of incremental complexity and present original features. The first two bijections Φ1,2 are correspondences on oriented planar maps. They can be considered as variations on the classical edge-poset construction for bipolar orientations on graphs, suitably adapted so as to operate only on the embeddings in a simple local way. In...

New bijective links on planar maps via orientations

This article presents new bijections on planar maps. At first a bijection is established between bipolar orientations on planar maps and specific “transversal structures” on triangulations of the 4-gon with no separating 3-cycle, which are called irreducible triangulations. This bijection specializes to a bijection between rooted non-separable maps and rooted irreducible triangulations. This yi...

Recurrence of bipartite planar maps ∗

This paper concerns random bipartite planar maps which are defined by assigning weights to their faces. The paper presents a threefold contribution to the theory. Firstly, we prove the existence of the local limit for all choices of weights and describe it in terms of an infinite mobile. Secondly, we show that the local limit is in all cases almost surely recurrent. And thirdly, we show that fo...

Multiplicative bijective maps on standard operator algebras

Bijective census and random generation of Eulerian planar maps with prescribed vertex degrees

We give a bijection between Eulerian planar maps with prescribed vertex degrees, and some plane trees that we call balanced Eulerian trees. To enumerate the latter, we introduce conjugation classes of planted plane trees. In particular the result answers a question of Bender and Canfield in [BC94] and allows uniform random generation of Eulerian planar maps with restricted vertex degrees. Using...