Title | ||
---|---|---|
A bijection for covered maps, or a shortcut between Harer–Zagierʼs and Jacksonʼs formulas |
Abstract | ||
---|---|---|
We consider maps on orientable surfaces. A map is called unicellular if it has a single face. A covered map is a map (of genus g ) with a marked unicellular spanning submap (which can have any genus in { 0 , 1 , … , g } ). Our main result is a bijection between covered maps with n edges and genus g and pairs made of a plane tree with n edges and a unicellular bipartite map of genus g with n + 1 edges. In the planar case, covered maps are maps with a marked spanning tree and our bijection specializes into a construction obtained by the first author in Bernardi (2007) [4] . Covered maps can also be seen as shuffles of two unicellular maps (one representing the unicellular submap, the other representing the dual unicellular submap). Thus, our bijection gives a correspondence between shuffles of unicellular maps, and pairs made of a plane tree and a unicellular bipartite map. In terms of counting, this establishes the equivalence between a formula due to Harer and Zagier for general unicellular maps, and a formula due to Jackson for bipartite unicellular maps. We also show that the bijection of Bouttier, Di Francesco and Guitter (2004) [8] (which generalizes a previous bijection by Schaeffer, 1998 [33] ) between bipartite maps and so-called well-labeled mobiles can be obtained as a special case of our bijection. |
Year | DOI | Venue |
---|---|---|
2011 | 10.1016/j.jcta.2011.02.006 | Journal of Combinatorial Theory Series A |
Keywords | Field | DocType |
Unicellular map,Tree-rooted map,Quasi-tree,Graph on orientable surfaces,Spanning submap,Spanning tree | Discrete mathematics,Combinatorics,Bijection,Bipartite graph,Equivalence (measure theory),Spanning tree,Mathematics,Special case | Journal |
Volume | Issue | ISSN |
118 | 6 | Journal of Combinatorial Theory, Series A |
Citations | PageRank | References |
4 | 0.54 | 13 |
Authors | ||
2 |
Name | Order | Citations | PageRank |
---|---|---|---|
Olivier Bernardi | 1 | 106 | 14.20 |
Guillaume Chapuy | 2 | 73 | 11.25 |