Geodesic
A bijection between planar constellations and some colored Lagrangian trees
Discrete mathematics & theoretical computer science, Tome 6 (2003-2004) no. 1.

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Constellations are colored planar maps that generalize different families of maps (planar maps, bipartite planar maps, bi-Eulerian planar maps, planar cacti, ...) and are strongly related to factorizations of permutations. They were recently studied by Bousquet-Mélou and Schaeffer who describe a correspondence between these maps and a family of trees, called Eulerian trees. In this paper, we derive from their result a relationship between planar constellations and another family of trees, called stellar trees. This correspondence generalizes a well known result for planar cacti, and shows that planar constellations are colored Lagrangian objects (that is objects that can be enumerated by the Good-Lagrange formula). We then deduce from this result a new formula for the number of planar constellations having a given face distribution, different from the formula one can derive from the results of Bousquet-Mélou and Schaeffer, along with systems of functional equations for the generating functions of bipartite and bi-Eulerian planar maps enumerated according to the partition of faces and vertices. 
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     author = {Chauve, Cedric},
     title = {A bijection between planar constellations and some colored {Lagrangian} trees},
     journal = {Discrete mathematics & theoretical computer science},
     publisher = {mathdoc},
     volume = {6},
     number = {1},
     year = {2003-2004},
     doi = {10.46298/dmtcs.340},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.340/}
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Chauve, Cedric. A bijection between planar constellations and some colored Lagrangian trees. Discrete mathematics & theoretical computer science, Tome 6 (2003-2004) no. 1. doi : 10.46298/dmtcs.340. http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.340/

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