## Discrete Mathematics & Theoretical Computer Science, Vol 6, No 1 (2003)

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DMTCS vol 6 no 1 (2003), pp. 13-40

# Discrete Mathematics & Theoretical Computer Science

## Volume 6 n° 1 (2003), pp. 13-40

author: Cedric Chauve A bijection between planar constellations and some colored Lagrangian trees Planar maps, trees, enumeration, bijection, Lagrange formula 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-Melou 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-Melou 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. If your browser does not display the abstract correctly (because of the different mathematical symbols) you can look it up in the PostScript or PDF files. Cedric Chauve (2003), Constellations are Lagrangian objects: a bijective proof, Discrete Mathematics and Theoretical Computer Science 6, pp. 13-40 For a corresponding BibTeX entry, please consider our BibTeX-file. dm060102.ps.gz (92 K) dm060102.ps (398 K) dm060102.pdf (260 K)