Geodesic
Counting gluings of octahedra
Valentin Bonzom1 ; Luca Lionni2
1LIPN, UMR CNRS 7030, Institut Galilée, Université Paris 13, Sorbonne Paris Cité, 99 avenue Jean-Baptiste Clément, 93430 Villetaneuse, France, EU
2Laboratoire de Physique Théorique, CNRS UMR 8627, Université Paris XI, 91405 Orsay Cedex, France, EU and LIPN, UMR CNRS 7030, Institut Galilée, Université Paris 13, Sorbonne Paris Cité, 99 avenue Jean-Baptiste Clément, 93430 Villetaneuse, France, EU
The electronic journal of combinatorics, Tome 24 (2017) no. 3
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Three-dimensional colored triangulations are gluings of tetrahedra whose faces carry the colors 0, 1, 2, 3 and in which the attaching maps between tetrahedra are defined using the colors. This framework makes it possible to generalize the notion of two-dimensional $2p$-angulations to three dimensions in a way which is suitable for combinatorics and enumeration. In particular, universality classes of three-dimensional triangulations can be investigated within this framework. Here we study colored triangulations obtained by gluing octahedra. Those which maximize the number of edges at fixed number of octahedra are fully characterized and are shown to have the topology of the 3-sphere. They are further shown to be in bijection with a family of plane trees. The enumeration is performed both directly and using this bijection.
DOI : 10.37236/6503
Classification : 13P10, 13P15, 05C15
Mots-clés : three-dimensional triangulations, combinatorial maps, random trees

Valentin Bonzom  1   ; Luca Lionni  2

1 LIPN, UMR CNRS 7030, Institut Galilée, Université Paris 13, Sorbonne Paris Cité, 99 avenue Jean-Baptiste Clément, 93430 Villetaneuse, France, EU
2 Laboratoire de Physique Théorique, CNRS UMR 8627, Université Paris XI, 91405 Orsay Cedex, France, EU and LIPN, UMR CNRS 7030, Institut Galilée, Université Paris 13, Sorbonne Paris Cité, 99 avenue Jean-Baptiste Clément, 93430 Villetaneuse, France, EU 
@article{10_37236_6503,
     author = {Valentin Bonzom and Luca Lionni},
     title = {Counting gluings of octahedra},
     journal = {The electronic journal of combinatorics},
     year = {2017},
     volume = {24},
     number = {3},
     doi = {10.37236/6503},
     zbl = {1390.13081},
     url = {http://geodesic.mathdoc.fr/articles/10.37236/6503/}
}
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Valentin Bonzom; Luca Lionni. Counting gluings of octahedra. The electronic journal of combinatorics, Tome 24 (2017) no. 3. doi: 10.37236/6503

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