Associahedra Via Spines

Abstract

An associahedron is a polytope whose vertices correspond to triangulations of a convex polygon and whose edges correspond to flips between them. Using labeled polygons, C. Hohlweg and C. Lange constructed various realizations of the associahedron with relevant properties related to the symmetric group and the permutahedron. We introduce the spine of a triangulation as its dual tree together with a labeling and an orientation. This notion extends the classical understanding of the associahedron via binary trees, introduces a new perspective on C. Hohlweg and C. Lange’s construction closer to J.-L. Loday’s original approach, and sheds light upon the combinatorial and geometric properties of the resulting realizations of the associahedron. It also leads to noteworthy proofs which shorten and simplify previous approaches.

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Correspondence to Carsten Lange.

Additional information

Vincent Pilaud was partially supported by grant MTM2011-22792 of the Spanish MICINN, by the French ANR grant EGOS (12 JS02 002 01), and by European Research Project ExploreMaps (ERC StG 208471).

Carsten Lange was supported by the TEOMATRO grant ANR-10-BLAN 0207.

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Lange, C., Pilaud, V. Associahedra Via Spines. Combinatorica 38, 443–486 (2018). https://doi.org/10.1007/s00493-015-3248-y

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Mathematics Subject Classification (2000)

  • 52B05
  • 52B11
  • 20F55